Cloud microphysics

Claudia Emde

Meteorological Institute, LMU, Munich, Germany

WS 2011/2012 Overview of cloud physics lecture

Atmospheric thermodynamics gas laws, hydrostatic equation 1st law of thermodynamics moisture parameters adiabatic / pseudoadiabatic processes stability criteria / cloud formation Microphysics of warm clouds nucleation of water vapor by condensation growth of cloud droplets in warm clouds (condensation, fall speed of droplets, collection, coalescence) formation of rain, stochastical coalescence Microphysics of cold clouds homogeneous nucleation heterogeneous nucleation contact nucleation crystal growth (from water phase, riming, aggregation) formation of precipitation Observation of cloud microphysical properties Parameterization of clouds in climate and NWP models

Cloud microphysics November 24, 2011 2 / 35 Growth rate and size distribution

growing droplets consume water vapor faster than it is made available by cooling and supersaturation decreases haze droplets evaporate, activated droplets continue to grow by condensation growth rate of water droplet dr 1 = G S dt l r

smaller droplets grow faster than larger droplets sizes of droplets in cloud become increasingly uniform, approach monodisperse distribution Figure from Wallace and Hobbs

Cloud microphysics November 24, 2011 3 / 35 Size distribution evolution

q 2 r = r0 + 2Gl St

0.8 1.4 t=0 t=0 0.7 t=10 t=10 1.2 t=30 t=30

0.6 t=50 t=50 1.0

0.5 0.8

0.4

0.6 0.3 n [normalized] n [normalized]

0.4 0.2

0.2 0.1

0.0 0.0 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 radius [arbitrary units] radius [arbitrary units]

Cloud microphysics November 24, 2011 4 / 35 Growth by collection

growth by condensation alone does not explain formation of larger drops other mechanism: growth by collection

Cloud microphysics November 24, 2011 5 / 35 Terminal fall speed

r 40µ . 0.6mm . r . 2mm 40µm . r . 0.6mm 2 g(ρ − ρ0)r 2 1 v = kr 2 v = v = kr 9 η 1   2 3 −1 3 ρ −1 k=8·10 s k=2.2·10 0 cms η – viscosity of air ρ

v r2 v r v r1/2 40 ∼ 500 ∼ 500 ∼

35 450 400 30 400 25 300 20 350

15 200 300 10 100

terminal fall speed [cm/s] 250 5

0 0 200 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.5 1.0 1.5 2.0 radius [mm] radius [mm] radius [mm]

Larger droplets approach constant value of about 10m/s.

Cloud microphysics November 24, 2011 6 / 35 Growth by collection

Figure from Wallace and Hobbs

Cloud microphysics November 24, 2011 7 / 35 Collision efficiency

Collision efficiency E increases when size of collector y 2 drop r1 increases E = 2 2 r1 + r2 E small for r1 < 20µm

r1  r2: E small because small droplets follow streamlines around collector drop

E increases with increasing r2 until r2/r1 ≈ 0.6–0.9

r2/r1 >0.6–0.9: E decreases because relative velocity between droplets becomes small

r2/r1 ≈1: strong interaction between droplets, E increases again

Figure from Wallace and Hobbs

Cloud microphysics November 24, 2011 8 / 35 Coalescence efficiency

Coalescence efficiency E’=fraction of collisions that result in coalescence

0 E large for r2  r1 0 E initially decreases as r2 increases 0 as r2 approaches r1, E increases sharply

Figure from Wallace and Hobbs

Cloud microphysics November 24, 2011 9 / 35 Collection efficiency

collection = collision + coalescence

Collection efficiency

0 Ec = E · E

Cloud microphysics November 24, 2011 10 / 35 Continuous collection model

collector drop with radius r1 and terminal velocity v1 drop falls in still air through cloud of equal sized droplets with r2 and v2 droplets are uniformly distributed and collected uniformly by all collector drops of a given size Figure from Wallace and Hobbs

Cloud microphysics November 24, 2011 11 / 35 Continuous collection model

dr v ω E 1 ≈ 1 l dt 4ρl

Z rH 4ρl w − v1 H ≈ dr1 ωl r0 v1E

Figure from Rogers Bowen (1955) assumed E=1.

Cloud microphysics November 24, 2011 12 / 35 Continuous collection model (Bowen model)

Figure from Rogers Figure from Rogers

Cloud microphysics November 24, 2011 13 / 35 Statistical growth: Telford model

local variations in droplet concentration in clouds

PDF for number of droplets in volume V (nV¯ )m p(m) = e−nV¯ m! Poisson probability law

n¯ – average droplet concentration in given size interval m – number of droplets in V “fortunate” drops fall through regions of locally high concentration ⇒ experience more than average number of collisions ⇒ grow more rapidly 1 of 105 or 106 droplet is sufficient to initiate rain

Cloud microphysics November 24, 2011 14 / 35 Statistical growth: Telford model

Assumptions:r 1=12.6µm, r2=10µm ⇒ V (r1) = 2V (r2), E=1 transformation of bimodal size distribution in broad size distribution happens in most cases within t<0.5min in few cases within t>30min PDF for time required for collector drop to experience given number of collisions (figure) most likely case corresponds to continuous collection model drops that grow faster than average may account for development of rain

Figure from Rogers Cloud microphysics November 24, 2011 15 / 35 Statistical growth: Telford model

Robertson (1974) expanded Telford model by including realistic E E not analytical ⇒Monte Carlo approach PDF for time required to experience given number of collisions PDFs approach limiting form standard deviation σ becomes negligible for r>40µm ⇒corresponds to continuous collection model r<20µm⇒E too small although σ large no drizzle or rain formation Figure from Rogers

Cloud microphysics November 24, 2011 16 / 35 Gap between condensational and collectional growth

condensational growth slows appreciably as droplet radius approaches ∼ 10µm tends to produce monodisperse size distribution droplets then have similar fall speeds ⇒ collisions become unlikely collectional growth conditions: a few reasonably efficient collector drops (i.e. r> 20µm) cloud deep enough and contains sufficient amount of water Question 1: How do the collector drops initially form

Cloud microphysics November 24, 2011 17 / 35 Broad size distributions

Question 2: How do the broad size distributions formed that are commonly measured?

