; Slide 1 - Stanford University
Documents
Resources
Learning Center
Upload
Plans & pricing Sign in
Sign Out
Your Federal Quarterly Tax Payments are due April 15th Get Help Now >>

Slide 1 - Stanford University

VIEWS: 0 PAGES: 40

  • pg 1
									             Presentation Slides
                         for
                  Chapter 13
                         of
Fundamentals of Atmospheric Modeling
             2nd Edition
                 Mark Z. Jacobson
  Department of Civil & Environmental Engineering
                Stanford University
             Stanford, CA 94305-4020
              jacobson@stanford.edu
                  March 29, 2005
 Sizes of Atmospheric Constituents

Mode              Diameter (mm)   Number (#/cm3)
Gas molecules         0.0005      2.45x1019
Aerosol particles
   Small              < 0.2       103-106
   Medium             0.2-2       1-104
   Large              1-100       <1 - 10
Hydrometeor particles
   Fog drops          10-20       1-1000
   Cloud drops        10-200      1-1000
   Drizzle            200-1000    0.01-1
   Raindrops          1000-8000   0.001-0.01


                                               Table 13.1
    Particles and Size Distributions
Particle
  Agglomerations of molecules in the liquid and / or solid
  phases, suspended in air. Includes aerosol particles, fog drops,
  cloud drops, and raindrops

Example 13.1. - Idealized particle size distribution
  10,000 particles of radius between 0.05 and 0.5 mm
  100 particles of radius between 0.5 and 5.0 mm
  10 particles of radius between 5.0 and 50 mm

Example 13.2. Number of size bins needs to be limited
  105 grid cells
  100 size bins
  100 components per size bin
  --> 109 words = 8 gigabytes to store concentration
       Volume Ratio Size Structure
Volume of particles in one size bin                    (13.1)

                        i  Vrati 1

                                                       (13.2)
                                      i1
                          i  1Vrat


Volume-diameter relationship for spherical particles

                        i  di3 6
       Volume Ratio Size Structure
Variation in particle sizes with the volume ratio size structure




                                                            Fig. 13.1
       Volume Ratio Size Structure
Volume ratio of adjacent size bins                               (13.3)
                                1 N B 1               3 N B 1
                      N B                 dN B   
              Vrat                              
                      1                    d1    

Example 13.3.

       d1         = 0.01 mm

       dNB        = 1000 mm
       NB         = 30 size bins
--->   Vrat       = 3.29
       Volume Ratio Size Structure
Number of size bins                                (13.4)
                               
                               
                            ln  d N d1
                                B
                                             
                                            3 
                                              
                   NB  1 
                                  ln Vrat
Example 13.4.
      d1      = 0.01 mm
       dNB    = 1000 mm
       Vrat   =4
--->   NB     = 26 size bins
       Vrat   =2
--->   NB     = 51 size bins
       Volume Ratio Size Structure
Average volume in a size bin                        (13.5)
                        1
                        2
                            
                    i  i,hi  i,lo    
Relationship between high- and low-edge volume      (13.6)

                      i,hi  Vrati,lo


Substitute (13.6) into (13.5) --> low edge volume   (13.7)
                                 2i
                       i,lo 
                               1  Vrat
       Volume Ratio Size Structure
Volume width of a size bin                             (13.8)


                          2i1   2i      2i Vrat  1
   i  i,hi  i,lo                 
                         1 Vrat 1 Vrat     1  Vrat


Diameter width of a size bin                           (13.9)



                                          
                                                          13
                        6 1 3 1 3
                                    13         1 3 Vrat  1
di  di,hi  di,lo         i,hi  i,lo  di 2
                                                            13
                                                      1  Vrat 
            Particle Concentrations
Number concentration in a size bin                (13.10)
                                  vi
                            ni 
                                  i
Number concentration in a size distribution       (13.11)
                                     NB
                            ND       ni
                                     i1
Volume concentration in a size bin                (13.12)
                                    NV
                             vi     v q,i
                                    q1
Surface area concentration in a size bin          (13.13)
                                    2         2
                      ai  ni 4ri  ni di
             Particle Concentrations
Mass concentration in a size bin                      (13.14)
        NV            NV                    NV
 mi    m q,i  cm qvq,i  cm p,i vq,i  cmp,ivi
        q1           q 1                  q 1

