1. CHAPTER 3: SIMPLE MODELS
The atmospheric evolution of a species X is given by the continuity equation
deposition
emission
transport
(flux divergence;
U is wind vector)
local change in concentration
with time
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
Design model; make assumptions needed
to simplify equations and make them solvable
Design observational system to test model
Define problem of interest
Evaluate model with observations
Apply model:
make hypotheses, predictions

2. Questions
1. The Badwater Ultramarathon held every July starts from the bottom of Death Valley (100 m below sea level) and finishes at the top of Mt. Whitney (4300 m above sea level).  This race is a challenge to the human organism! By what percentage does the oxygen number density decrease between the start and the finish of the race?
2. Consider a pollutant emitted in an urban airshed of 100 km dimension. The pollutant can be removed from the airshed by oxidation, precipitation scavenging, or export. The lifetime against oxidation is 1 day. It rains once a week. The wind is 20 km/h. Which is the dominant pathway for removal?

3. 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
Lifetimes add in parallel:
Loss rate constants add in series:

4. NO2 has atmospheric lifetime ~ 1 day: strong gradients away from combustion source regions
Satellite observations of NO2 columns

5. CO has atmospheric lifetime ~ 2 months: mixing around latitude bands
Satellite observations

6. CO2 has atmospheric lifetime ~ 100 years: global mixing
Assimilated observations

7. Using a box model to quantify CO2 sinks
Pg C yr-1
On average, only 60% of emitted CO2 remains in the atmosphere – but there is large interannual variability in this fraction

8. 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
If S, k are constant over t >> t, then dm/dt g 0 and mg S/k: quasi steady state

9. LATITUDINAL GRADIENT OF CO2 , 2000-2012
Illustrates long time scale for interhemispheric exchange;
use 2-box model to constrain CO2 sources/sinks in each hemisphere

10. 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)

11. 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

12. 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

13. 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,

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

15. LAGRANGIAN RESEARCH MODELS FOLLOW LARGE NUMBERS OF INDIVIDUAL “PUFFS”
C(x, to+Dt)
Individual puff trajectories
over time Dt
ADVANTAGE OVER EULERIAN MODELS:
Computational performance (focus puffs on region of interest)
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

16. LAGRANGIAN RECEPTOR-ORIENTED MODELING
Run Lagrangian model backward from receptor location,
with points released at receptor location only
backward in time
Efficient cost-effective quantification of source influence distribution on receptor (“footprint”)