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SIMPLE MODELS
Defineproblem ofinterest
Design model; makeassumptions neededto simplify equationsand make them solvable
Evaluatemodel withobservations
Apply model:make hypotheses,predictions Improve model, characterize its error
The atmospheric evolution of a species X is given by the
continuity equation
This equation cannot be solved exactly
need to construct
model
(simplified representation of complex system)
Designobservationalsystem to testmodel
[
]
(
[
])
X
X
X
X
X
E
X
P
L
D
t
U
local change inconcentration
with time
transport(flux divergence;U is wind vector)
chemical production and loss(depends on concentrationsof other species)
emission
deposition
CLAUSIUS-CLAPEYRON EQUATION:
P
H2O, SAT
f(T)
2
,
1
1
exp[
(
)]
H
O SAT
o
P
A
B
T
T
A
= 6.11 hPa B
= - 5310 K
T
o
= 273 K
(hPa)^ H2O,SAT P
T
(K)
EXAMPLE: GLOBAL BOX MODEL FOR CO
2
(Pg C yr
IPCC [2001]
IPCC [2001]
SPECIAL CASE:
SPECIES WITH CONSTANT SOURCE, 1
st
ORDER SINK
( )
(0)
(
)
kt
kt
dm
S
S
km
m t
m
e
e
dt
k
Steady statesolution(
dm/dt = 0)
Initial condition
m(0)
Characteristic time
k
for
reaching steady state
decay of initial condition
If
S, k
are constant over
t >>
,^
then
dm/dt
0 and
m
S/k:
quasi steady state
Illustrates long time scale for interhemispheric exchange; can use 2-box modelto place constraints on CO
2
sources/sinks in each hemisphere
LATITUDINAL GRADIENT OF CO
2
EULERIAN
RESEARCH MODELS SOLVE MASS BALANCE
EQUATION IN 3-D ASSEMBLAGE OF GRIDBOXES
Solve continuity equationfor individual gridboxes
Models can presently afford~ 10
6
gridboxes
In global models, this implies a horizontal resolution of 100-500 kmin horizontal and ~ 1 km in vertical •
Drawbacks: “numerical diffusion”, computational expense
The mass balance equation is then the finite-difference approximationof the continuity equation.
PUFF MODEL: FOLLOW AIR PARCEL MOVING WITH WIND C
X
x
o
,^
t
o
C
X
x,
t
wind
In the moving puff,
X
dC
E
P
L
D
dt
…no transport terms! (they’re implicit in the trajectory)
Application to the chemical evolution of an isolated pollution plume:
C
X
C
X,b
,
(
)
X
dilution
X
X b
dC
E
P
L
D
k
C
C
dt
In pollution plume,
COLUMN MODEL FOR TRANSPORT ACROSS
URBAN AIRSHED
Temperature inversion(defines “mixing depth”)
Emission
E
In column moving across city,
X
X
dC
E
k
C
dx
Uh
U
C
X
L
x docsity.com
LAGRANGIAN RECEPTOR-ORIENTED MODELING
Run Lagrangian model backward from receptor location,with points released at receptor location only
Efficient cost-effective quantification of source influence distribution on receptor (“footprint”)
backward in time