Volume-averaged mass density (g cm-3) of particle of size i
                                                       (13.15)
                             NV
                              vi,q q 
                            q1
                      p,i  N
                                V
                                v i,q
                              q 1
            Particle Concentrations
Example 13.5
      mq,i = 3.0 mg m-3 for water
      mq,i = 2.0 mg m-3 for sulfate
       di      = 0.5 mm
       q      = 1.0 g cm-3 for water
       q      = 1.83 g cm-3 for sulfate
--->   v q,i   = 3 x 10-12 cm3 cm-3 for water
--->   v q,i   = 1.09 x 10-12 cm3 cm-3 for sulfate
--->   mi      = 5.0 mg m-3
--->   vi      = 4.09 x 10-12 cm3 cm-3
--->   i      = 6.54 x 10-14 cm3
--->    ni     = 62.5 partic. cm-3
--->    ai     = 4.8 x 10-7 cm2 cm-3
           Lognormal Distribution
              Bell-curve distribution on a log scale


Geometric mean diameter
  50% of area under a lognormal curve lies below it



Geometric standard deviation
  68% of area under a lognormal curve lies between +/-1 one
  geometric standard deviation around the mean diameter
                   Lognormal Distribution
                   Describes particle concentration versus size
                         2
                    10
/ d log10 Dpp
  ) / d log
         10


                    101

                    100
              -3
   dv (mm33cm-3)
     dv m m cm




                      -1
                    10
                      -2
                    10
        (




                                    D    D
                      -3             1    2
                    10
                      0.001        0.01        0.1      1
                              Particle diameter (D, mm)
                                                  p
                                                                  Fig. 13.2a
              Lognormal Distribution
   The lognormal curve drawn on a linear scale

                        2
/ d log10 D p
             p
                   10
  ) / d log
         10


                   101

                   100
   dv (mm3 cm-3)
    dv ( m3 cm-3




                     -1
                   10
                     -2
                   10
    m




                   10-3
                            0       0.05        0.1       0.15
                                Particle diameter (D, mm)
                                                   p
                                                                 Fig. 13.2b
Lognormal Parameters From Data
   Low-pressure impactor -- 7 size cuts
   0.05   - 0.075 mm                0.5 - 1.0 mm
   0.075 - 0.12 mm                  1.0 - 2.0 mm
   0.12   - 0.26 mm                 2.0- 4.0 mm
   0.26   - 0.5 mm
 Lognormal Parameters From Data
Natural log of geometric mean mass diameter        (13.16)

                                     7
                                  m j ln d j 
                          1
                  ln DM 
                          ML
                                 j1


Total mass concentration of particles (mg m-3)
                                 7
                        ML      mj
                                j 1
  Lognormal Parameters From Data
Natural log of geometric mean volume diameter             (13.17)
                                    7
                                    v j ln d j 
                               1
                      ln DV 
                              VL
                                   j 1
Total volume concentration of particles (cm3 cm-3)
                7
                                                  mj
        VL    vj                        vj 
                                                 c m j
               j 1
 Lognormal Parameters From Data

Natural log of geometric mean area diameter                (13.18)
                                  7
                                  a j ln d j 
                            1
                    ln DA 
                            AL
                                 j1
Total area concentration of particles (cm2 cm-3)
                7                               3m j
        AL    a j                     aj 
                                               c m j rj
               j1
 Lognormal Parameters From Data
Natural log of geometric mean number diameter              (13.19)

                             1 7
                    ln DN 
                            NL
                                    
                                n j ln d j    
                                 j1


Total number concentration of particles (partic. cm-3)
               7
                                                   mj
       NL    n j                      nj 
                                               cm j  j
              j1
 Lognormal Parameters From Data
Natural log of geometric standard deviation           (13.20)

