Presentation Slides for

Chapter 20of

Fundamentals of Atmospheric Modeling 2nd Edition

Mark Z. JacobsonDepartment of Civil & Environmental Engineering

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

March 10, 2005

Particle Sedimentation

Fig. 20.1

Drag

Gravity

Vertical forces acting on a particle

Drag and Gravitational ForcesDrag during Stokes flow (20.1)

Where particle radius > mean free path of air molecule (e.g., 68 nm) but small enough so its inertial force < viscous force.

Drag during slip flow (20.2)Particle radius < mean free path of an air molecule

FD =6πri ηaVf,i

FD =6πri ηaVf,i

Gi=

6πriηaVf,i

1+Kna,i ′ A + ′ B exp− ′ C Kna,i−1

( )⎡ ⎣

⎤ ⎦

Kna,i =λari

Knudsen number of particle in air

Particle Sedimentation

Equate gravity with drag to estimate fall speed (20.4)

Small particlesLess resistance to motion ---> diffusion and fall speed enhanced at small particle sizes

Large particlesFall speed decreases due to physical properties effect --> need to correct fall speed for large particles

Vf,iest=

2ri2 ρp−ρa( )g

9ηaGi

Gravitational force (20.3)

FG =43

πri3 ρp −ρa( )g

Estimated Reynolds NumberEstimate Reynolds number from estimated fall speed. (20.4)

Recalculate Reynolds number for three flow regimes

• slip flow around a rigid sphere ( «1-20 m diameter)

• continuum flow around a rigid sphere (20 m - 1 mm)

• continuum flow around equilibrium-shaped drop (1-5 mm)

Reiest=

2riVf,iest

νa

Final Reynolds Number (20.6)

Parameters affected by physical properties (e.g., temperature, density, viscosity, surface tension, gravity) (20.7)

Reifinal=

2riVf,iest νa Rei

est<0.01

Gi expB0 +B1X +B2X2 +...( ) 0.01<Reiest<300

NP16Gi expE0+E1Y +E2Y2+...( ) Rei

est>300

⎧

⎨

⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪

X =ln32ri

3 ρp −ρa( )ρag

3ηa2

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

Y =ln43

N BoNP16⎡

⎣ ⎢ ⎤ ⎦ ⎥

Physical Properties CorrectionPhysical property number (20.8)

Bond number (20.8)

Final fall speed from final Reynolds number (20.9)

N p =σw/a

3 ρa2

ηa4 ρp −ρa( )g

N Bo=4ri

2 ρp−ρa( )g

σw/a

Vf,i =Vf,ifinal=

Reifinalνa2ri

Sedimentation Times

Table 20.2

Time for a particle (or gas molecule for the smallest size) to fall 1 km in the atmosphere due to sedimentation

Diameter

(m)

Time to Fall 1 km

Diameter

(m)

Time to Fall 1 km

0.0005 9630 y 4 23 d

0.02 226 y 5 14.5 d

0.1 36 y 10 3.6 d

0.5 3.2 y 20 23 h

1 326 d 100 1.1 h

2 89 d 1000 4 m

3 41 d 5000 1.8 m

Dry DepositionDry deposition

Removal of gas molecules or particles from the air when they stick to or react with a surface

Gas dry deposition speed (20.10)

Particle dry deposition speed (20.11)

Vd,gas= Ra+Rb +Rs( )−1

Vd,part,i = Ra +Rb+RaRbVf,i( )−1

+Vf,i

Dry Deposition Resistances

Fig. 20.2

zr

Ra

Rb

Rs

Dry Deposition ResistancesAerodynamic resistance (20.12)

Resistance to diffusion in laminar sublayer (20.14)

Particle and gas Schmidt numbers, Prandtl number (15.36)

Ra =

φhdzzz0,q

zr∫ku*

Rb =lnz0,mz0,q

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Sc Pr( )2 3

ku*

Scpi =νaDpi

Scq =νaDq

Pr =ηacp,m

κa

Surface ResistanceSurface resistance due to biological interactions (20.15)

Stomatal resistance (20.16)Resistance to entering openings in leaf surfaces

Leaf mesophyll resistance (20.17)Resistance to dissolving in or reacting with water within leaves

Rs =1

Rstom+Rmeso+

1Rcut

+1

Rconv+Rexp+

1Rcanp+Rsoil

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

−1

Rstom,q =Rmin1+200

Sf +0.1

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

400

Ta,c 40−Ta,c( )

DvDq

Rmeso,q =Hq

*

3000+100f0,q

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

−1

Surface ResistanceResistance to deposition on leaf cuticles (waxy surface) (20.18)

Resistance to buoyant convection in canopy (20.19)

Resistance to deposition on bark, exposed surfaces (20.20)

Rconv=1001+1000

Sf +10

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

11+1000st

Rsurf,q =10−5Hq

*

Rsurf,SO2+

f0,qRsurf,O3

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

−1

Rcut=Rcut,dry,q =Rcut,0 10−5Hq* + f0,q( )

−1

Surface ResistanceIn-canopy resistance (20.21)

