Download Chapter 18 - Stanford University

Presentation Slides
for
Chapter 18
of
Fundamentals of Atmospheric Modeling
2nd Edition
Mark Z. Jacobson
Department of Civil & Environmental Engineering
Stanford University
Stanford, CA 94305-4020
[email protected]
April 1, 2005
Cloud Formation
Altitude range (km) of different cloud-formation étages
Étage
High
Middle
Low
Polar
3-8
2-4
0-2
Temperate
5-13
2-7
0-2
Tropical
6-9
2-8
0-2
Table 18.1
Fog
Cloud touching the ground
Radiation Fog
Forms as the ground cools radiatively at night, cooling the air
above it to below the dew point.
Advection Fog
Forms 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.
Fog
Evaporation Fog
Forms when water evaporates in warm, moist air, then mixes
with cooler, drier air and re-condenses.
Steam Fog
Occurs when warm surface water evaporates, rises into
cooler air, and recondenses, giving the appearance of rising
steam.
Frontal Fog
Occurs 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 Classification
Low clouds (0-2 km)
stratus
Stratus (St)
cumulus
Stratocumulus (Sc)
cirrus
Nimbostratus (Ns)
nimbus
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)
= "layer"
= "clumpy"
= "wispy"
= "rain"
Low Clouds
Stratus
A low, gray uniform cloud layer composed of water droplets
that often produces drizzle.
Stratocumulus
Low, lumpy, rounded clouds with blue sky between them.
Nimbostratus
Dark, gray clouds associated with continuous precipitation.
Not accompanied by lightning, thunder, or hail.
Middle Clouds
Altostratus
Layers 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 Clouds
Cirrus
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.
Cirrocumulus
High, puffy, rounded, ice crystal clouds that often form in
ripples.
Clouds of Vertical Development
Cumulus
Clouds with flat bases and bulging tops. Appear in individual,
detached domes, with varying degrees of vertical growth.
Cumulus humilis
Limited vertical development
Cumulus congestus
Extensive vertical development
Cumulonimbus
Dense, 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 Formation
Cloud Formation Mechanisms
free convection
forced convection
orography
frontal lifting
Formation of clouds along a cold and warm front, respectively
Fig. 18.1
Pseudoadiabatic Process
Condensation, latent heat release occurs during adiabatic ascent
Adiabatic process
dQ = 0
Pseudoadiabatic process
(18.1)
dQ   Led v,s
Saturation mass mixing ratio of water vapor over liquid water
pv,s
 v,s 
pd
Pseudoadiabatic Process
Differentiate v,s= pv,s/pd with respect to altitude, substitute
dpv,s  Le pv,sdT Rv T
2
 v,s  pv,s pd
R Rv
pd z   pd g RT

(18.5)
 v,s
z
 pv,s pv,s pd  Le  v,s T  v,s g




 
2
pd  z
pd z 
z
RT
RT
Pseudoadiabatic Process
Substitute (18.5) and d,m=g/cp,m into (18.4)
 Le v,s 
T 
   w  d,m 1 

 z w
R

T


Example 18.1
pd
T
--->
pv,s
--->
v,s
--->
w
T
--->
w
= 950 hPa
= 283 K
= 12.27 hPa
= 0.00803 kg kg-1
= 5.21 K km-1
= 293 K
= 4.27 K km-1
(18.6)

L2e v,s 
1

 Rc T 2 
p,m


Dry or Moist Air Stability Criteria
(18.7)
e  d,m

e  d,m


d,m  e  w

e  w

e  w
absolutely unstable
unsaturated neutral
conditionally uns table
s aturated neutral
absolutely s table
Stability in Dry or Moist Air
2
Altitude (km)
Altitude (km)
2.2
d,m
1.8
w
Absolutely
unstable
1.6
1.4
1.2
1
Conditionally
unstable
2
Absolutely
3 stable
1
4
0.8
-2
0
2
4
6
8 10
Temperature (o C)
12
14
Fig. 18.2
Stability in Multiple Layers
3
Saturated neutral
Altitude
Altitude(km)
(km)
2.5
Saturated neutral
2
e
d
w
1.5
1
0.5
Conditionally unstable
Unsaturated neutral
Absolutely stable
Absolutely unstable
0
0
5
10
15
20
Temperature ( oC)
25
Fig. 18.3
Equivalent Potential Temperature
Potential 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)
 L

