Presentation Slides for
Chapter 18of
Fundamentals of Atmospheric Modeling 2nd Edition
Mark Z. JacobsonDepartment of Civil & Environmental Engineering
Stanford UniversityStanford, CA [email protected]
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.
AltocumulusPatches of wavy, rounded rolls, made of water droplets and ice crystals.
High CloudsCirrus
High, thin, featherlike, wispy, ice crystal clouds.
CirrostratusHigh, 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 airWarm air
Cold airCold front Warm air
Warm 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 = ε
pd
∂pv,s∂z − pv,s
pd
∂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 unstableΓe=Γd,m unsaturated neutralΓd,m>Γe >Γw conditionally unstableΓe=Γw saturated neutralΓe<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)
Absolutly
stabl
Absolutly
unstabl
Conditionally
unstabl
1 4
32
d,m
Alti
tude
(km
)
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
ltitu
de (k
m)
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≈θpexp Lecp,dT ωv,s
⎛ ⎝ ⎜ ⎜
⎞ ⎠ ⎟ ⎟
θp,e≈θpexp Lecp,dTLCL
ωv⎛ ⎝ ⎜ ⎜
⎞ ⎠ ⎟ ⎟
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
d,m
d
p
p,Alti
tude
(km
)
Cumulus Cloud Development
0
0.5
1
1.5
2
2.5
3
5 10 15 20 25 30 35
Altitude (km)
Temperature (oC)
d
Γw
Alti
tude
(km
)
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 ≈4880.357−29.66ln ωvpd,0
εTICT0
⎛ ⎝ ⎜
⎞ ⎠ ⎟ 1κ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
19.48−ln ωvpd,0ε
TICT0
⎛ ⎝ ⎜
⎞ ⎠ ⎟ 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 aa=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 vTv
+Le ωv,s − ′ ω v( )
cp,dTv
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
dMcMc
−Ledωv,scp,dTv
∂Tv∂z =− g
cp,d− Tv − ˆ T v( )+ Le
cp,dωv,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 =θv
Tv
∂Tv∂z −κ θv
pa
∂pa∂z
∂Tv∂z =Tv
θv
∂θv∂z + ′ R Tv
cp,dpa
∂pa∂z =Tv
θv
∂θv∂z − g
cp,d
∂θv∂z =−θv
TvTv − ˆ T v( )+ Le
cp,dωv,s− ˆ ω v( )
⎡ ⎣ ⎢ ⎢
⎤ ⎦ ⎥ ⎥
1Mc
∂Mc∂z −θv
Tv
Lecp,d
∂ωv,sdz
Cloud Thermodynamic Energy Eq.Multiply through by dz and dividing through by dt (18.22)
Entrainment rate (18.23)
dθvdt =−θv
TvTv − ˆ T v( )+ Le
cp,dωv,s− ˆ ω v( )
⎡ ⎣ ⎢ ⎢
⎤ ⎦ ⎥ ⎥ E − θvLe
cp,dTv
dωv,sdt
E = 1Mc
dMcdt ≈ 3
4πrt3ddt
4πrt33
⎛ ⎝ ⎜
⎞ ⎠ ⎟
Cloud Thermodynamic Energy Eq.Add terms to (18.22)
--> thermodynamic energy equation in a cloud (18.24)
dθvdt =−θv
TvTv − ˆ T v( )+ Le
cp,dωv,s− ˆ ω v( )
⎡ ⎣ ⎢ ⎢
⎤ ⎦ ⎥ ⎥ E + 1
ρa∇ •ρaKh∇( )θv
+ θvcp,dTv
−Ledωv,s
dt −LmdωLdt −Ls
dωv,Idt +dQsolar
dt +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 + 1
ρa∇ •ρaKm∇( )w
∂ˆ p a ∂z =−̂ ρ ag
dwdt =−gρa −ˆ ρ a
ρa− 1
ρa
∂ pa − ˆ p a( )∂z + 1
ρa∇ •ρaKm∇( )w
Cloud Vertical Momentum EquationBuoyancy factor (18.27)
Adjust buoyancy factor for condensate (18.28)
B=−ρa −ˆ ρ aρa
=−pa ˆ T v − ˆ p aTvpa ˆ T v
=−ˆ T v −Tv
ˆ T v+ Tv
ˆ T v
⎛ ⎝ ⎜
⎞ ⎠ ⎟
ˆ p a −papa
≈−ˆ θ v −θv
ˆ θ v
B=−ρa −ˆ ρ aρa
=−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 =g θv −ˆ θ v
ˆ θ v−ωL
⎛ ⎝ ⎜
⎞ ⎠ ⎟ − 1
ρa
∂ pa − ˆ p a( )∂z + 1
ρa∇ •ρaKm∇( )w
1ρa
∂pa∂z =−g =−∂Φ
∂z =cp,dθv∂P∂z
dwdt =g θv −ˆ θ v
ˆ θ v−ωL
⎛ ⎝ ⎜
⎞ ⎠ ⎟ −cp,dθv
∂ P − ˆ P ( )∂z + 1
ρa∇•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 =dw
dzdzdt =dw
dz w =g θv −ˆ θ vˆ θ v
−ωL⎛ ⎝ ⎜
⎞ ⎠ ⎟ =gB
w =dzdt
wdw=gBdz
w2 =wa2 +2g θv −ˆ θ v
ˆ θ v−ωL
⎛ ⎝ ⎜
⎞ ⎠ ⎟ za
z∫ dz =wa2 +2g Bza
z∫ dz
Convective Available Potential Energy(18.34)