Accounts for canopy leaf density

One-sided leaf area index (LT)Integrate foliage area density from surface to height hc

Foliage area densityArea of plant surface per unit volume of space. Thus, the leaf-area index measures canopy area density

Rcanp=bchcLT

u*

Resistance to deposition on soil and leaf litter at ground (20.22)

Rsoil,q =10−5Hq

*

Rsoil,SO2

+f0,q

Rsoil,O3

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

−1

Dry Deposition, Sedimentation Speeds

Fig. 20.3

10

-6

10

-4

10

-2

10

0

10

2

10

4

0.01 0.1 1 10 100 1000

Speed (cm s

-1

)

Particle diameter ( )m

-Dry deposition

speed without

sedimentation

Sedimentation speed

-Total dry deposition

speed

Spe

ed (

cm/s

)

Several Parameters Versus Size

Fig. 20.4

10

-11

10

-7

10

-3

10

1

10

5

0.01 0.1 1 10 100 1000

Particle diameter ( )m

Knudsen

number

- ( Total dry deposition speed cm s

-1

)

Reynolds

number

(Diffusion coefficient cm

2

s

-1

)

Gas Dry Deposition Speeds

Fig. 20.5a,b

10

-3

10

-2

10

-1

10

0

10

1

0 10 20 30 40 50

10 g/mol, 10 m/s

130 g/mol, 10 m/s

10 g/mol, 0 m/s

130 g/mol, 0 m/s

Dry deposition speed (cm s

-1

)

Surface resistance (s cm

-1

)

(a) z0,m=3 m

Dry

dep

osit

ion

spee

d (c

m/s

)

10

-2

10

-1

10

0

10

1

0 10 20 30 40 50

10 g/mol, 10 m/s

130 g/mol, 10 m/s

10 g/mol, 0 m/s

130 g/mol, 0 m/s

Dry deposition speed (cm s

-1

)

Surface resistance (s cm

-1

)

(b) z0,m=0.01 m

Dry

dep

osit

ion

spee

d (c

m/s

)

Air-Sea FluxesChange in concentration of a gas at the air-sea interface (20.23)

Mole concentration of a gas (20.24)

Mole concentration of a gas dissolved in seawater (20.25)

Cq,t =Cq,t−h +hVd,gas,q

Δza

cq,T,t−h′ H q

−Cq,t−h⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Cq =pq

R*T

cq,T =ρdwmq,T

Air-Sea FluxesDissolution and dissociation of carbon dioxide (20.26)

Dimensionless effective Henry’s constant (20.27)

Surface resistance of gas over the ocean (20.34)=chemical reactivity (1 for CO2; large for HCl)

Rs,q =1

αr,q ′ H qkw,q

CO2

(g) + H2

O(aq) H2

CO3

(aq) H+

+ HCO3

-

2H+

+ CO3

2-

HK

1K

2

′ H q =ρdwR*THqs 1+

K1,qs

mH+1+

K2,qs

mH+

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

Air-Sea FluxesAir-sea gas transfer speed (two parameterizations) (20.35,7)

Schmidt number ratio in water (20.36)Schmidt number is kinematic viscosity / diffusion coefficient

Srw,q =Scw,CO2,20oC

Scw,q,Ts,c

kw,q =0.31vh,10

2Srw,q12

3600

kw,q =1

3600

0.17vh,10Srw,q2 3 vh,10 ≤3.6 m s-1

0.612Srw,q23 + 2.85vh,10 −10.262( )Srw,q

12 3.6 m s-1<vh,10 ≤13 m s-1

0.612Srw,q23 + 5.9vh,10 −49.912( )Srw,q

12 13 m s-1<vh,10

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪

Solution to Air-Sea Flux EquationsImplicit equation for atmosphere-ocean transfer (20.23)

Solution to gas concentration (20.39)

Cq,t =Cq,t−h +hVd,gas,q

Δza

cq,T,t′ H q

−Cq,t⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Cq,t =

Cq,t−h +hVd,gas,q

Δza

cq,T,t′ H q

1+hVd,gas,q

Δza

Solution to Air-Sea Flux EquationsSubstitute into mass balance equation (20.40)

Solution to ocean concentration (20.41)

cq,T,tDl +Cq,tΔza =cq,T,t−hDl +Cq,t−hΔza

cq,T,t =

cq,T,t−h +hVd,gas,qCq,t−h

Dl1+

hVd,gas,qΔza

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

1+hVd,gas,q

Dl ′ H q1+

hVd,gas,qΔza

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Stability TestAir-sea transfer plus chemistry of CO2 with time steps of 6 h to 1 y

7.4

7.6

7.8

8

8.2

0.01 0.1 1 10 100

6 h

1 d

10 d

100 d

1 y

pH

Year of simulation

(c)

pH

1-D Ocean, 2-Box Atmosphere Case

Ocean Chemistry System

Na+

Ca2+

Mg2+

K+

H+

Sr2+

Li+

NH4+

Cl-

Br-

OH-

HSO4-

HCO3-

CO32-

B(OH)4-

SiO(OH)3-

H 2PO4-

HPO42-

PO43-

HNO3-

H2O(aq)H2CO4(aq)H2SO4(aq)H3PO4(aq)HF(aq)H2S(aq)

