Presentation Slides for Chapter 18 of Fundamentals of Atmospheric Modeling 2 nd Edition

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Presentation Slides for Chapter 18 of Fundamentals of Atmospheric Modeling 2 nd Edition. Mark Z. Jacobson Department of Civil & Environmental Engineering Stanford University Stanford, CA 94305-4020 jacobson@stanford.edu April 1, 2005. Cloud Formation. - PowerPoint PPT Presentation

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Presentation Slides for

Chapter 18of

Fundamentals of Atmospheric Modeling 2nd Edition

Mark Z. JacobsonDepartment of Civil & Environmental Engineering

Stanford UniversityStanford, CA 94305-4020jacobson@stanford.edu

April 1, 2005

Cloud Formation

Table 18.1

Altitude range (km) of different cloud-formation étages

Étage Polar Temperate TropicalHigh 3-8 5-13 6-9Middle 2-4 2-7 2-8Low 0-2 0-2 0-2

FogCloud touching the ground

Radiation FogForms as the ground cools radiatively at night, cooling the air above it to below the dew point.

Advection FogForms when warm, moist air moves over a colder surface and cools to below the dew point.

Upslope Fog Forms when warm, moist air flows up a slope, expands, and cools to below the dew point.

FogEvaporation Fog

Forms when water evaporates in warm, moist air, then mixes with cooler, drier air and re-condenses.

Steam FogOccurs when warm surface water evaporates, rises into cooler air, and recondenses, giving the appearance of rising steam.

Frontal FogOccurs when water from warm raindrops evaporates as the drops fall into a cold air mass. The water then recondenses to form a fog. Warm over cold air appears ahead of an approaching surface front.

Cloud ClassificationLow clouds (0-2 km)

Stratus (St)Stratocumulus (Sc)Nimbostratus (Ns)

Middle clouds (2-7 km)Altostratus (As) Altocumulus (Ac)

High clouds (5-18 km)Cirrus (Ci)Cirrostratus (Cs)Cirrocumulus (Cc)

Clouds of vertical development (0-18 km)Cumulus (Cu)Cumulonimbus (Cb)

stratus = "layer"cumulus = "clumpy"cirrus = "wispy"nimbus = "rain"

Low Clouds

StratusA low, gray uniform cloud layer composed of water droplets that often produces drizzle.

StratocumulusLow, lumpy, rounded clouds with blue sky between them.

NimbostratusDark, gray clouds associated with continuous precipitation. Not accompanied by lightning, thunder, or hail.

Middle Clouds

AltostratusLayers of uniform gray clouds composed of water droplets and ice crystals. The sun or moon is dimly visible in thinner regions.

Altocumulus

Patches of wavy, rounded rolls, made of water droplets and ice crystals.

High CloudsCirrus

High, thin, featherlike, wispy, ice crystal clouds.

Cirrostratus

High, thin, sheet-like, ice crystal clouds that often cover the sky and cause a halo to appear around the sun or moon.

CirrocumulusHigh, puffy, rounded, ice crystal clouds that often form in ripples.

Clouds of Vertical DevelopmentCumulus

Clouds with flat bases and bulging tops. Appear in individual, detached domes, with varying degrees of vertical growth.

Cumulus humilisLimited vertical development

Cumulus congestusExtensive vertical development

CumulonimbusDense, vertically developed cloud with a top that has the shape of an anvil. Can produce heavy showers, lightning, thunder, and hail. Also known as a thunderstorm cloud.

Cloud FormationCloud Formation Mechanisms

free convectionforced convectionorographyfrontal lifting

Fig. 18.1

Formation of clouds along a cold and warm front, respectively

.

Cold air

Warm air

Cold air

Cold frontWarm airWarm front

Pseudoadiabatic ProcessCondensation, latent heat release occurs during adiabatic ascent

Pseudoadiabatic process (18.1)

Saturation mass mixing ratio of water vapor over liquid water

Adiabatic process dQ = 0

dQ=−Ledωv,s

ωv,s =εpv,spd

Pseudoadiabatic Process

(18.5)

Differentiate v,s=pv,s/pd with respect to altitude, substitute

dpv,s =Lepv,sdT RvT2

ωv,s =εpv,s pd

′ R =εRv

∂pd ∂z=−pdg ′ R T

∂ωv,s∂z

=εpd

∂pv,s∂z

−pv,spd

∂pd∂z

⎝ ⎜

⎠ ⎟ =

Leεωv,s

′ R T2∂T∂z

+ωv,sg

′ R T

Pseudoadiabatic ProcessSubstitute (18.5) and d,m=g/cp,m into (18.4) (18.6)

