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```ONE-BOX MODEL
Chemical
production
Inflow Fin
P
X
Atmospheric “box”;
spatial distribution of X
Chemical within box is not resolved
loss
Outflow Fout
L
D
E
Deposition
Emission
dm
Mass balance equation:
  sources - sinks  Fin  E  P  Fout  L  D
dt
m
Fout
Fout  L  D Fraction lost by export: f  F  L  D
out
1
Fout L D
1
1
1

  



m m m  export  chem  dep
Loss rate constants add in series:
k
1

 kexport  kchem  kdep
EXAMPLE: GLOBAL BOX MODEL FOR CO2 (Pg C yr-1)
IPCC [2001]
IPCC [2001]
SPECIAL CASE:
SPECIES WITH CONSTANT SOURCE, 1st ORDER SINK
dm
S
 kt
 S  km  m(t )  m(0)e  (1  e  kt )
dt
k
solution
(dm/dt = 0)
Initial condition m(0)
Characteristic time  = 1/k for
• decay of initial condition
TWO-BOX MODEL
defines spatial gradient between two domains
F12
m2
m1
F21
Mass balance equations:
dm1
 E1  P1  L1  D1  F12  F21
dt
(similar equation for dm2/dt)
If mass exchange between boxes is first-order:
dm1
 E1  P1  L1  D1  k12m1  k21m2
dt
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
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
~ 106 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
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
CX(x, t)
In the moving puff,
dC x
 PL
dt
wind
CX(xo, to)
…no transport terms! (they’re implicit in the trajectory)
Application to the chemical evolution of an isolated pollution plume:
CX,b
CX
In pollution plume,
dCx
 P  L  k dilution (C x  C x ,b )
dt
COLUMN MODEL FOR TRANSPORT ACROSS
URBAN AIRSHED
Temperature inversion
(defines “mixing depth”)
Emission E
Udt = dx
In column moving across city,
dC X
E k

 CX
dx
Uh U
CX
0
L
x
LAGRANGIAN RESEARCH MODELS FOLLOW
LARGE NUMBERS OF INDIVIDUAL “PUFFS”
C(x, toDt)
Individual puff trajectories
over time Dt
• Computational performance (focus puffs
on region of interest)
• No numerical diffusion
C(x, to)
Concentration field at time t
defined by n puffs
• Can’t handle mixing between puffs a
can’t handle nonlinear processes
• Spatial coverage by puffs may be
SHORT QUESTIONS
1.
2.
3.
•
•
•
The Montreal Protocol has banned worldwide production of CFC-12.
CFC-12 is removed from the atmosphere by photolysis with a lifetime
of 100 years. Assuming compliance with the Protocol, and
neglecting residual emissions from existing stocks, how long will it
take for CFC-12 concentrations to drop to half of present-day values?
Consider a 2-box model for the atmosphere where one box is the
troposphere (1000-150 hPa) and the other is the stratosphere (150-1
hPa). Assume a 2-year residence time for air in the
stratosphere. What is the corresponding residence time of air in the
troposphere?
Of the simple models presented in chapter 3, which one would be
most appropriate for answering the following questions:
Will atmospheric releases of a new industrial gas harm the
stratospheric ozone layer?
What areas will be affected by the radioactive plume from a nuclear
accident?
An air pollution monitoring site suddenly detects high
concentrations of a toxic gas. Where is this gas coming from?
```