e
 p,e   p exp 
c T  v,s 

 p,d

Equivalent potential temperature in unsaturated air
 L

e
 p,e   p exp 
v 
c T

 p,d LCL

(18.9)
Equivalent Potential Temperature
3.5
3
Altitude (km)
Altitude (km)
Relationship between potential temperature and equivalent
potential temperature
2.5
2
d,m
w
1.5
1
p
LCL
p,e
d
0.5
0
0
5
10
15
20
25
Temperature (K)
30
35
Fig. 18.4
Cumulus Cloud Development
3
Altitude
Altitude(km)
(km)
2.5
e
w
2 d
Cloud top
Cloud temperature
1.5
LCL
1
0.5
0
5
Dew point of
rising bubble
10
15 20 25 30
Temperature ( oC)
Temperature of
rising bubble
35
Fig. 18.5
Isentropic Condensation
Temperature
Temperature at the base of a cumulus cloud
Occurs at the lifting condensation level (LCL), which is that
altitude at which the dew point meets parcel temperature.
Isentropic condensation temperature
(18.11)
1  
 p
TIC 
v
d,0

4880.357  29.66ln 



T0  




TIC 
1  
 p
v d,0 TIC  

19.48 ln
 T 


 0  


Entrainment
Mixing 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)
*
ˆ
dQ  c
T  T dM
1
p,d
v
v

c
2) Energy loss from cloud due to evaporation of liquid water in
the cloud to ensure entrained, ambient air is saturated
(18.13)
*
ˆ v dMc
dQ 2   Le  v,s  


3) Energy gained by cloud during condensation of rising air
(18.14)
*
dQ3  Mc Led v,s
Entrainment
Sum the three sources and sinks of energy



(18.15)

*
ˆ v dMc  M c Led v,s
dQ  c p,d Tv  Tˆv dMc  Le  v,s  
First law of thermodynamics
*
(18.16)

dQ  Mc c p,d dTv   a dpa

Subtract (18.16) from (18.15) and rearrange
 
 
(18.17)

dM c
ˆ
ˆ
c p,d dTv   a dpa   c p,d Tv  Tv  Le  v,s  v
 Le dv,s
Mc
Entrainment
Divide by cp,d Tv and substitute a=R’Tv/pa
(18.18)

dMc  Led v,s
T  Tˆ
Le v,s  
dTv
R dpa
v
v
v




Tv
c p,d pa
c p,d Tv

 Tv

 Mc
Rearrange and differentiate with respect to height

Tv
g
Le
ˆ
ˆv

  Tv  Tv 
 v,s  
z
c p,d 
c p,d




c p,d Tv
(18.19)
 1 M
Le  v,s
c


c p,d z

 Mc z

No entrainment (dMc = 0) --> pseudoadiabatic temp. change
Cloud Vertical Temperature Profile
Change of potential virtual temperature with altitude
 v  v Tv


z
Tv z
(2.103)
v pa
pa z
Rearrange
(18.20)
Tv Tv v
RTv pa Tv  v
g




z
v z
c p,d pa z
v z
c p,d
Substitute into (18.19)
--> change of potential virtual temperature in entrainment region
 v
v

z
Tv

Le
ˆ
ˆv
 Tv  Tv 
 v,s  
c p,d





 1 M
v Le  v,s
c


Tv c p,d dz

 Mc z

Cloud Thermodynamic Energy Eq.
Multiply through by dz and dividing through by dt
d v
v

dt
Tv

Le
ˆ
ˆv
 Tv  Tv 
 v,s  
c p,d





(18.22)

v Le d v,s
E 

 c p,d Tv dt

Entrainment rate
1 dMc
3 d 4rt3 
E



3
Mc dt
4rt dt  3 
(18.23)
Cloud Thermodynamic Energy Eq.
Add terms to (18.22)
--> thermodynamic energy equation in a cloud
d v
v

dt
Tv

Le
ˆ
ˆv
 Tv  Tv 
 v,s  
c p,d





(18.24)