             1 7       2 d j          1 7       2 d j 
 ln g 
            ML
                m j ln D           VL
                                             v j ln D 
               j 1       M             j 1       V 


          1 7       2 d j         1 7      2 d j 
     
         AL
             a j ln D       
                                     NL
                                         n jln D 
            j1         A           j1       N 
Redistribute With Lognormal Parameter
 Redistribute mass concentration in model size bin     (13.21)

                    M Ldi          ln2 di DM 
            mi                exp         2
                                                     
                  di 2 ln g       2 ln  g 
                                                   
 Redistribute volume concentration                      (13.22)
                    VLdi          ln 2 di DV 
            vi               exp        2
                                                  
                 di 2 ln g       2 ln  g 
                                                
 Redistribute area concentration                       (13.23)

                   A L di       ln2 di DA 
            ai              exp
                                        2
                                               
                 di 2 ln g     2 ln  g 
                                             
Redistribute With Lognormal Parameter
 Redistribute number concentration                   (13.24)


                    N L di      ln2  i D N 
                                        d
            ni              exp
                                        2
                                                
                 di 2 ln g     2 ln  g 
                                              


 Exact volume concentration in a mode                (13.25)

                       
                                   3      9 2 
     VL   vd dd       ndd dd  6 D N exp2 ln  gN L
                             3
                    6                               
           0            0
                                          Lognormal Modes
Number (partic. cm-3), area (cm2 cm-3), and volume (cm3 cm-3)
          concentrations distributed lognormally
                   (x=n, a, v)




                                      5
dx / d log10 Dp (x=n,a,v)



                                 10
                                                         D
                                                          N
                                      3
                                 10
                                                         DA
                      p




                                              n
             D
     dx/d log




                                      1
                                 10
           10




                                                             D
                                  -1                          V
                                 10           a
                                              v
                                  -3
                                 10
                                      0.001       0.01            0.1     1
                                              Particle diameter (D, mm)
                                                                 p            Fig. 13.3
Lognormal Param. for Cont. Particles

                     Nucleation   Accumulation   Coarse
 Parameter             Mode          Mode        Mode
 g                    1.7           2.03        2.15
 NL (particles cm-3)   7.7x104       1.3x104     4.2
 DN (mm)               0.013         0.069       0.97
 AL (mm2 cm-3)         74            535         41
 DA (mm)               0.023         0.19        3.1
 VL (mm3 cm-3)         0.33          22          29
 DV (mm)               0.031         0.31        5.7



                                                    Table 13.2
 Quadramodal Size Distribution
Size distribution at Claremont, California, on the morning of
                       August 27, 1987
     6
   10                                                              300
                                2   -3
   105                     da (mm cm )/d log D
                                           10 p                    250
     4
   10
                                                  3      -3        200
   10
     3                  -3
              dn (No. cm )                dv (mm cm )
                /dlog D                     /d log D               150
   102              10 p                              10 p


   101                                                             100
     0                                                             50
   10
   10-1                                                            0
       0.01             0.1             1                     10
                     Particle diameter (D, mm)                     Fig. 13.4
                                        p
      Marshall-Palmer Distribution
Raindrop number concentration between di and di+di (13.30)

                    ni  di n 0 e  r di

        din0   = value of ni at di = 0
        n0      = 8.0 x 10-6 partic. cm-3 mm-1
        r      = x Rmm-1
        R       = rainfall rate (1-25 mm hr-1)


Total number concentration and liquid water content

                                             6           4
     nT  n0 r                 w L  10          w n0 r
Marshall-Palmer Distribution
  Example 13.6.