Example 18.1 pd = 950 hPaT = 283 K

---> pv,s = 12.27 hPa---> v,s = 0.00803 kg kg-1

---> w = 5.21 K km-1

T = 293 K---> w = 4.27 K km-1

∂T∂z

⎛ ⎝ ⎜

⎞ ⎠ ⎟

w=−Γw =−Γd,m 1+

Leωv,s′ R T

⎝ ⎜

⎠ ⎟ 1+

Le2εωv,s

′ R cp,mT2

⎝ ⎜ ⎜

⎠ ⎟ ⎟

Dry or Moist Air Stability Criteria(18.7)

Γe>Γd,m absolutely unstable

Γe=Γd,m unsaturated neutral

Γd,m>Γe >Γw conditionally unstable

Γe=Γw saturated neutral

Γe<Γw absolutely stable

⎪ ⎪ ⎪

⎪ ⎪ ⎪

Stability in Dry or Moist Air

Fig. 18.2

0.8

1

1.2

1.4

1.6

1.8

2

2.2

-2 0 2 4 6 8 10 12 14

Altitude (km)

Temperature (

o

C)

Absolutely

stable

Absolutely

unstable

Conditionally

unstable

1 4

32

,d m w

Alt

itud

e (k

m)

Stability in Multiple Layers

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20 25

Altitude (km)

Temperature (oC)

e

Γd

ΓwA

ltit

ude

(km

)

Absolutely unstable

Absolutely stable

Unsaturated neutral

Conditionally unstable

Saturated neutral

Saturated neutral

Fig. 18.3

Equivalent Potential TemperaturePotential temperature a parcel of air would have if all its water vapor were

condensed and the resulting latent heat were released and used to heat the parcel

Equivalent potential temperature in unsaturated air (18.8)

Equivalent potential temperature in unsaturated air (18.9)

θp,e≈θpexpLe

cp,dTωv,s

⎝ ⎜ ⎜

⎠ ⎟ ⎟

θp,e≈θpexpLe

cp,dTLCLωv

⎝ ⎜ ⎜

⎠ ⎟ ⎟

Equivalent Potential Temperature

Fig. 18.4

Relationship between potential temperature and equivalent potential temperature

0

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25 30 35

Altitude (km)

Temperature (K)

LCL

θ

θ

w ,d m

d

p

,p eAlt

itud

e (k

m)

Cumulus Cloud Development

0

0.5

1

1.5

2

2.5

3

5 10 15 20 25 30 35

Altitude (km)

Temperature (oC)

e

d

Γw

Alt

itud

e (k

m)

Dew point ofrising bubble

Temperature ofrising bubble

LCL

Cloud temperature

Cloud top

Fig. 18.5

Isentropic Condensation Temperature

Temperature at the base of a cumulus cloudOccurs at the lifting condensation level (LCL), which is that altitude at which the dew point meets parcel temperature.

Isentropic condensation temperature (18.11)

TIC ≈

4880.357−29.66lnωvpd,0

εTICT0

⎝ ⎜

⎠ ⎟

1κ⎡

⎢ ⎢

⎥ ⎥

19.48−lnωvpd,0

εTICT0

⎝ ⎜

⎠ ⎟

1κ⎡

⎢ ⎢

⎥ ⎥

EntrainmentMixing of relatively cool, dry air from outside the cloud with warm, moist air inside the cloud

Factors affecting the temperature inside a cloud

1) Energy loss from cloud due to warming of entrained, ambient air by the cloud (18.12)

2) Energy loss from cloud due to evaporation of liquid water in the cloud to ensure entrained, ambient air is saturated (18.13)

3) Energy gained by cloud during condensation of rising air (18.14)

dQ1* =−cp,d Tv − ˆ T v( )dMc

dQ2* =−Le ωv,s− ˆ ω v( )dMc

dQ3* =−McLedωv,s

EntrainmentSum the three sources and sinks of energy (18.15)

First law of thermodynamics (18.16)