Figure from Wallace and Hobbs

Cloud microphysics November 24, 2011 18 / 35 Possible explanations

Giant cloud condensation nuclei (GCCN) Turbulence Radiative broadening Stochastic collection

Cloud microphysics November 24, 2011 19 / 35 Stochastic collection

continuous collection model collector drop collides in continuous and uniform fashion with smaller cloud droplets which are uniformly distributed in space therefore collector drops of the same size grow at the same rate if they fall through the same cloud of droplets

stochastic (statistical) collection model treats collisions as individual events, distributed statistically in time and space

Cloud microphysics November 24, 2011 20 / 35 Stochastic collection

Line 1: 100 droplets start at the same size Line 2: after some time 10 droplets have collided with other droplets Line 3: second collisions produce 3 sizes

Figure from Wallace and Hobbs Stochastical model results in broad size distributions

Cloud microphysics November 24, 2011 21 / 35 Statistical growth: stochastic coalescence equation

evolution of entire droplet spectrum needs to be taken into account development of droplet size spectrum in time:

Stochastic coalescence equation

∂ n(V )dV = 1 R V H(δ, V 0)n(δ)n(V 0)dδdV 0 ∂t 2 0 R ∞ −n(V )dV 0 H(V1, V )n(V1)dV1 (Melzak and Hitchfeld, 1953)

first term: growth of smaller droplets to gain volume V second term: all possible captures of droplets with V by larger drops, as well as capture of smaller droplets

Cloud microphysics November 24, 2011 22 / 35 First solution of stochastic coalescence equation

Warshaw (1968)

solution corresponds to average value of n(V,t) over many realizations n(V,t=0): Gaussian distribution σ/¯r = 0.15, LWC=1g/m3 various values for E fastest evolution for E=1 (geometrical sweep-out)

Figure from Rogers

Cloud microphysics November 24, 2011 23 / 35 Development of size distribution

radius of 100th largest droplet – measure of the development of the size distribution initial conditions 50 droplets cm−3 ⇒maritime cloud 200 droplets cm−3 ⇒continental cloud n(V,t=0): Gaussian, σ/¯r = 0.15, LWC=1g/m3

Figure from Rogers

Cloud microphysics November 24, 2011 24 / 35 Statistical growth: stochastic coalescence equation Kovetz and Olund (1969)

modifications compared to Warshaw: gravitational settling of droplets considered condensational growth included initial condition as Warshaw, maritime case after 600s ⇒r≈200µm (logr=2.3) (Warshaw only 52µm) faster growth when condensational growth is included

CloudFigure microphysics from Rogers November 24, 2011 25 / 35 Condensation + stochastic coalescence

When condensation growth is included in stochastic growth model, droplet growth is accelerated!

Condensation growth alone narrows size distribution

Condensation + stochastic coalescence As small droplets grow, their collision efficiency relative to large drops increase at rapid rate ⇒growth by coalescence accelerates

(relative velocity between collector drop and collected drop does not change due to condensational growth)

Cloud microphysics November 24, 2011 26 / 35 Droplet development in large convective clouds

Leighton and Rogers (1974)

strong updraft v=7.5 m/s approximations: fall out of droplets neglected condensation rate determined by amount of condensed water (pseudoadiabatic ascend) drizzle drop size reached at t>10.5min peak due to condensational growth remains stable without condensation ⇒negligible growth

Figure from Rogers

Cloud microphysics November 24, 2011 27 / 35 Growth of droplets in maritime clouds

Yang (1974) calculation accounts for supersaturation and breakup of large drops t<15min mainly growth by condensation (N=const) t>15min coalescence, rapid growth: N decreases supersaturation increases (reduced number of droplets do not accommodate all water vapor released more droplets become activated, Figure from Rogers but are quickly consumed by coalescence.

Cloud microphysics November 24, 2011 28 / 35 Size distribution of raindrops

Measurements of the size distribution of raindrops that reach the ground can often be fitted to the same size distribution function:

Marshall-Palmer distribution

N(D) = N0 exp −ΛD

N(D)dD – number of drops per unit volume with diameters between D and D + dD N0 and Λ – empirical fitting parameters

N0 almost const., Λ varies with rainfall rate

can be explained by breakup of large drops

Cloud microphysics November 24, 2011 29 / 35 Efficient solution of stochastic coalescence equation

published in 2001

Cloud microphysics November 24, 2011 30 / 35 Formation of rain in warm clouds

Cloud microphysics November 24, 2011 31 / 35 Formation of rain in warm clouds

Cloud microphysics November 24, 2011 32 / 35 Formation of rain in warm clouds

Cloud microphysics November 24, 2011 33 / 35 Formation of rain in warm clouds

Cloud microphysics November 24, 2011 34 / 35 Formation of rain in warm clouds

Formation of rain in warm clouds still not completely understood, mainly because observed growth rates are larger than theoretical predictions !!

Cloud microphysics November 24, 2011 35 / 35