QUESTIONS
1.
The definition of "1 atmosphere" is 1013 hPa, the average atmospheric pressure at sea level. But when we computed the mass of the atmosphere, we used a mean atmospheric pressure of
984 hPa. Why?
2.
Torricelli used mercury for his barometer, but a column barometer could be constructed using any fluid, with the height of the column measuring atmospheric pressure given by h = P/ r fluid g
At sea level, with mercury r fluid
= 13.6 g cm -3 e h = 76 cm with water r fluid
= 1.0 g cm -3 e h = 1013 cm
Now what about using air as a fluid?
with air r fluid
= 1.2 kg m -3 e h = 8.6 km which means that the atmosphere should extend only to 8.6 km, with vacuum above! WHAT IS THE FLAW IN THIS REASONING?

CHAPTER 3: SIMPLE MODELS
The atmospheric evolution of a species X is given by the continuity equation

 t

E
X
  
U

P
X

L
X

D
X deposition local change in concentration with time emission transport
(flux divergence;
U is wind vector) chemical production and loss
(depends on concentrations of other species)
This equation cannot be solved exactly e need to construct model
(simplified representation of complex system)
Improve model, characterize its error
Define problem of interest
Design model; make assumptions needed to simplify equations and make them solvable
Design observational system to test model
Evaluate model with observations
Apply model: make hypotheses, predictions

TYPES OF SOURCES
Natural Surface: terrestrial and marine highly variable in space and time, influenced by season, T, pH, nutrients… eg. oceanic sources estimated by measuring local supersaturation in water and using a model for gasexchange across interface =f(T, wind velocity….)
Natural In situ: eg. lightning (NO x
) N
2
 NO x
, volcanoes (SO
2
, aerosols)
 generally smaller than surface sources on global scale but important b/c material is injected into middle/upper troposphere where lifetimes are longer
Anthropogenic Surface: eg. mobile, industry, fires
 good inventories for combustion products (CO, NO x
, SO
2
) for US and EU
Anthropogenic In situ: eg. aircraft, tall stacks
Secondary sources: tropospheric photochemistry
Injection from the stratosphere : transport of products of UV dissociation (NO x
, O
3
) transported into troposphere (strongest at midlatitudes, important source of NO x in the
UT)

TYPES OF SINKS
Wet Deposition: falling hydrometeors (rain, snow, sleet) carry trace species to the surface
• in-cloud nucleation (depending on solubility)
• scavenging (depends on size, chemical composition)
 Soluble and reactive trace gases are more readily removed
 Generally assume that depletion is proportional to the conc (1 st order loss)
Dry Deposition : gravitational settling; turbulent transport particles > 20 µm  gravity (sedimentation) particles < 1 µm  diffusion
 rates depend on reactivity of gas, turbulent transport, stomatal resistance and together define a deposition velocity (v d
)
F d
 v C d x
Typical values v d
:
Particles:0.1-1 cm/s
In situ removal: chain-terminating rxn: OH ● +HO
2 change of phase: SO
2
 SO
4
2-
●  H
2
O + O
2
(gas  dissolved salt)
Gases: vary with srf and chemical nature
(eg. 1 cm/s for SO
2
)

RESISTANCE MODEL FOR DRY DEPOSITION
Deposition Flux: F d
= -v d
C V d
= deposition velocity (m/s)
C = concentration
Use a resistance analogy, where r
T
=v d
-1
C
3
C
2
C
1
Aerodynamic resistance
= r a
Quasi-laminar layer resistance = r b
Canopy resistance = r c
For gases at steady state can relate overall flux to the concentration differences and resistances across the layers:
F

C
3
 r a
C
2

C
2
 r b
C
1

C
1

C
0
0

C
3

C
0 r c r
T
0

C
3

(Fr b

C )
 r a
C
3

(Fr b

Fr ) c r a
C
0
=0
F

C
3 r a
  b r c v d

1    a b r c
For particles , assume that canopy resistance is zero (so now
C
1
=0), and need to include particle settling (settling velocity=v s
) which operates in parallel with existing resistances. End result: v d

1 r a
  b r r v a b s
 v s
Reference: Seinfeld & Pandis, Chap 19

Inflow F in
ONE-BOX MODEL
Chemical production
Chemical loss
Atmospheric “box”; spatial distribution of X within box is not resolved
Outflow F out
P X L
D
E
Flux units usually
[mass/time/area]
Deposition
Emission
Mass balance equation:
Atmospheric lifetime:
(turnover time)
 dm

  sources dt m
F out
 
D
 sinks

F in
  
F out
 
D
Fraction lost by export: f

F out
F out
 
Lifetimes add in parallel:
(because fluxes add linearly)

1

F out

L

D m m m

1
 export
Loss rate constants add in series: k
 k export

1
 chem
 k chem
 k dep

1
 dep

ASIDE: LIFETIME VS RADIOACTIVE HALF-LIFE
Both express characteristic times of decay, what is the relationship?
½ life:
[ ]( )

