Master Course „Environmental Physics“ (MKEP4)
http://www.iup.uni-heidelberg.de/institut/studium/lehre/MKEP4/
19. Model Concepts, Box Models,
Lumped-Parameter Models
Summer Term 2011
Werner Aeschbach-Hertig
Institut für Umweltphysik
Lecture Program of MKEP4
Part 4: Methods and Models (6 sessions)
16.
17.
18
18.
19.
20.
21.
Fundamentals of isotope methods
Isotope methods in environmental physics
Measurement methods of environmental physics
Model concepts, box models, lumped-parameter models
Complex models and numerical methods
General circulation models and inversion techniques
Part 5: Climate (5 sessions)
22.
23.
24
24.
25.
26.
The carbon cycle
Radiative transfer
Aerosols and clouds
Climate sensitivity, feedbacks, and predictions
Climate variability and palaeoclimate reconstruction
Last week: Written Exam (19.7.2011)
2
1
Contents of Today's Lecture
Model concepts, box models, lumped-parameter models
• Types of models, motivation for models
• Environmental
E i
l system analysis:
l i B
Box models
d l
• Linear 1-box model
• Linear 1-box model: Variable input
• Lumped-parameter models, transit time distribution
3
Literature on Modeling / System Analysis
Box Models, System Analysis
• Imboden, D.M., and Koch, S., 2003: Systemanalyse. Springer. IUP 1876
• Schwarzenbach, R.P., Gschwend, P.M., Imboden, D.M., 2003:
Environmental organic chemistry. Wiley. IUP 1865
• Gruber, N., 2010. Lecture "Systemanalyse" at ETH Zurich, see
http://www.up.ethz.ch/education/system_analysis/index_DE
Lumped Parameter Models, Groundwater Modeling
• Mook, W.G. (ed.), 2001: UNESCO/IAEA Series on Environmental
Isotopes in the Hydrological Cycle. Vol. 6. available at http://wwwnaweb.iaea.org/napc/ih/IHS_resources_publication_hydroCycle_en.html
Climate Modeling
• Stocker, T. 2009. Introduction to Climate Modelling. Lecture notes,
University of Bern. See http://www.climate.unibe.ch/main/courses/
klimamodellierung_hs09/stocker09climmod.pdf
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2
Environmental Systems
•
•
•
•
Many parts of the environment can be understood as (complex) systems
System: Boundary, internal relations, and external relations
Complexity calls for description by models
Only internal processes are modeled (def
(def. of system/model boundary)
external forcing
(input)
x
no external
feedback
(output)
IInfluence
fl
on
environment
(internal)
5
Types of Models
• Physical or scale model – physical representation of the studied system
(at smaller scale)
• Conceptual model – simplified, abstract description of a system, defining
its
ts boundaries,
bou da es, internal
te a a
and
de
external
te a relations,
e at o s, etc
etc.
• Mathematical model – a conceptual model translated into mathematical
equations
• Numerical (computer) model – numerical solution of mathematical model
• Compare with a map – depiction of nature:
1. flat
2. miniaturized
3. simplified
4. explained
– you can not use the same map for all purposes (e.g. scale), same is
true for models: pick the appropriate one for the problem at hand
6
3
A Physical Model
• physical model - wind tunnel model of flow in a city
Univ. Hamburg
In the following: Mathematical / numerical models …
7
Benefits of Models
•
•
•
•
•
•
•
•
Models summarise our understanding of processes in an
environmental system
Model – data comparisons: test our current understanding
off the
th processes involved,
i
l d ttestt new ideas
id
Upscaling from local experiments  bigger picture
Models provide 4D info which experiments can never do
Analysis of complex system behaviour
Prediction (weather, climate, contaminant plumes,…)
“what
what ifif..” scenarios/numerical experiments:
• switching off different processes
• future/past scenarios
• 2xCO2, 4xCO2, more/less aerosol, ..
…
8
4
Limitations of Models
•
•
•
•
•
•
•
•
Models are never a perfect description of reality
Models can only represent current knowledge
Not all processes can be included
Resolution in time and space is limited (computing time)
Approximations are necessary