Subtract (18.16) from (18.15) and rearrange (18.17)

dQ* =−cp,d Tv − ˆ T v( )dMc −Le ωv,s− ˆ ω v( )dMc −McLedωv,s

dQ* =Mc cp,ddTv −αadpa( )

cp,ddTv −αadpa =−cp,d Tv − ˆ T v( )+Le ωv,s− ˆ ω v( )[ ]dMcMc

−Ledωv,s

EntrainmentDivide by cp,d Tv and substitute a=R’Tv/pa (18.18)

Rearrange and differentiate with respect to height (18.19)

No entrainment (dMc = 0) --> pseudoadiabatic temp. change

dTvTv

−′ R

cp,d

dpapa

=−Tv − ˆ T v

Tv+

Le ωv,s − ′ ω v( )

cp,dTv

⎣ ⎢ ⎢

⎦ ⎥ ⎥

dMcMc

−Ledωv,scp,dTv

∂Tv∂z

=−g

cp,d− Tv − ˆ T v( )+

Lecp,d

ωv,s− ˆ ω v( )⎡

⎣ ⎢ ⎢

⎦ ⎥ ⎥

1Mc

∂Mc∂z

−Le

cp,d

∂ωv,s∂z

Cloud Vertical Temperature ProfileChange of potential virtual temperature with altitude (2.103)

Rearrange (18.20)

Substitute into (18.19) --> change of potential virtual temperature in entrainment region

∂θv∂z

=θvTv

∂Tv∂z

−κθvpa

∂pa∂z

∂Tv∂z

=Tvθv

∂θv∂z

+′ R Tv

cp,dpa

∂pa∂z

=Tvθv

∂θv∂z

−g

cp,d

∂θv∂z

=−θvTv

Tv − ˆ T v( )+Le

cp,dωv,s− ˆ ω v( )

⎣ ⎢ ⎢

⎦ ⎥ ⎥

1Mc

∂Mc∂z

−θvTv

Lecp,d

∂ωv,sdz

Cloud Thermodynamic Energy Eq.Multiply through by dz and dividing through by dt (18.22)

Entrainment rate (18.23)

dθvdt

=−θvTv

Tv − ˆ T v( )+Le

cp,dωv,s− ˆ ω v( )

⎣ ⎢ ⎢

⎦ ⎥ ⎥ E −

θvLecp,dTv

dωv,sdt

E =1

Mc

dMcdt

≈3

4πrt3

ddt

4πrt3

3

⎝ ⎜

⎠ ⎟

Cloud Thermodynamic Energy Eq.Add terms to (18.22)

--> thermodynamic energy equation in a cloud (18.24)

dθvdt

=−θvTv

Tv − ˆ T v( )+Le

cp,dωv,s− ˆ ω v( )

⎣ ⎢ ⎢

⎦ ⎥ ⎥ E +

1ρa

∇ •ρaKh∇( )θv

+θv

cp,dTv−Le

dωv,sdt

−LmdωLdt

−Lsdωv,I

dt+

dQsolardt

+dQirdt

⎝ ⎜

⎠ ⎟

Cloud Vertical Momentum EquationVertical momentum equation in Cartesian / altitude coordinates (18.25)

Add hydrostatic equation, for air outside cloud (18.26)

dwdt

=−g−1

ρa

∂pa∂z

+1ρa

∇ •ρaKm∇( )w

∂ˆ p a ∂z =−̂ ρ ag

dwdt

=−gρa −ˆ ρ a

ρa−

1ρa

∂ pa− ˆ p a( )∂z

+1ρa

∇ •ρaKm∇( )w

Cloud Vertical Momentum EquationBuoyancy factor (18.27)

Adjust buoyancy factor for condensate (18.28)

B=−ρa −ˆ ρ a

ρa=−

paˆ T v − ˆ p aTvpa ˆ T v

=−ˆ T v −Tv

ˆ T v+

Tvˆ T v

⎝ ⎜

⎠ ⎟

ˆ p a−papa

≈−ˆ θ v −θv

ˆ θ v

B=−ρa −ˆ ρ a

ρa=−

ˆ θ v 1+ωL( )−θv 1+ ˆ ω L( )ˆ θ v

≈θv −ˆ θ v

ˆ θ v−ωL

Cloud Vertical Momentum EquationSubstitute (18.28) into (18.26) (18.29)

Rewrite pressure gradient term (18.30)

Substitute (18.30) and (18.29)