1
E +

  a K h  v

  a

d v,s
dv,I dQ solar dQ ir 
v 
d L

 Lm
 Ls


 Le

c p,d Tv 
dt
dt
dt
dt
dt 
Cloud Vertical Momentum Equation
Vertical momentum equation in Cartesian / altitude coordinates
(18.25)
dw
1 pa
1
 g 

  aKmw
dt
 a z  a
ˆ ag for air outside cloud
Add hydrostatic equation, pˆ a z  
(18.26)
ˆa
dw
 a  
1 pa  pˆ a  1
 g



   a Km w
dt
a
a
z
a
Cloud Vertical Momentum Equation
Buoyancy factor
(18.27)
ˆ 
ˆa
a  
pa Tˆv  pˆ a Tv
Tˆv  Tv Tv  pˆ a  pa

v
B 


  
 v
ˆ
a
pa Tˆv
Tˆv

Tˆv  pa
v
Adjust buoyancy factor for condensate
(18.28)
ˆ 1    1 
ˆ L    
ˆ
ˆa

 a  
v
L
v
v 
B 

 v
L
ˆ
ˆ
a


v
v
Cloud Vertical Momentum Equation
Substitute (18.28) into (18.26)
(18.29)
ˆ
  
 1 pa  pˆ a  1
dw
v
v
 g ˆ
  L  


   a Km  w
dt
z
a
 v
 a
Rewrite pressure gradient term
(18.30)
1 pa

P
 g  
 c p,d  v
 a z
z
z
Substitute (18.30) and (18.29)
--> vertical momentum equation in a cloud


(18.31)
ˆ
 P  Pˆ
v  

dw
1
v
 g ˆ
  L   c p,d v


  a K m w
dt
z
a
 v

Simplified Vertical Velocity in Cloud
Simplify (18.31) for basic calculations
Ignore pressure perturbation and the eddy diffusion term (18.32)
ˆ
  

dw dw dz dw
v
v


w  g ˆ
  L   gB
dt
dz dt
dz
 v

where
dz
w
dt
Rearrange (18.32)
wdw  gBdz
Integrate over altitude --> vertical velocity in a cloud
w
2
ˆ

z  v  
2
2
v
 wa  2g  ˆ
  L dz  wa  2g
z a  

v

(18.33)
z
zBa dz
Convective Available Potential
Energy
(18.34)
ˆ 
z LNB  v  
CAPE  g
Bdz  g
 ˆ v dz
z LFC
z LFC  
v 

z LNB

Cloud Microphysics
Assume clouds form on multiple aerosol particle size distributions
Each aerosol distribution consists of multiple discrete size bins
Each size bin contains multiple chemical components
Three cloud hydrometeor distributions can form
Liquid
Ice
Graupel
Each hydrometeor distribution contains multiple size bins.
Each size bin contains the chemical components of the aerosol
distribution it originated from
Cloud Microphysics
Processes considered
Condensation/evaporation
Ice deposition/sublimation
Hydrometeor-hydrometeor coagulation
Large liquid drop breakup
Contact freezing of liquid drops
Homogeneous/heterogeneous freezing
Drop surface temperature
Subcloud evaporation
Evaporative freezing
Ice crystal melting
Hydrometeor-aerosol coagulation
Gas washout
Lightning
Condensation and Ice Deposition
Condensation/deposition onto multiple aerosol distributions
(18.35)
dc L,Ni,t
 k L,Ni,t h Cv,t  SL,Ni,t

 hC L,s,t h
dt
(18.36)
dc I ,Ni,t
 k I ,Ni,t h Cv,t  S
I,Ni,t h CI ,s,t h
dt




Water vapor-hydrometeor mass balance equation


(18.37)