Initial and boundary conditions are crucial but often not
available as needed – not enough data to use the model
as exact depiction of the environment
Sometimes: good results for the wrong reasons
Models are only as good as modeler / user –
“garbage in, garbage out”
9
Different Model Structures
Dynamic models of environmental systems
Describing temporal evolution of system variables yi(t)
1. Models without detailed spatial structure
Box models ("0-dimensional")
•
•
•
•
1-box model (1 ore more variables, linear and non-linear)
n-box models (n coupled 1-box models)
Lumped parameter models (pseudo-structure: TTDs)
Description by ordinary differential equations (ODEs)
2. Models with continuous spatial extension
•
•
•
•
•
1-dimensional (e.g. vertical models of atmosphere, lakes)
2-dimensional (e.g. horizontal layers, vertical sections)
3-dimensional (full spatial extension of studied systems)
Description by partial differential equations (PDEs)
Numerical solution on discrete grid ( link to box-models)
10
5
0-D: “Box Model”
Independent variable:
time
Example:
Concentration c(t) of a
chemical in a wellmixed air volume
Graedel and Crutzen
11
1-D: Vertically Resolved Profile
Independent variables:
time, altitude
Example:
Concentration profile
c(z,t) of a chemical in
a column of air
Graedel and Crutzen
12
6
2-D: Horizontal Groundwater Flow
Independent variables:
time, E and N coordinate
Example:
Horizontal flow field
v(x,y,t) of groundwater
with transport:
concentration c(x,y,t)
Wollschläger et al., Aquat. Sci. Vol. 69, 2007
13
3D: General Circulation Models
Independent variables:
time, altitude, E and N
Example:
Global distribution of
system variables
y(x,y,z,t) in the
atmosphere and ocean
http://www.bom.gov.au/info/GreenhouseEffectAndClimateChange.pdf
14
7
Why use Box Models?
Does it make sense to use simple box models, when 3-D
models are obviously more complete descriptions?
Yes! Simple models have advantages:
• Th
They provide
id iinsights
i ht iinto
t th
the characteristic
h
t i ti b
behaviour
h i
off a system
t
• They can often be solved analytically
• They are adequate if few data are available
Some problems that can be addressed with box models
•
•
•
•
•
General: Input and reaction of substances in simple compartments
Chemicals in lakes (contaminants, nutrients, oxygen etc.)
Nitrate in groundwater (residence time, concentration in wells)
Accumulation of anthropogenic trace gases in the atmosphere
Carbon cycle: CO2 in atmosphere, ocean, biosphere
15
3-Box Model of the Carbon Cycle
Fossil fuel C is
released into the
atmosphere (box 1).
The atmosphere
Th
t
h
exchanges with the
ocean (box 2) and the
biosphere (box 3).
System analytical
question:
The CO2 level in air
shall be stabilised at
8% above the present.
How do the emissions
have to change?
Gruber, 2010
16
8
One Box or Mixed Reactor Model
Simple model for a substance in an environmental system:
Fully mixed reservoir with input, output, and 1st order reaction
Mass balance:
Q,cin
V
M, c
Q,c
mixed
V: System volume [L3]
Q: Flow rate [L3T-1]
kf = Q/V: Flushing rate [T-1]
M: Mass of substance [M]
c = M/V: Concentration [ML-3]
cin: Input concentration [ML-3]
kr: Reaction rate constant [T-1]
dM
 Qcin  Qc  krM
dt
dc
 k f cin   k f  kr  c  k f cin  k totc
dt
1st order linear inhomogeneous ODE
General solution for constant coefficients (cin
const ) and c(0) = c0:
i , kf, kr = const.)
c  t    c0  c   ektot t  c 
with the stationary concentration:
c 
kf
kf
cin 
cin
k tot
k f  kr
17
One Box or Mixed Reactor Model
Generalisation: Input and output do not have to be tied to
volume fluxes of a fluid. Mass in/output per time Jin, Jout.
Mass balance:
dM
 Jin  Jout  kM  J  kM
dt
J  Jin  Jout
dc J
  kc
dt V
1st order linear inhomogeneous ODE
General solution for constant coefficients (J
(J, k = const.)
const ) and c(0) = c0:
V: System
V
S t
volume
l
[L3]
M: Mass of substance [M]
c  t    c0  c   ekt  c 
-3
c = M/V: Concentration [ML ]
with the stationary concentration:
Jin: Input of substance [MT-1]
Jout: Output of substance [MT-1]
J
j
J