--> vertical momentum equation in a cloud (18.31)

dwdt

=gθv −ˆ θ v

ˆ θ v−ωL

⎝ ⎜

⎠ ⎟ −

1ρa

∂ pa− ˆ p a( )∂z

+1

ρa∇ •ρaKm∇( )w

1ρa

∂pa∂z

=−g =−∂Φ∂z

=cp,dθv∂P∂z

dwdt

=gθv −ˆ θ v

ˆ θ v−ωL

⎝ ⎜

⎠ ⎟ −cp,dθv

∂ P − ˆ P ( )

∂z+

1ρa

∇•ρaKm∇( )w

Simplified Vertical Velocity in CloudSimplify (18.31) for basic calculations

Ignore pressure perturbation and the eddy diffusion term (18.32)

where

Integrate over altitude --> vertical velocity in a cloud (18.33)

Rearrange (18.32)

dwdt

=dwdz

dzdt

=dwdz

w =gθv −ˆ θ v

ˆ θ v−ωL

⎝ ⎜

⎠ ⎟ =gB

w =dzdt

wdw=gBdz

w2 =wa2 +2g

θv −ˆ θ vˆ θ v

−ωL⎛

⎝ ⎜

⎠ ⎟ za

z∫ dz =wa

2 +2g Bza

z∫ dz

Convective Available Potential Energy

(18.34)

CAPE =g BzLFC

zLNB∫ dz≈gθv −ˆ θ v

ˆ θ v

⎝ ⎜

⎠ ⎟ zLFC

zLNB∫ dz

Cloud MicrophysicsAssume clouds form on multiple aerosol particle size distributionsEach aerosol distribution consists of multiple discrete size binsEach size bin contains multiple chemical componentsThree cloud hydrometeor distributions can form

LiquidIceGraupel

Each hydrometeor distribution contains multiple size bins.Each size bin contains the chemical components of the aerosol distribution it originated from

Cloud MicrophysicsProcesses considered

Condensation/evaporationIce deposition/sublimationHydrometeor-hydrometeor coagulationLarge liquid drop breakupContact freezing of liquid dropsHomogeneous/heterogeneous freezingDrop surface temperatureSubcloud evaporationEvaporative freezingIce crystal meltingHydrometeor-aerosol coagulationGas washoutLightning

Condensation and Ice DepositionCondensation/deposition onto multiple aerosol distributions

(18.35)

(18.36)

dcL,Ni,tdt

=kL,Ni,t−h Cv,t − ′ S L,Ni,t−hCL,s,t−h( )

dcI ,Ni,tdt

=kI ,Ni,t−h Cv,t − ′ S I,Ni,t−hCI ,s,t−h( )

dCv,tdt

=−kL,Ni,t−h Cv,t − ′ S L,Ni,t−hCL,s,t−h( )+

kI,Ni,t−h Cv,t − ′ S I,Ni,t−hCI,s,t−h( )

⎢ ⎢

⎥ ⎥

i=1

NB

∑N=1

NT

Water vapor-hydrometeor mass balance equation (18.37)

Vapor-Hydrometeor Transfer Rates

(18.38,9)

kL,Ni =nlq,Ni4πrNiDvωv,L,NiFv,L,Ni

mvDvωv,L,NiFv,L,NiLe ′ S L,NiCL,sκaωh,NiFh,L,NiT

LemvR*T

−1⎛

⎝ ⎜

⎠ ⎟ +1

kI ,Ni =nic,Ni4πχNiDvωv,I ,NiFv,I ,Ni

mvDvωv,I ,NiFv,I,NiLs ′ S I,NiCI ,sκaωh,NiFh,I ,NiT

LsmvR*T

−1⎛

⎝ ⎜

⎠ ⎟ +1

Köhler EquationsLiquid (18.40)

Ice (18.41)

Rewrite as (18.42)

′ S L,Ni,t−h ≈1+2σ L,Ni,t−hmv

rNiR*TρL

−3mv

4πrNi3ρLnNi,t−h

cq,Ni,t−hq=1

Ns

′ S I,Ni,t−h ≈1+2σI,Ni,t−hmv

rNiR*TρI

′ S L,Ni,t−h ≈1+aL,Ni,t−h

rNi−

bL,Ni,t−h

rNi3

Köhler Equations(18.43)

aL,Ni,t−h =2σL,Ni,t−hmv

R*TρL

bL,Ni,t−h =3mw

4πρLnNi,t−hcq,Ni,t−h

q=1

Ns

Solve for critical radius and critical saturation ratio (18.44)

rL,Ni,t−h* =

3bL,Ni,t−haL,Ni,t−h

SL,Ni,t−h* =1+

4aL,Ni,t−h3

27bL,Ni,t−h

CCN and IDN ActivationCloud condensation nuclei (CCN) activation (18.45)

