Evolution of the cloud droplet size distribution under the

effect of turbulence

Charmaine Franklin

Bureau of Meteorology Research Centre

Melbourne, Australia

Outline

Why is an accurate collision-coalesence rate of cloud droplets important?

Development of turbulent collision kernel (work from postdoc at McGill University with Paul Vaillancourt and Peter Yau)

Effects of turbulent collision kernel on evolution of the drop size distribution

Development of autoconversion parameterisation that includes the effect of turbulence

Comparison with other autoconversion models

Motivation Autoconversion of cloud droplets to rain drops is

one of the key process that determines cloud liquid water path, precipitation and cloud cover - implications for both NWP and climate

Warm rain accounts for 31% of total rain in tropics and 72% rain area

Autoconversion also important for mixed phase clouds as impacts the formation of hail/graupel

Theoretical droplet growth times are too slow to describe observed onset of precipitation

Turbulence has been recognised to play a role in autoconversion process for over half a century and may reduce the growth time of rain drops

Turbulent collision kernel Empirical model based on DNS of turbulent flows

and explicit tracking of large numbers of droplets Flow dissipation rates of TKE 100 – 1500 cm2 s-3

Collector droplets in size range 10 – 30 m radius

N=80, R=33 N=240, R=55

95 cm2 s-3

0= 0.45 s u’=13 cm s-1 k=0.04 s vk=2.0 cm s-1

1535 cm2 s-3

0= 0.45 s u’=39 cm s-1 k=0.01 s vk=4.0 cm s-1

details in Franklin, Vaillancourt and Yau (2006) JAS, in press

Collision kernel- measures the rate of collisions between two

particle size groups normalised by their number concentrations

132212 scmwrr rsph

= relative radial velocity

R

RRw rw

Average collision kernel is described as the average volume of fresh fluid entering the

collision sphere per unit time

Preferential concentration (clustering)

)(2 2 RgwR rsph

m radius m radius

- cloud droplets have finite inertia, interactions with flow produce spatial

correlations

- g(R) is the radial distribution function

- similar to clustering index, calculates the mean-variance ratio

- g(R)=1 for random distribution

- g(R)>1 if clustering

Turbulent collision kernel parameterisation

21.021Re

008.002.0Re

2

21015.0

v

vv

gravrturbr

tt

eww

039.0014.0Re)(2

21333.0

21

v

vv tt

errg

)(2 212

21 rrgwrr r

where is the droplet terminal velocity, is the Kolmogorov velocity, is the eddy dissipation rate of TKE (cm3 s-2)

tv v

cm3 s-1

cm s-1

Stochastic collection equation Using Bott’s (1998) SCE solver Mass doubles after 4 grid cells 160 bins with drop radii from 0.6 m to 104 m Hydrodynamic kernel of Hall (1980)

• Terminal velocities of Beard (1976)• Collision efficiencies:

collector < 30m Davis (72) Jonas (72) 40 < collector < 300m, radius ratio < 0.6 Schlamp et al. (76),

Lin and Lee (75), Shafrir and Gal-Chen (71)

40 < collector < 300m, radius ratio > 0.6 Klett and Davis (73)

Turbulent kernel used for collector drops 10-30 m but still use the gravitational collision efficiencies

Coalescence efficiency equal to 1 Expect turbulence effects on drops > 30 m to start

to diminish

Increases in collision kernel

100 cm2s-3

1000

500

1500

m

m

gravturb

Temporal evolution of mass weighted mean radius

time at which rg = 200m

time (sec)

gravity 30.9 min

100 cm2s-3 28.9

500 27.2

1000 26.5

1500 26.0

gravity 100

500 1000

1500

m

lwc = 1 g m-3 no. conc = 240 cm-3

dispersion = 0.5 mean vol radius ~ 10m

Temporal evolution of effective radius

time when effective radius equals 40 m

time (min)

m

gravity 43.2 min

100 cm2s-3 38.9

500 35.3

1000 33.7

1500 32.8

gravity 100

500 1000

1500Slingo (90) showed that

reducing re by 2m can

offset the warming effect caused by doubling CO2

Effect of turbulent collision efficiencyAssume collision efficiency increases as a function of

eddy dissipation rate of TKETest two cases: moderate increase and large increase•ec1 increases the gravitational collision efficiencies by

1.1 – 1.3 times as fn of •ec2 increases by 1.1 – 2.0 times ec gravity

ec increase

dissipation rate of TKE (cm2 s-3)

ec1

ec2

gravity 100

500 1000

1500

Time (min) dBZ=20

ec: 1.0 ec1: 1.1-1.3

ec2: 1.1-2

grav 34.2 34.2 34.2

100 cm2s-3 32.1 31.4 31.4

500 30.3 29.0 27.4

1000 29.4 27.9 25.7

1500 28.0 27.3 24.8

ec1 increased by 1.1 - 1.3

ec2 increased by 1.1 – 2.0

Effect of turbulent collision efficiency on reflectivity

dBZ

ec original

time (min)

gravity 100

500 1000

1500

100 cm2s-3

500

1000 1500

gravity turbulence rel. velocity

clustering

fraction of mass > 40m

time (min)

