1. X ONE-BOX MODEL Atmospheric “box”;
spatial distribution of X within box is not resolved
Chemical
production
Chemical
loss
Inflow Fin
Outflow Fout P X L D E
Deposition
Emission
be explicit with the units here !! E = mass emission rate, versus flux (mass/area/s)
Lifetimes add in parallel:
Loss rate constants add in series:

2. EXAMPLE: GLOBAL BOX MODEL FOR CO2 (Pg C yr-1)
IPCC [2001]
IPCC [2001]

3. SPECIAL CASE: SPECIES WITH CONSTANT SOURCE, 1st ORDER SINK
Steady state solution (dm/dt = 0)
Initial condition m(0)
Characteristic time t = 1/k for
reaching steady state
decay of initial condition

4. TWO-BOX MODEL defines spatial gradient between two domains
Mass balance equations:
(similar equation for dm2/dt)
If mass exchange between boxes is first-order:
e system of two coupled ODEs (or algebraic equations if system is assumed to be at steady state)

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

6. 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 km in horizontal and ~ 1 km in vertical
Drawbacks: “numerical diffusion”, computational expense

7. 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

8. PUFF MODEL: FOLLOW AIR PARCEL MOVING WITH WIND
CX(x, t)
In the moving puff,
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,

9. COLUMN MODEL FOR TRANSPORT ACROSS URBAN AIRSHED
Temperature inversion
(defines “mixing depth”)
Emission E
Udt = dx
In column moving across city, CX L x

10. LAGRANGIAN RESEARCH MODELS FOLLOW LARGE NUMBERS OF INDIVIDUAL “PUFFS”
C(x, to+Dt)
Individual puff trajectories
over time Dt
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
C(x, to)
Concentration field at time t defined by n puffs

11. SHORT QUESTIONS
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 ( 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?