Ice deposition nuclei (IDN) activation (18.46)

rNi >rL,Ni* and Cv,t−h > ′ S L,Ni,t−hCL,s,t−h

or

rNi ≤rL,Ni* and Cv,t−h >SL,Ni,t−h

* CL,s,t−h

⎨ ⎪ ⎪

⎩ ⎪ ⎪

Cv,t−h > ′ S I,Ni,t−hCI ,s,t−h{

Solution to Growth EquationsAerosol mole concentrations (18.47,8)

Mole balance equation (18.49)

cL,Ni,t =cL,Ni,t−h +hkL,Ni,t−h Cv,t − ′ S L,Ni,t−hCL,s,t−h( )

cI,Ni,t =cI,Ni,t−h+hkI,Ni,t−h Cv,t− ′ S I ,Ni,t−hCI,s,t−h( )

Cv,t + cL,Ni,t +cI,Ni,t( )i=1

NB

∑N=1

NT

=Cv,t−h + cL,Ni,t−h+cI,Ni,t−h( )i=1

NB

∑N=1

NT

∑ =Ctot

Solution to Growth Equations

Final gas mole concentration (18.50)

Cv,t =

Cv,t−h+hkL,Ni,t−h ′ S L,Ni,t−hCs,L,t−h +

kI,Ni,t−h ′ S I,Ni,t−hCs,I ,t−h

⎝ ⎜

⎠ ⎟

i=1

NB

∑N=1

NT

1+h kLi,t−h +kIi ,t−h( )i =1

NB

∑N=1

NT

Growth in Multiple Layers

Fig. 18.6

Dual peaks when grow on multiple size distributions, each with different activation characteristic

0

200

400

600

800

1000

1200

1400

1600

10 100

dn (No. cm

-3

) / d log

10

D

p

Particle diameter (D

p

, μ )m

788hPa

656hPa

872hPa

729hPa

835hPa

dn (

No.

cm

-3)

/ d lo

g 10 D

p

Growth in Multiple Layers

Fig. 18.6

Single peaks when size distribution homogeneous

0

200

400

600

800

1000

1200

1400

1600

10 100

dn (No. cm

-3

) / d log

10

D

p

Particle diameter (D

p

, μ )m

788hPa

656hPa

872hPa

729hPa

835hPa

dn (

No.

cm

-3)

/ d lo

g 10 D

p

Hydrometeor-Hydrometeor CoagulationFinal volume concentration of component or total particle

(18.53)

vx,Yk,t =vx,Yk,t−h +h Tx,Yk,t,1+Tx,Yk,t,2( )

1+hTx,Yk,t,3

Tx,Yk,t,1 = PY,M nMj,t−h fYi,Mj,YkβYi,Mj,t−hvx,Yi,ti=1

k−1

∑⎛

⎝ ⎜ ⎜

⎠ ⎟ ⎟

j=1

k

∑⎡

⎢ ⎢

⎥ ⎥

M=1

NH

Tx,Yk,t,2 = QI,M,Y nMj,t−h fIi ,Mj,YkβIi ,Mj,t−hvx,Ii,ti=1

k

∑⎛

⎝ ⎜ ⎜

⎠ ⎟ ⎟

j=1

k

∑⎡

⎢ ⎢

⎥ ⎥

I=1

NH

∑M=1

NH

Tx,Yk,t,3 = 1−LY,M( )1−fYk,Mj,Yk( )+LY,M[ ]βYk,Mj,t−hnMj,t−hM=1

NH

∑⎡

⎣ ⎢ ⎢

⎦ ⎥ ⎥ j =1

NC

Hydrometeor-Hydrometeor CoagulationFinal number concentration (18.54)