NT N B k

dCv,t
L,Ni,t h Cv,t  S
L,Ni,t h CL,s,t h 



dt
k I,Ni,t h Cv,t  S

C
I, Ni,t h I,s,t h
N 1 i1 



Vapor-Hydrometeor Transfer Rates
(18.38,9)
k L,Ni 
nlq,Ni 4rNi Dv v,L,Ni Fv,L,Ni
mv Dv v,L,Ni Fv,L, Ni Le SL,Ni
 CL,s L emv

 * 1 1
 a h,Ni Fh,L,Ni T
 R T

nic ,Ni 4 Ni Dv v,I ,Ni Fv,I ,Ni
k I ,Ni 
m v Dv v,I ,Ni Fv,I, Ni L s S

I,Ni C I ,s  Ls m v
 * 1 1
 a  h,Ni Fh,I ,Ni T
 R T

Köhler Equations
Liquid
(18.40)
S 
L,Ni,t h  1 
2 L,Ni,t h mv
*
rNi R TL

3m v
Ns

cq,Ni,t h
3
4rNi Ln Ni,t h q1
Ice
(18.41)
S 
I,Ni,t h  1
2 I,Ni,t h mv
rNi R*TI
Rewrite as
(18.42)
a L,Ni,t h b L,Ni,t h
S 

L,Ni,t h  1 
3
rNi
r
Ni
Köhler Equations
a L,Ni,t h 
2 L,Ni,th mv
R*TL
3mw
bL, Ni,t h 
4 L nNi,t h
Ns
 c q,Ni,t h
q 1
Solve for critical radius and critical saturation ratio
*
rL,Ni,th

*
S L,Ni,th  1 
(18.43)
3bL,Ni,t h
a L,Ni,th
4a3L,Ni,t h
27bL,Ni,th
(18.44)
CCN and IDN Activation
Cloud condensation nuclei (CCN) activation
(18.45)
*
rNi  rL,Ni
and C v,t h  S 
L,Ni,t h C L,s,t  h


or

*
*

r

r
and
C

S
 Ni L,Ni
v,t h
L,Ni,t h C L,s,t  h
Ice deposition nuclei (IDN) activation
Cv,t h  SI,Ni,t hCI,s,t h
(18.46)
Solution to Growth Equations
Aerosol mole concentrations
(18.47,8)


cI, Ni,t  cI, Ni,th  hkI,Ni,t hCv,t  SI,Ni,th

CI,s,th 
cL,Ni,t  cL,Ni,th  hkL,Ni,th Cv,t  S 
L,Ni,thCL,s,t h
Mole balance equation
Cv,t 
(18.49)
NT N B
  cL, Ni,t  cI, Ni,t 
N1 i1
 Cv,t h 
NT N B
  cL, Ni,th  cI,Ni,t h Ctot
N 1 i1
Solution to Growth Equations
Final gas mole concentration
(18.50)
NT NB k

L,Ni,t h SL,Ni,t
h Cs,L,t h  
Cv,t h  h 
C v,t 
 k I,Ni,t h S I,Ni,t h Cs,I ,t h
N 1 i1
N T NB
1 h 
 k Li,t h  kIi,t  h
N 1 i 1


Growth in Multiple Layers
10
dn (No. cm ) / d log
-3
D
-3
dn (No. cm ) / d log 10D
p
Dual peaks when grow on multiple size distributions, each with
different activation characteristic
1600
1400
1200
1000
800
600
400
200
0
872 hPa
835 hPa
788 hPa
729 hPa
656 hPa
10
Particle diameter (D p , m)
100
Fig. 18.6
Growth in Multiple Layers
dn (No. cm ) / d log
-3
D
-3
10
dn (No. cm ) / d log10 Dpp
Single peaks when size distribution homogeneous
1600
1400
1200
1000
800
600
400
200
0
872 hPa
835 hPa
788 hPa
729 hPa
656 hPa
10
Particle diameter (D p , m)
100
Fig. 18.6
Hydrometeor-Hydrometeor Coagulation
Final volume concentration of component or total particle
(18.53)
v x,Yk,t h  h Tx,Yk ,t,1  T x,Yk ,t,2
v x,Yk,t 
1 hTx,Yk,t,3
N H 
k 
k1

