CHAPTER 3: SIMPLE MODELS
The atmospheric evolution of a species X is given by the continuity equation
[ X ]
 E X    (U[ X ])  PX  LX  DX
t
local change in
concentration
with time
emission
transport
(flux divergence;
U is wind vector)
deposition
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
Apply model:
make hypotheses,
predictions
Evaluate
model with
observations
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?
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
Atmospheric lifetime:  
Fout  L  D Fraction lost by export: f  F  L  D
out
Lifetimes add in parallel:
1
Fout L D
1
1
1

  



m m m  export  chem  dep
Loss rate constants add in series:
k
1

 kexport  kchem  kdep
NO2 has atmospheric lifetime ~ 1 day:
strong gradients away from combustion source regions
Satellite observations of NO2 columns
CO has atmospheric lifetime ~ 2 months:
mixing around latitude bands
Satellite observations
CO2 has atmospheric lifetime ~ 100 years:
global mixing
Assimilated observations
Pg C yr-1
Using a box model to quantify CO2 sinks
On average, only 60% of emitted CO2 remains in the atmosphere – but
there is large interannual variability in this fraction
SPECIAL CASE:
SPECIES WITH CONSTANT SOURCE, 1st ORDER SINK
dm
S
 kt
 S  km  m(t )  m(0)e  (1  e  kt )
dt
k
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 mg
S/k: quasi steady state
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
http://www.esrl.noaa.gov/gmd/ccgg/globalview/
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)
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
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
 EPLD
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,
dC X
 E  P  L  D  kdilution (C X  C X ,b )
dt
COLUMN MODEL FOR TRANSPORT ACROSS
URBAN AIRSHED
Temperature inversion
(defines “mixing depth”)
Emission E
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
ADVANTAGE OVER EULERIAN MODELS:
• Computational performance (focus puffs
on region of interest)
C(x, to)
Concentration field at time t
defined by n puffs
DISADVANTAGES:
• Can’t handle mixing between puffs a
can’t handle nonlinear processes
• Spatial coverage by puffs may be
inadequate
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”)