Volume fraction of coagulated pair partitioned to a fixed bin (18.55)

nlq,k,t =vT,lq,k,tυlq,k

fIi ,Mj,Yk =

υYk+1−VIi ,MjυYk+1−υYk

⎝ ⎜

⎠ ⎟

υNkVIi,Mj

υYk≤VIi,Mj <υYk+1 k <NC

1−fIi,Mj,Yk−1 υYk-1<VIi,Mj <υYk k >1

1 VIi,Mj ≥υYk k =NC

0 all othercases

⎪ ⎪ ⎪

⎪ ⎪ ⎪

Drop Breakup Size Distribution

Fig. 18.7

Drops breakup when they reach a given size

0

0.5

1

1.5

2

2.5

0 1000 2000 3000 4000 5000 6000

dM / M

T

d log

10

D

p

Particle diameter (D

p

, μ )m

Breakup distribution

dM /

MT d

log 10

Dp

Contact FreezingFinal volume concentration of total liquid drop or its components (18.59)

Final volume concentration of a graupel particle in a size bin or of an individual component in the particle (18.60)

(18.61)

vx,lq,k,t =vx,lq,k,t−h1+hTx,k,t,3

Tx,k,t,3 =FT βYk,Nj,t−hFICN,NjnNj,t−hN=1

NT

∑⎡

⎣ ⎢ ⎢

⎦ ⎥ ⎥ j =1

NC

vx,gr,k,t =vx,gr,k,t−h +vx,lq,k,thTx,k,t,3

Contact FreezingFinal number concentrations (18.62)

Temperature-dependence parameter (18.64)

(18.63)

nlq,k,t =vT,lq,k,tυlq,k

ngr,k,t =vT,gr,k,tυgr,k

FT =

0 T >−3oC

−T +3( ) 15 −18<T <−3oC

1 T <−18oC

⎨ ⎪ ⎪

⎩ ⎪ ⎪

Homogeneous/Heterogeneous FreezingFractional number of drops of given size that freeze (18.65)

Median freezing temperature (18.66)

FFr,k,t =minυlq,k exp−B Tc −Tr( )[ ],1{ }

Tmf =Tr −1B

ln0.5

υlq,k

⎝ ⎜ ⎜

⎠ ⎟ ⎟

B=0.475oC−1; Tr =0oC Tm≤−15oC

B=1.85oC−1; Tr =−11.14oC −15oC ≤Tm<−10oC

⎧ ⎨ ⎪

⎩ ⎪

Homogeneous/Heterogeneous Freezing

Fig. 18.8

Fitted versus observed median freezing temperatures

-28

-24

-20

-16

-12

10 100 1000 10

4

Median Freezing Temperature (

o

C)

Particle radius ( μ )m

Med

ian

free

zing

tem

pera

ture

(o C

)

Homogeneous/Heterogeneous FreezingTime-dependent freezing rate (18.67)

Final number conc. of drops and graupel particles after freezing (18.68)

(18.69)

dngr,k,tdt

=nlq,k,t−hυlq,kAexp−B Tc −Tr( )[ ]

nlq,k,t =nlq,k,t−h 1−FFr ,k,t( )

ngr,k,t =ngr,k,t−h+nlq,k,t−hFFr,k,t

Homogeneous/Heterogeneous Freezing

Fractional number of drops that freeze (18.70)

Time-dependent median freezing temperature (18.71)

FFr,k,t =1−exp−hAυlq,k exp−B Tc −Tr( )[ ]{ }

Tmf =Tr −1B

lnln0.5

hAυlq,k

⎝ ⎜ ⎜

⎠ ⎟ ⎟

Homogeneous/Heterogeneous Freezing

Fig. 18.9

Simulated liquid and graupel size distributions with and without homogeneous/heterogeneous freezing after one hour

10

-8

10

-6

10

-4

10

-2

10

0

10

2

1 10 100 1000 10

4

dn (No. cm

-3

) / d log

10

D

p

Particle diameter (D

p

, μ )m

,Liquid

baseline

( )with HHF

Layer below

Cloud top

236.988K

214hPa

,Liquid

no HHF

,Graupel

no HHF ,Graupel

baseline

( )with HHF

dn (

No.

cm

-3)

/ d lo

g 10 D

p

Drop Surface TemperatureIterate for drop surface temperature at sub-100 percent RH

(18.72)

ps,n = pv,s Ts,n( )

Δpv,n =0.3 ps,n−pv,n[ ]

pf,n =0.5 ps,n +pv,n( )

Tf,n =0.5 Ts,n+Ta( )

Ts,n+1 =Ts,n −DvLe

κa 1−pf,n pa( )