PY,M n Mj,t  h

Tx,Yk,t,1 
f

v
Yi,Mj,Yk
Yi,Mj,t
h
x,Yi,t




M 1 
j1
i1
NH N H 
k 
k

QI,M,Y n Mj,t h

Tx,Yk,t,2 
f

v
Ii,Mj,Yk
Ii,Mj,th
x,Ii,t




M1 I 1
j1
i1







NC N H




Tx,Yk,t,3 
1  LY,M 1  fYk,Mj,Yk  LY,M  Yk,Mj,th nMj,t h 

M 1

j 1 
 


Hydrometeor-Hydrometeor Coagulation
Final number concentration
nlq,k,t 
(18.54)
v T,lq,k ,t
lq,k
Volume fraction of coagulated pair partitioned to a fixed bin
(18.55)
 Yk 1  VIi,Mj   Nk
Yk  VIi,Mj  Yk 1
k  NC


 Yk 1  Yk  VIi,Mj

Yk-1  VIi,Mj  Yk
k 1
fIi,Mj,Yk  1  f Ii,Mj,Yk 1
1
VIi,Mj  Yk
k  NC

0
all other cas es

Drop Breakup Size Distribution
Drops breakup when they reach a given size
dM / M d log
D
10 p
T
dM / MT d log10 Dp
2.5
Breakup distribution
2
1.5
1
0.5
0
0
1000
2000 3000 4000 5000
Particle diameter (D p , m)
6000
Fig. 18.7
Contact Freezing
Final volume concentration of total liquid drop or its components
(18.59)
v
v x,lq,k,t 
x,lq,k,t h
1  hTx,k,t,3
(18.61)
NC  NT

  Yk,Nj,th FICN,Nj n Nj,t h 
Tx,k,t,3  FT

N1

j 1
 
Final volume concentration of a graupel particle in a size bin or of
an individual component in the particle
(18.60)
vx,gr,k,t  vx,gr,k,t h  vx,lq,k,thTx,k,t,3
Contact Freezing
Final number concentrations
nlq,k,t 
(18.62)
v T,lq,k ,t
lq,k
(18.63)
ngr ,k ,t 
vT ,gr ,k ,t
gr,k
Temperature-dependence parameter
0


FT   T  3 15


1
(18.64)
T  3o C
18  T  3o C
o
T  18 C
Homogeneous/Heterogeneous Freezing
Fractional number of drops of given size that freeze

(18.65)


FFr,k,t  min lq,k exp BTc  Tr  ,1
Median freezing temperature
(18.66)
1  0.5 
Tmf  Tr  ln 


B lq,k 

o C 1 ; T  0 o C

B

0.475

r

o 1
o

B

1.85
C
;
T

11.14
C

r
Tm  15 o C
o
o
15 C  Tm  10 C
Homogeneous/Heterogeneous Freezing
Median freezing temperatureo(oC)
C)
Fitted versus observed median freezing temperatures
-12
-16
-20
-24
-28
10
100
1000
Particle radius (m)
10 4
Fig. 18.8
Homogeneous/Heterogeneous Freezing
Time-dependent freezing rate
dngr,k,t
dt
(18.67)


 nlq ,k,t h lq,k Aexp BTc  Tr 
Final number conc. of drops and graupel particles after freezing
(18.68)

nlq,k,t  nlq,k,th 1 FFr ,k,t

(18.69)
ngr ,k,t  ngr,k,th  nlq,k,t h FFr,k,t
Homogeneous/Heterogeneous Freezing
Fractional number of drops that freeze

(18.70)


FFr,k,t  1  exp hAlq,k exp BTc  Tr 
Time-dependent median freezing temperature
1  ln 0.5 
Tmf  Tr  ln 