c 
with j  ML3T1 
k: Total 1st order removal rate [T-1]
kV k
V
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9
One Box Model: Temporal Evolution
Discussion of solution for constant input

c  t    c0  c   ektot t  c   c0ektot t  c  1 ektot t
Concentrattion
dilution/decay of c0

ingrowth of c∞
Ingrowth
g
to C∞
Washing out of C0
Gruber, 2010
19
One Box Model: Equilibration Time
How quickly does the system reach its steady state?
Depends on how small the deviation shall be.
c     c     c    c0  c 
 
 ln 
k tot
E.g.:
1/ 2 
 ektot   
ln2
k tot
1/ e 
1
k tot
0.05 
3
k tot
e-folding time, system time constant
Steady state
concentration
Initial disturbance
Equilibration time
Initial
concentration
Gruber, 2010
20
10
One Box Model with Variable Input
Generalisation: External input (forcing) can depend on time
Mass balance:
f
dc
 k f cin  t   k totc
dt
f
1st order linear inhomogeneous ODE
with time-variable inhomogeneous term
General solution for constant rate coefficients (kf, kr = const.) and c(0) = c0:
t
c  t   c 0 e k tot t   k f c in  t   e
V: System volume [L3]
kf = Q/V: Flushing rate [T-1]
c(t): Concentration [ML-3]
cin(t): Input concentration [ML-3]
kr: reaction rate constant [T-1]
 k tot  t  t 
dt
0
decay of c0
ingrowth to c∞(t)
Time-dependent "stationary state":
c  t  
kf
cin  t 
k tot
Equilibrium conc. at
momentary input
21
One Box Model: Temporal Evolution (2)
Discussion of solution for variable input
"Ingrowth term"
t
 k c  t   e
f
in
 k tot  t  t  
dt
0
past inputs weighted according to age t - t'
Superposition of decay
curves from a series of
"input
p events".
Gruber, 2010
22
11
Variable Input: Exponential Forcing
Exponentially increasing input c in  t   c in0  et
t
c  t   c 0 e k tot t   k f c in0  et  e
 k tot  t  t 
dt
0
c  t   c 0 e k tot t 
k f c in0
et  e k tot t
k tot  


for  ≠ -ktot
fast growing forcing:  ≈ ktot
slow (adiabatic) forcing:  << ktot
C∞(t)
t
System follows the forcing
Gruber, 2010
t
System lags behind the forcing
23
Fast Exponential Forcing
System with fast exponentially growing forcing:
What happens, if input is stabilised? (e.g.: keep CO2 emissions constant)
Answer: Option 4
After tstab we have the
case with constant
input  exponential
ingrowth to equilibrium
Stabilising emissions
does not stabilise
system concentration
immediately
Gruber, 2010
24
12
Variable Input: Periodic Forcing
Periodically oscillating input
c in  t   c in0  c1in  sin  t 
t
 k t  t
c  t   c 0 e k tot t   k f c in0  c1in  sin  t    e tot   dt
0

kfc
k c0 
k f c1in
k c1 
  c 0  f in  ek tot t 
sin  t     2f in 2 e k tot t
  k tot
k tot 
k tot 
2  k 2tot
0
in
c t 
  
  arctan 

 k tot 
terms remaining for ktott >> 1
c
 ≈ ktot
 << ktot
 >> ktot
C(t)
C∞(t)
t
25
Gruber, 2010
1-Box Model as Lumped Parameter Model
Re-Interpretation of the box model with variable input:
Not a mixed reactor but a system with transit time distribution
Q
General solution for system starting at
t = 0 and e
evolving
ol ing to time tt:
cin((t))
t
V
cout(t)
 
c  t   c 0 e k tot t   k f c in t *  e
  dt *
 k tot t  t *
0
t*: forward running time between 0 and t
t' = t - t*: backward running "age" before t
t
V: System volume [L3]
Q: Flow rate [L3T-1]
cin(t): Input concentration [ML-3]
cout(t): Output concentration [ML-3]
kf = Q/V: Flushing rate [T-1]
 = 1/kf: Mean transit time [T]
c  t   c 0 e k tot t   k f c ini  t  t   ek tot tdt
0
Put system start to t = -∞ or maximum
"age" t' to ∞ and use  = 1/kf:

1
c  t    c in  t  t   e kr t  e  t  dt

0
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13
Lumped Parameter Models: Interpretation
Discussion of generalised lumped parameter model equation

c  t    c in  t  t   e t  g  t  dt
0
input of age t't before present
weight of input at t't , tranist time distribution (TTD)
decay for radioactive tracer
g(t'): Transfer or response function, TTD
Special case: Exponential model (= mixed reactor)
Input
1
g  t   e  t 

Environ. System = Black Box
Output
27
Lumped Parameter Models: Mathematics
• Time axis: t = (present) time, t' = time span before t, age
• Input in the past: c in  t  t 
• Transit time distribution (TTD) g  t  with

 g  t dt  1
0
Depends on mean transit time  + other model parameters

• Mean age or transit time:
   t  g  t  dt
0
• Output (convolution of input and transfer function + decay):

c out  t    c in  t  t   e t  g  t  dt
0
28
14
Piston-flow Model (PM)
• Transfer function: g  t     t   
– Parameter : Age

• Output: c out  t    c in  t  t   e t    t    dt  c in  t     e 
0
• Picks out the input of one specific age
• Describes closed system, no mixing
• Piston-flow age  = apparent tracer age
29
c out  t   c in  t     e 
30
15
Exponential Model (EM)

1 t
g  t   e 

– Parameter : Mean age
• Transfer function:

• Output:

1 t
c out  t    c in  t  t   e t  e  dt

0
• Weight of input events decreases exponentially with age
– recent input (t' << ): highest contribution to output
– very old input (t
(t' >> ): very small contribution to output
• Describes completely mixed system (mixed reactor)
• Model age  = mean residence time in mixed system
31
32
16
Dispersion Model (DM)
g  t  
• Transfer function:
– Parameter : Mean age


e
3
4t
 t  2
4 t 
– Parameter : Dispersion

• Output:
c out  t    c in  t  t   e
0
t 



e
3
4t
 t 2
4 t 
dt
• Weight of input events: ~ Gauss-type distribution
– highest
g
contribution to output
p for certain input
p age
g t'
– decreasing contribution to output for younger/older input
• Describes throughflow system with limited mixing
(dispersion)
• Model age  = mean transit time
33
34
17
Transit Time Distributions
tran
nsfer function g(t')
0.4
Exponential Model (EM)
0.35
0.3
=3
 = 10
 = 30
0.25
0.2
0.15
0.1
0.05
0
0
10
20
30
40
50
t' [yr]
0.4
0.4
Dispersion Model (DM),  = 1
transfer functio
on g(t')
=3
 = 10
 = 30
0.25
Dispersion Model (DM),  = 0.1
0.35
0.3
0.2
0.15
0.1
0.3
=3
 = 10
 = 30
0.25
0.2
0.15
0.1
0.05
0.05
0
0
0
10
20
30
40
0
50
10
20
30
40
t' [yr]
t' [yr]
50
35
Example: Tritium Output of Expon. Model
1000
Tritium [TU]
transfer functio
on g(t')
0.35
100
input
decay
exp modd  = 3
exp mod  = 7
exp mod  = 15
10
1952
1960
1968
1976
1984
1992
2000
Year
36
18
Example: Tritium Time Series in Rivers
Mixing of fast and slow runoff components in a river
from Mook, 2001
37
Tritium Time Series from a Spring
80
70
=3
=6
Input
Samples
 = 10
Tritium [TU]
60
50
40
decay
30
20
10
0
1980
1984
1988
1992
1996
Year
38
19
Summary
• Models are simplified representations of complex systems.
Different types of models:
 Physical, conceptual, mathematical, numerical models
• Numerical models can be 0-D (box models), 1-D, 2-D, 3-D
• Box models can help to analyse system characteristics
• 1-Box or mixed reactor model:
 1st order linear inhomogeneous ODE
 Exponential approach to stationary concentration (equilibrium)
 System time constant  = 1/ktot determines reaction to forcing
• Lumped-parameter models/ Transit-time distributions (TTD)
 Input – output description of (tracers in) environmental systems
 Internal dynamics described by TTD with few "lumped" parameters
 TTDs can be derived from conservative tracers
39
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