Δpv,nRvTf,n

pv,n+1=pv,n+Δpv,n

Drop Surface Temperature vs. RH

Fig. 18.10

270

275

280

285

0

5

10

15

20

0 0.2 0.4 0.6 0.8 1

Temperature (K)

Vapor pres. (hPa) and final RH x 10

Initial relative humidity (fraction)

Initial and final T

a

and initial T

s

Final T

s

Final p

v

= final p

s

Initial p

v

Final RHx10

Initial p

s

Tem

pera

ture

(K

)

Vapor pressure (hP

a) and final RH

x 10Air temperature = 283.15 K

Drop Surface Temperature vs. RH

Fig. 18.10

Air temperature = 245.94 K

240

241

242

243

244

245

246

247

248

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Temperature (K)

Vapor pres. (hPa) and final RH

Initial relative humidity (fraction)

Initial and final T

a

and initial T

s

Final T

s

Final p

v

= final p

s

Initial p

v

Final RH

Initial p

s

Tem

pera

ture

(K

)V

apor pressure (hPa) and final R

H x 10

Drop Surface Temperature vs. RH

Fig. 18.10

222

222.5

223

223.5

0

0.02

0.04

0.06

0.08

0.1

0 0.2 0.4 0.6 0.8 1

Temperature (K)

Vapor pres. (hPa) and final RHx0.01

Initial relative humidity (fraction)

Initial and final T

a

and initial T

s

Final T

s

Final p

v

= final p

s

Initial p

v

Final RH x 0.01

Initial p

s

Vapor pressure (hP

a) and final RH

x 10T

empe

ratu

re (

K)

Air temperature = 223.25 K

EvaporationReduction in volume due to evaporation/sublimation (18.73)

vL,lq,k,t,m=MAX vL,lq,k,t−h −nlq,k4πrkDv

1−pf,nf pa( )

pv,s,0 −pv,nf( )

ρLRvTf,nf

ΔzVf,lq,k

,0⎡

⎢ ⎢

⎥ ⎥ m

Reduction in precipitation size due to evaporation below cloud base

Fig. 18.11

10

-4

10

-3

10

-2

10

-1

10

0

10

1

10

2

10

3

1 10 100 1000 10

4

dn (No. cm

-3

) / d log

10

D

p

Particle diameter (D

p

, μ )m

Cloud base

(872 )hPa

,Surface

=75%RH

below base

,Surface

=99%RH

below base

dn (

No.

cm

-3)

/ d lo

g 10 D

p

Evaporative Freezing

Fig. 18.12

Incremental homogeneous/heterogeneous freezing due to evaporative cooling below a cloud base

10

-8

10

-6

10

-4

10

-2

10

0

10 100

dn (No. cm

-3

) / d log

10

D

p

Particle diameter (D

p

, μ )m

Liquid distribution

=100%,at RH

p

a

=214hPa

T

a

=236.988K

Additonal

portion of

. .liq distrib that

freezes due to

. evap cooling at

=80%RH

dn (

No.

cm

-3)

/ d lo

g 10 D

p

When drops fall into regions of sub-100 percent RH below cloud base, they start to evaporate and cool. If the temperature is below the freezing temperature, the cooling increases the rate of drop freezing.

Ice Crystal MeltingWhen an ice crystal melts in sub-100 percent relative humidity air, simultaneous evaporation of the liquid meltwater cools the

particle surface, retarding the rate of melting. Thus, the melting temperature must be higher than that of bulk ice in saturated air.

Melting point (18.74)

Time-dependent change in particle mass due to melting (18.75)

Tmelt=T0 +MAXDvLeκaRv

pv,s T0( )T0

−pvTa

⎣ ⎢

⎦ ⎥ ,0

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

mic,Ni,t =mic,Ni,t−h -MAX h4πrNiLm

κa Ta −T0( )Fh,I,Ni −

DvLeRv

pv,s T0( )T0

−pvTa

⎝ ⎜

⎠ ⎟ Fv,I,Ni

⎢ ⎢ ⎢

⎥ ⎥ ⎥ ,0

⎨ ⎪ ⎪

⎩ ⎪ ⎪

⎬ ⎪ ⎪

⎭ ⎪ ⎪

Aerosol-Hydrometeor CoagulationFinal volume conc. of total aerosol particle or its components (18.76)

vx,Nk,t =vx,Nk,t−h

1+hTx,Nk,t,3

Tx,Nk,t,3 = βNk,Mj,t−hnMj,t−hM=1

NH

∑⎡

⎣ ⎢ ⎢

⎦ ⎥ ⎥ j=1

NC

Aerosol-Hydrometeor CoagulationFinal volume conc. of total hydrometeor or aerosol inclusions (18.77)

vx,Yk,t =vx,Yk,t−h +h Tx,Yk,t,1+Tx,Yk,t,2( )