B hAlq,k 

(18.71)
Homogeneous/Heterogeneous Freezing
dn (No. cm ) / d log
-3
Dp
dn (No. cm-3) / d log1010D
p
Simulated liquid and graupel size distributions with and without
homogeneous/heterogeneous freezing after one hour
10 2
Layer below
Cloud top
236.988 K
214 hPa
10 0
10 -2
Graupel,
no HHF
Graupel,
baseline
(with HHF)
Liquid,
baseline
(with HHF)
10 -4
10 -6
Liquid,
no HHF
10 -8
1
10
100
1000
Particle diameter (D p , m)
104
Fig. 18.9
Drop Surface Temperature
Iterate for drop surface temperature at sub-100 percent RH
(18.72)
ps,n  pv,s Ts,n
 


p f ,n  0.5ps,n  pv,n 
Tf ,n  0.5Ts,n  Ta
pv,n  0.3 ps,n  pv,n
pv,n
Ts,n1  Ts,n 
a 1  p f ,n pa RvT f,n

Dv Le

pv,n1  pv,n  pv,n
Temperature (K)
285
Initial and final T
Initial p
280
a
20
and initial T
Final T
s
15
s
s
10
Final RHx10
275
Final p = final p
Initial p
270
0
v
v
0.8
0.6
0.4
0.2
Initial relative humidity (fraction)
5
s
0
1
Vapor pressure (hPa) and final RH x 10
Air temperature = 283.15 K
Vapor pres. (hPa) and final RH
Temperature (K)
Drop Surface Temperature vs. RH
Fig. 18.10
Temperature (K)
248
247
246
245
244
243
242
241
240
Final RH
Initial and final T
Final T
s
Initial p
1
a
and initial T
s
Final p = final p
Initial p
0
v
v
0.2
0.4
0.6
0.8
Initial relative humidity (fraction)
0.8
s
0.6
0.4
0.2
s
0
1
Vapor pressure (hPa) and final RH x 10
Air temperature = 245.94 K
Vapor pres. (hPa) and final R
Temperature (K)
Drop Surface Temperature vs. RH
Fig. 18.10
Drop Surface Temperature vs. RH
Temperature (K)
Vapor pressure (hPa) and final RH x 10
Air temperature = 223.25 K
Fig. 18.10
Evaporation
Reduction in volume due to evaporation/sublimation

nlq,k 4rk Dv
v L,lq,k ,t,m  MAXv L,lq ,k ,t h 

1 p f ,nf pa


(18.73)
pv,s,0  pv,nf  z , 0 
 L RvT f ,nf V f ,lq,k m
dn (No.
10
p
dn (No. cm ) / d log
-3
-3
cm ) / d log 10DD
p
Reduction in precipitation size due to evaporation below cloud base
10 3
2
10
1
10
10
Cloud base
(872 hPa)
0
10 -1
10 -2
Surface,
RH=99%
below base
Surface,
RH=75%
below base
-3
10
10 -4
1
10
100
1000
Particle diameter (D p , m)
10
4
Fig. 18.11
Evaporative Freezing
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.
dn
p
dn (No. cm ) / d log
-3
D Dp
(No. cm-3) / d log1010
Incremental homogeneous/heterogeneous freezing due to
evaporative cooling below a cloud base
Liquid distribution
at RH=100%,
10 0
10-2
10-4
p =214 hPa
10-6
10
a
T =236.988 K
Additonal
port ion of
liq. distrib. that
freezes due t o
evap. cooling at
RH=80%
a
-8
10
100
Particle diameter (D p , m)
Fig. 18.12
Ice Crystal Melting
When 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

 Dv Le pv,s T0 
Tmelt  T0  MAX



 a Rv  T0
(18.74)

pv  
, 0
Ta  

Time-dependent change in particle mass due to melting (18.75)

 4r
Ni
mic ,Ni,t  mic,Ni,t h - MAXh
 L m


 a Ta  T0 Fh,I,Ni 
 

 
Dv Le  pv,s T0  pv 
, 0
 Fv,I,Ni  
 R  T
Ta 
0
 v 
 

Aerosol-Hydrometeor Coagulation
Final volume conc. of total aerosol particle or its components
(18.76)
v x,Nk,t h
v x, Nk,t 
1 hTx,Nk,t,3
NC N H