CaCO3(s)& other solids

Chemicals treated in simulations discussed next

Modeled CO2(g) and Modeled v Measured Ocean pH 1751-2003

280

300

320

340

360

380

8.1

8.15

8.2

8.25

8.3

1750 1800 1850 1900 1950 2000

Data

Model

pH-500Tg-C/yBBemis

CO

2

mixing ratio (ppmv)

Surface ocean pH

Year

CO

2(g)

mix

ing

rati

o (p

pmv)

Surface ocean pH

Fig. 20.6

Modeled Ocean Profiles 1751; 2004

8 8.1 8.2 8.3

0

500

1000

1500

2000

1751

2004

pH

Depth (m)

(a)

Dep

th (

m)

1.9 2 2.1 2.2

0

500

1000

1500

2000

1751

2004

Total inorganic carbon (mmol/kg)

Depth (m)

(b)

Dep

th (

m)

0 5 10 15 20

0

500

1000

1500

2000

S-30 (1751)

S-30 (2004)

Tc

S-30 (ppth-mass) and T

c

(

o

C)

Depth (m)

(d)

Dep

th (

m)

Jacobson, JGR 2005

Modeled Ocean Profiles 2004; 2104 Under SRES A1B Emission Scenario

7.8 7.9 8 8.1 8.2

0

500

1000

1500

2000

2004

2104

pH

Depth (m)

(a)

1.9 2 2.1 2.2 2.3

0

500

1000

1500

2000

2004

2104

Total inorganic carbon (mmol/kg)

Depth (m)

(b)

Dep

th (

m)

Dep

th (

m)

Sensitivity of Future Results

300

400

500

600

700

800

900

7.8

7.9

8

8.1

8.2

2000 2020 2040 2060 2080 2100

T=292.25

T=289.25

T=286.25

pH

CO

2

mixing ratio (ppmv)

Surface ocean pH

Year

(b)

CO

2(g)

mix

ing

rati

o Surface ocean pH

To temperature (K)

300

400

500

600

700

800

900

7.8

7.9

8

8.1

8.2

2000 2020 2040 2060 2080 2100

u=1

u=3

u=5

CO

2

mixing ratio (ppmv)

Surface ocean pH

Year

(a)

Surface ocean pH

To wind speed (m/s)

CO

2(g)

mix

ing

rati

o

300

400

500

600

700

800

900

1000

7.7

7.8

7.9

8

8.1

8.2

2000 2020 2040 2060 2080 2100

D=0.00001

D=000063

D=0.0001

CO

2

mixing ratio (ppmv)

Surface ocean pH

Year

(c)

To mean oceandiffusion (m2/s)

CO

2(g)

mix

ing

rati

o Surface ocean pH

300

400

500

600

700

800

900

7.7

7.8

7.9

8

8.1

8.2

2000 2020 2040 2060 2080 2100

BB=0

BB=500

CO

2

mixing ratio (ppmv)

Surface ocean pH

Year

(d)

To biomass burningemission (Tg-C/yr)

CO

2(g)

mix

ing

rati

o Surface ocean pH

Sensitivity of Future Results

Effect of CO2(g) on Atmospheric Acids

10

-18

10

-16

10

-14

10

-12

10

-10

10

-8

10

-6

2000 2020 2040 2060 2080 2100

Ammonia x (-1)

Nitric acid

Hydrochloric acid

Sulfur dioxide

Mixing ratio (ppbv)

Year of simulation

Diff. between A1B base and today

(d)

Mix

ing

rati

o (p

pbv)

Assumes trace gases initialized but not emitted

Atmospheric NH3 Without and With Ocean Acidification

0

0.2

0.4

0.6

0.8

1

1.2

2000 2020 2040 2060 2080 2100

Current CO2, u=3

Future CO2, u=3

Current CO2, u=8

Future CO2, u=8

Mixing ratio (ppbv)

Year

NH3

Mix

ing

rati

o (p

pbv)

Assumes NH3 initialized and continuously emitted

Air-Sea Exchange Summary• Globally-averaged surface ocean pH may have decreased

from about 8.25 to 8.14 between 1751 and 2004

• Under the SREAS A1B emission scenario, pH may decrease to 7.85 by 2100, for an increase in the hydrogen ion by a factor of 2.5 from1751 to 2100.

• Ocean acidification may slightly increase concentrations of atmospheric acids and more significantly decrease those of bases, thereby affecting cloud and radiative properties and ocean nutrient availability.

Effect of Calcite and AragonitePrecipitation reaction forming calcite/aragonite (20.42)

Formation of solid when (20.43)

Molality of carbonate ion (20.44)

Ca2++CO32-CaCO3(s)

mCa2+mCO32−γ

Ca2+,CO23−

2 >Keq T( )

mCO32− =

cCO2,T

ρsw

K1,CO2K2,CO2

mH+mH++K1,CO2mH+ +K1,CO2

K2,CO2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