1+hTx,Yk,t,3

Tx,Yk,t,1 = nNj,t−h fYi,Nj,YkβYi,Nj,t−hvx,Yi,ti=1

k−1

∑⎛

⎝ ⎜ ⎜

⎠ ⎟ ⎟

j =1

k

∑⎡

⎢ ⎢

⎥ ⎥

N=1

NT

Tx,Yk,t,2 = nYj,t−h fNi,Yj,YkβNi,Yj,t−hvx,Ni,ti=1

k

∑⎛

⎝ ⎜ ⎜

⎠ ⎟ ⎟

j=1

k

∑⎡

⎢ ⎢

⎥ ⎥

N=1

NT

Tx,Yk,t,3 = 1−fYk,Nj,Yk( )βYk,Nj,t−hnNj,t−hN=1

NT

∑⎡

⎣ ⎢ ⎢

⎦ ⎥ ⎥ j =1

NB

Aerosol-Hydrometeor CoagulationFinal number concentrations (18.78)

(18.79)

nNk,t =vT,Nk,tυNk

nYk,t =vT,Yk,tυYk

Aerosol-Hydrometeor Coagulation

Fig. 18.13

Below-cloud aerosol number and volume concentration before (solid lines) and after (short-dashed lines) aerosol-hydrometeor coagulation

0

500

1000

1500

2000

0

5

10

15

20

25

30

35

0.001 0.01 0.1 1 10 100

dn (No. cm

-3

) / d log

10

D

p

dV(

μ

m

3

cm

-3

)/ d log

10

D

p

(Particle diameter D

p

, μ )m

Aerosol

volume

Aerosol

number

Below cloud base

(902 )hPa

dn (

No.

cm

-3)

/ d lo

g 10 D

pdv (μ

m3 cm

-3) / d log10 D

p

Gas WashoutGas-hydrometeor equilibrium relation (18.80)

Gas-hydrometeor mass-balance equation (18.81)

cq,lq,t,mCq,t,m

= ′ H qR*T pL,lq,t,mk=1

NC

Cq,t,m+cq,lq,t,m=Cq,t−h,m+cq,lq,t,m−1Δzm−1Δzm

Gas WashoutFinal gas concentration in layer m (18.82)

Final aqueous mole concentration (18.83)

Cq,t,m=

Cq,t−h,m+cq,lq,t,m−1Δzm−1Δzm

1+ ′ H qR*T pL,lq,t,mk=1

NC

cq,lq,t,m=Cq,t−h,m+cq,lq,t,m−1Δzm−1Δzm

−Cq,t,m

LightningCoulomb’s law (18.84)

Electric field strength (18.86)

Rate coefficient for bounceoff (18.87)

Fe =kCQ0Q1

r012

Ef =Fe,0iQ0i

∑ =kCQi

r0i2

i∑

BIi,Jj ,m= 1−Ecoal,Ii ,Jj ,m( )KIi ,Jj ,m

LightningCharge separation rate per unit volume of air (18.88)

Overall charge separation rate (18.91)

dQb,mdt

= BIi,Jjυ IinIi ,tnJj ,t−h +υ Jj nIi ,t−hnJj ,t( )

υIi +υ JjΔQIi ,Jj

i=j

NC

∑I=J

NH

∑j =1

NC

∑J =2

NH

∑⎡

⎢ ⎢

⎥ ⎥ m

dQb,cdt

=FcAcelldQb,m

dtΔzm

m=Ktop

Kbot

LightningTime-rate-of-change of the in-cloud electric field strength

(18.92)

Summed vertical thickness of layers (18.93)

Horizontal radius of cloudy region (18.94)

dE fdt

=2kC

Zc Zc2+Rc

2

dQb,cdt

Zc = Δzmm=K top

Kbot

Rc = FcAcell π

LightningNumber of intracloud flashes per centimeter per second

(18.95)

Number of NO molecules per cubic centimeter per second

(18.96)

dFrdt

=1

ZcEth

dEfdt

ENO =El FNOAcell

dFrdt