         R        = 5 mm hr-1
         di       = 1 mm
         di+di = 2 mm
  --->   ni       = 0.00043 partic. cm-3
  --->   nT       = 0.0027 partic. cm-3
  --->   wL       = 0.34 g m-3
     Modified Gamma Distribution
Number concentration (partic. cm-3) of drops in size bin i (13.30)



                                       g 
                                             
                        g    g  ri  
           ni  ri Ag ri exp
                               g rc,g  
                                        
      Modified Gamma Distribution
               Parameters
                                                       Liquid   Number
    Cloud Type          Ag       g    g      rcg     Water     Conc.
                                              (mm)    Conten    (partic.
                                                          t      cm-3)
                                                      (g m-3)
Stratocumulus base    0.2823     5.0   1.19   5.33     0.141      100
Stratocumulus top     0.19779    2.0   2.46   10.19    0.796      100
Stratus base          0.97923    5.0   1.05   4.70     0.114      100
Stratus top           0.38180    3.0   1.3    6.75     0.379      100
Nimbostratus base     0.08061    5.0   1.24   6.41     0.235      100
Nimbostratus top      1.0969     1.0   2.41   9.67     1.034      100
Cumulus congestus     0.5481     4.0   1.0     6.0     0.297      100
Light rain           4.97x10-8   2.0   0.5    70.0     1.17       0.01
                                                                Table 13.3
     Modified Gamma Distribution
Example 13.7.
  Find number concentration of droplets between 14 and 16 mm
  in radius at base of a stratus cloud

           --->   ri     = 15 mm
           --->   ri    = 2 mm
           --->   ni     = 0.1506 partic. cm-3
      Full-Stationary Size Structure
 Average single-particle volume in size bin (i) stays constant.
 When growth occurs, number concentration in bin (ni) changes.

Advantages:
      • Covers wide range in diameter space with few bins
      • Nucleation, emissions, transport treated realistically

Disadvantages:
      • When growth occurs, information about the original
        composition of the growing particle is lost.
      • Growth leads to numerical diffusion
       Full-Stationary Size Structure
Demonstration of a problem with the full-stationary size bin structure




                                                           Fig. 13.5
             Full-Moving Structure
Number concentration (ni) of particles in a size bin does not change
  during growth; instead, single-particle volume (i) changes.

Advantages:
      • Core volume preserved during growth
      • No numerical diffusion during growth

Disadvantages:
      • Nucleation, emissions, transport treated unrealistically
      • Reordering of size bins required for coagulation
            Full-Moving Structure
Preservation of aerosol material upon growth and evaporation in a
                        moving structure




                                                        Fig. 13.6
Full-Moving Structure
Particle size bin reordering for coagulation




                                               Fig. 13.7
           Quasistationary Structure
Single-particle volumes change during growth like with full-moving
 structure but are fit back onto a full-stationary grid each time step.




 Advantages and Disadvantages:
       • Similar to those of full stationary structure
       • Very numerically diffusive
           Quasistationary Structure
After growth, particles in bin i have volume i’, which lies
  between volumes of bins j and k
                                 
                           j     k
                                 i
Partition volume of i between bins j and k while conserving
  particle number concentration                         (13.32)
                        ni  n j  nk
and particle volume concentration                        (13.33)
                        
                    ni   n j  j  nk k
                        i
Solution to this set of two equations and two unknowns (13.34)
               k  
                                                    
                                                        j
                                                      i
     n j  ni       i                     nk  ni
               k  j                              k   j
          Moving-Center Structure
  Single-particle volume (i) varies between i,hi and i,lo during
           growth, but i,hi, i,lo, and di remain fixed.

Advantages:
      • Covers wide range in diameter space with few bins
      • Little numerical diffusion during growth
      • Nucleation, emission, transport treated realistically

Disadvantages:
      • When growth occurs, information about the original
        composition of the growing particle is lost
                                      Moving-Center Structure
Comparison of moving-center, full-moving, and quasistationary size
structures during water growth onto aerosol particles to form cloud
                              drops.
          dv (mm cm ) / d log10 p p


                                                 Moving-center
                            10 D D




                                        7
                                      10
                                                  Full-moving
     ) /d log




                                      105
                                                     Quasi-
                                                   stationary
                                      103
                  -3
                 -3
        dv m m3 cm




                                                     Initial
           ( 3




                                      101

                                      10-1
                                         0.1       1              10       100
                                               Particle diameter (D, mm)
                                                                   p             Fig. 13.8

								
To top