  Nk,Mj,th nMj,th 
Tx,Nk,t,3 

M 1

j1 
 
Aerosol-Hydrometeor Coagulation
Final volume conc. of total hydrometeor or aerosol inclusions
(18.77)
v x,Yk,t h  h Tx,Yk ,t,1  T x,Yk ,t,2
v x,Yk,t 
1 hTx,Yk,t,3

N T  k 


 nNj,t h

Tx,Yk,t,1 
f

v
Yi,Nj,Yk
Yi,
Nj,th
x,Yi,t

 

N1 j 1
i1
 
NT  k 
k1


 nYj,t h

Tx,Yk,t,2 
f

v
Ni,Yj,Yk
Ni,Yj,t
h
x,Ni,t

 

N1 j1
i1
NB N T


Tx,Yk,t,3 
1 fYk, Nj,Yk  Yk, Nj,th nNj,th 

N1

j 1 
 
 
k


Aerosol-Hydrometeor Coagulation
Final number concentrations
(18.78)
v T,Nk ,t
n Nk,t 
 Nk
(18.79)
v T,Yk,t
nYk,t 
Yk
Aerosol-Hydrometeor Coagulation
dn (No.
10 p
-3
cm-3
)
Aerosol
number
20
15
10
5
/d
500
25
p
Dp
log10 D
1000
30
10
) / d log
dn (No. cm
1500
Below cloud base
Aerosol
(902 hPa)
volume
35
dv
2000
3
-3-3
)
(m
3 cm
dV( m cm
) / d log
D Dp
/ d log10
Below-cloud aerosol number and volume concentration before (solid
lines) and after (short-dashed lines) aerosol-hydrometeor coagulation
0
0.001
0
0.01
0.1
1
10
Particle diameter (D , m)
p
100
Fig. 18.13
Gas Washout
Gas-hydrometeor equilibrium relation
c q,lq,t,m
Cq,t,m
(18.80)
NC
 HqR*T  pL,lq,t,m
k1
Gas-hydrometeor mass-balance equation
(18.81)
zm1
Cq,t,m  c q,lq,t,m  Cq,t h,m  c q,lq,t,m1
zm
Gas Washout
Final gas concentration in layer m
(18.82)
z
Cq,t h,m  cq,lq,t,m1 m1
zm
Cq,t,m 
NC
1 HqR*T  pL,lq,t,m
k 1
Final aqueous mole concentration
(18.83)
zm 1
c q,lq,t,m  Cq,t h,m  c q,lq,t,m1
 Cq,t,m
zm
Lightning
Coulomb’s law
(18.84)
Fe 
k CQ0 Q1
2
r01
Electric field strength
(18.86)
Fe,0i
k CQi
Ef  
 2
Q0
r
i
i
0i
Rate coefficient for bounceoff

(18.87)

BIi,Jj,m  1 Ecoal,Ii,Jj,m KIi,Jj,m
Lightning
Charge separation rate per unit volume of air
dQb,m
dt

(18.88)

N H NC NH NC

 Ii nIi,t nJj,t h   Jj nIi,t h n Jj,t
 
BIi,Jj
QIi,Jj 

Ii   Jj

J

2
j
1
I

J
i
j

m
  
Overall charge separation rate
dQb,c
 Fc Acell
dt
(18.91)
K b ot

m Kto p
dQb,m
zm
dt
Lightning
Time-rate-of-change of the in-cloud electric field strength
dE f
(18.92)
dQb,c
2k C

dt
Zc Z c2  Rc2 dt
Summed vertical thickness of layers
Zc 
(18.93)
Kbo t
 zm
m K to p
Horizontal radius of cloudy region
Rc  Fc Acell 
(18.94)
Lightning
Number of intracloud flashes per centimeter per second
(18.95)
dFr
1 dE f

dt
Z cEth dt
Number of NO molecules per cubic centimeter per second
(18.96)
El FNO dFr
ENO 
Acell dt