Contribution of turbulence effects - clustering and velocity

Autoconversion parameterisation that includes turbulent collision kernel

Solve SCE for liquid water contents 0 < lwc 2 g m-3, number concentrations 500 cm-3 and relative dispersion coefficients of the initial DSD 0.25 dispersion 0.4

Gamma function

is calculated as the rate of change at which the cloud droplets are colliding to form raindrops

Threshold radius 40 m – good agreement with exponents from Wood (05) data

Difference to Beheng (94) is that we cover smaller lwc and also include turbulent collision kernels as well as a different solver

By covering a broad range of lwc, no. conc. and initial spread of DSD, we increase the range of applicability and statistical meaning of the results

autor tq

rerNrf

10

Empirical model of autoconversion

where is rain water, is cloud water, is cloud

number concentration

bc

ac

auto

r Nqct

q

rq cq

cN

gravity 20.0 x 103 2.89 -1.32

100 cm2 s-3 86.1 x 102 2.74 -1.35

500 26.7 x 102 2.60 -1.27

1000 17.8 x 102 2.57 -1.22

1500 12.6 x 102 2.53 -1.18

c a b

38.023.0 Re3.5Re4.33.613 )9.1Re105.6(

ccauto

r nqt

q

Empirical modelwhere is rain water, is drop mean volume radius

avc

auto

r rct

q

rq vcr

gravity 1.37 x 10-13 7.18

100 cm2 s-3 2.6 x 10-13 6.90

500 5.0 x 10-13 6.61

1000 6.7 x 10-13 6.47

1500 8.2 x 10-13 6.38

c aLarger power reflects a sharper gradient and more of a Kessler type Heaviside function

))ln(Re109.5105.2( 1313Re5.8

vcauto

r rt

q

Empirical model

where is rain water, is cloud water, is cloud number concentration

31

37

ccauto

r Nqct

q

rq cqcN

gravity 3.94 0.071

100 cm2 s-3 4.39 0.079

500 4.95 0.089

1000 5.24 0.095

1500 5.46 0.099

c cE is the mean collision efficiency, usually taken to be ~ 0.5

Baker (93) estimated that this can overestimate the rate by 1-2 orders

cE

(Manton and Cotton 1977) includes the collision

efficiency, Stokes constant, drop mean volume radius

c

SCE data

new gravity param.

Khairoutdinov&Kogan(00)

Beheng (94)

Seifert&Beheng (01)

Liu&Daum (06) =0.1

Liu&Daum (06) =1

Liu&Daum (06) =100

Comparison with other models

dt

dqauto

dt

dqautoSCE data (kg m-3 s-1)

lwc 1 g m-3, 50 drops, dispersion 0.4

parameterised

SCE data

new gravity param.

Khairoutdinov&Kogan(00)

Beheng (94)

Seifert&Beheng (01)

Liu&Daum (06) =0.1

Liu&Daum (06) =1

Liu&Daum (06) =100

Comparison with other modelslwc 2 g m-3, 300 drops, dispersion 0.4

dt

dqauto

dt

dqauto

parameterised

SCE data (kg m-3 s-1)

SCE data

new gravity param.

Khairoutdinov&Kogan(00)

Beheng (94)

Seifert&Beheng (01)

Liu&Daum (06) =0.1

Liu&Daum (06) =1

Liu&Daum (06) =100

Comparison with other modelslwc 0.5 g m-3, 50 drops, dispersion 0.4

dt

dqauto

dt

dqautoparameterised

SCE data (kg m-3 s-1)

SCE data

new gravity param.

Khairoutdinov&Kogan(00)

Beheng (94)

Seifert&Beheng (01)

Liu&Daum (06) =0.1

Liu&Daum (06) =1

Liu&Daum (06) =100

Comparison with other modelslwc 0.2 g m-3, 50 drops, dispersion 0.25

dt

dqauto

dt

dqautoparameterised

SCE data (kg m-3 s-1)

Sensitivity to turbulence

lwc (g m-3)

gravity 100

500 1000

1500dt

dqauto

no. conc. = 50 cm-3 no. conc. = 100 cm-3

no. conc. = 500 cm-3no. conc. = 300 cm-3

(kg m-3 s-1)

Sensitivity to turbulence

lwc (g m-3)

gravity

100 500 1000

1500

dt

dqauto

no. conc. = 500 cm-3no. conc. = 300 cm-3

spread = 0.4 no. conc. = 50 cm-3 no. conc. = 100 cm-3

K&K (00)

B (94)

S&B (01)

L&D (06) 0.1

L&D (06) 1

L&D (06)100

(kg m-3 s-1)

Effect of turbulence is to offset the aerosol indirect effects

Use same data to find accretion parameterisation Implement new autoconversion-accretion

parameterisation into BAM SCM to investigate the impact on the liquid water path, precipitation etc

Summary and future work

gravity 0.9%

100 cm2s-3 21.4%

500 41.2%

1000 51.9%

1500 58.3%

Turbulence can accelerate the evolution of the DSD by reducing time to formation of rg

= 200 m by up to 20%More significant is percentage of mass transferred to drop sizes >40 m in 20 minutes

– see table