CAPE =g BzLFC
zLNB∫ dz≈g θv −ˆ θ vˆ 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 ,NimvDvωv,I ,NiFv,I,NiLs ′ S I,NiCI ,s
κaωh,NiFh,I ,NiTLsmvR*T
−1⎛ ⎝ ⎜ ⎞
⎠ ⎟ +1
Köhler EquationsLiquid (18.40)
Ice (18.41)
Rewrite as (18.42)
′ S L,Ni,t−h ≈1+2σ L,Ni,t−hmvrNiR*TρL
− 3mv4πrNi
3ρLnNi,t−hcq,Ni,t−h
q=1
Ns∑
′ S I,Ni,t−h ≈1+2σI,Ni,t−hmvrNiR*TρI
′ S L,Ni,t−h ≈1+aL,Ni,t−hrNi
−bL,Ni,t−hrNi3
Köhler Equations(18.43)
aL,Ni,t−h =2σL,Ni,t−hmvR*TρL
bL,Ni,t−h = 3mw4πρLnNi,t−h
cq,Ni,t−hq=1
Ns∑Solve for critical radius and critical saturation ratio (18.44)
rL,Ni,t−h* = 3bL,Ni,t−h
aL,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
orrNi ≤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+h
kL,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
, m m)
788hPa
656hPa
872hPa
729hPa
835hPa
dn (N
o. c
m-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
, m m)
788hPa
656hPa
872hPa
729hPa
835hPa
dn (N
o. c
m-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
⎛ ⎝ ⎜
⎞ ⎠ ⎟ υNk
VIi,MjυYk≤VIi,Mj <υYk+1 k <NC
1−fIi,Mj,Yk−1 υYk-1<VIi,Mj <υYk k >11 VIi,Mj ≥υYk k =NC0 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
, m m)
Brakupdistribution
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 >−3o C−T +3( ) 15 −18<T <−3oC1 T <−18o C
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
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 ln 0.5
υlq,k
⎛ ⎝ ⎜ ⎜
⎞ ⎠ ⎟ ⎟
B=0.475oC−1; Tr =0oC Tm≤−15oCB=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 ( m 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 ln ln0.5
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
, m m)
Liθuid,
baslin
(ithHHF)
Layrblo
Cloudtop
236.988K
214hPa
Liθuid,
noHHF
raupl,
noHHF raupl,
baslin
(ithHHF)
dn (N
o. c
m-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 − DvLeκa 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 (hPa) and final R
H x 10
Air 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)
Vapor 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 (hPa) and final R
H x 10
Tem
pera
ture
(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,k4πrkDv1−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. 18.1110
-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
, m m)
Cloudbas
(872hPa)
Surfac,
RH=75%
blobas
Surfac,
RH=99%
blobas
dn (N
o. c
m-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
, m m)
Liθuiddistribution
atRH=100%,
p
a
=214hPa
T
a
=236.988K
Additonal
portionof
liθ.distrib.that
frzsduto
vap.coolingat
RH=80%
dn (N
o. c
m-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 +MAX DvLeκaRv
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−h1+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(
m
m
3
cm
-3
)/dlog
10
D
p
Particldiamtr(D
p
, m m)
Arosol
volum
Arosol
numbr
Blocloudbas
(902hPa)
dn (N
o. c
m-3) /
d lo
g 10 D
pdv (mm
3 cm-3) / d log
10 Dp
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−1
Δzm−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 =kCQ0Q1r012
Ef = Fe,0iQ0i
∑ = kCQir0i2
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,Jj
υ IinIi,tnJj ,t−h +υ Jj nIi,t−hnJj ,t( )υIi +υ Jj
ΔQIi ,Jji=j
NC∑I=J
NH∑j =1
NC∑J =2
NH∑⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥ m
dQb,cdt =FcAcell
dQb,mdt Δzm
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
2dQb,c
dt
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