[ ](0) e
 kt
A t
1/ 2
)

1
2
A t
1/ 2
 ln(2) k

0.7


EXAMPLE: GLOBAL BOX MODEL FOR CO
2 reservoirs in PgC, flows in Pg C yr -1 atmospheric content (mid 80s)
= 730 Pg C of CO
2
(now ~816 PgCO annual exchange land = 120 Pg C yr -1
2
) annual exchange ocean= 90 Pg C yr-1
Human Perturbation
IPCC [2001]

SPECIAL CASE:
SPECIES WITH CONSTANT SOURCE, 1 st ORDER SINK dm dt
S km

( )
 m (0) e
 kt 
S
(1
 k e
 kt
)
Steady state solution
( dm/dt = 0)
Initial condition m(0)
Characteristic time

= 1/ k for
• reaching steady state
• decay of initial condition
If S, k are constant over t >>

, then dm/dt g
0 and m g
S/k: " steady state"

TWO-BOX MODEL defines spatial gradient between two domains
F
12 m
1 m
2
F
21
Mass balance equations: dm
1 dt

E
1
  
1 1
D
1

F
12

F
21
(similar equation for dm
2
/dt )
If mass exchange between boxes is first-order: dm
1 dt

E
1
  
1 1
D
1
 k m
12 1
 k m
21 2 e system of two coupled ODEs (or algebraic equations if system is assumed to be at steady state)

Illustrates long time scale for interhemispheric exchange; can use 2-box model to place constraints on sources/sinks in each hemisphere

Q m
1
TWO-BOX MODEL
(with loss)
T m
2 m o
= m
1
+m
2
Lifetimes:
S1

1
 m
1
T

S
1

2
 m
2
S
2

0

S2 m
1
 m
2
S
1

S
2
If at steady state sinks=sources, so can also write:

1
 m
Q
1

2
 m
T
2

0
 m
1

Q m
2
Now if define: α=T/Q, then can say that:
   o 2
Maximum α is 1 (all material from reservoir 1 is transferred to reservoir
2), and therefore turnover time for combined reservoir is the sum of turnover times for individual reservoirs. For other values of
α, the turnover time of the combined reservoir is reduced.

EULERIAN RESEARCH MODELS SOLVE MASS BALANCE
EQUATION IN 3-D ASSEMBLAGE OF GRIDBOXES
The mass balance equation is then the finite-difference approximation of the continuity equation.
Solve continuity equation for individual gridboxes
• Models can presently afford
~ 10 6 gridboxes
• In global models, this implies a horizontal resolution of 100-500 km in horizontal and ~ 1 km in vertical
• Drawbacks: “numerical diffusion”, computational expense

EULERIAN MODEL EXAMPLE
Summertime Surface Ozone Simulation
Here the continuity equation is solved for each 2
 x2.5
 grid box.
They are inherently assumed to be well-mixed
[Fiore et al., 2002]

IN EULERIAN APPROACH, DESCRIBING THE
EVOLUTION OF A POLLUTION PLUME REQUIRES
A LARGE NUMBER OF GRIDBOXES
Fire plumes over southern California,
25 Oct. 2003
A Lagrangian “puff” model offers a much simpler alternative

PUFF MODEL: FOLLOW AIR PARCEL MOVING WITH WIND
[X](x o
, t o
) wind
[X](x, t)
In the moving puff,
E P L D dt
…no transport terms! (they’re implicit in the trajectory)
Application to the chemical evolution of an isolated pollution plume:
[X] b
[X]
In pollution plume, dt
E P L D k dilution
([ ] [ ] ) b

COLUMN MODEL FOR TRANSPORT ACROSS
URBAN AIRSHED
Temperature inversion
(defines “mixing depth”)
[ ]

E dt h



U dx dt dx
Emission E
In column moving across city,
Solution:
[X]
[ ]( )

0
0
[ ]

E
 k dx Uh U
[ ]( )

[ ]( )

(

)/

E hk

1
 e


L x

LAGRANGIAN RESEARCH MODELS FOLLOW
LARGE NUMBERS OF INDIVIDUAL “PUFFS”
C( x , t o
D t

C( x , t o
)
Concentration field at time t defined by n puffs
Individual puff trajectories over time
D t
ADVANTAGES OVER EULERIAN MODELS:
• Computational performance (focus puffs on region of interest)
• No numerical diffusion
DISADVANTAGES:
• Can’t handle mixing between puffs a can’t handle nonlinear processes
• Spatial coverage by puffs may be inadequate

FLEXPART: A LAGRANGIAN MODEL
Retroplume (20 days):
Trinidad Head, Bermuda x
Emissions Map (NO x
)
=
Region of Influence
But no chemistry, deposition, convection here [Cooper et al., 2005]