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Max Planck Institutefor Biogeochemistry
Modelling Global Biogeochemical Cycles Part I
Carlos A. Sierra
Max Planck Institute for Biogeochemistry
December 3, 2019
Modelling Global Biogeochemical Cycles Sierra, C.A. 1 / 44
Max Planck Institutefor Biogeochemistry
Assigned reading
Chapter 4: Modeling Biogeochemical Cycles
Modelling Global Biogeochemical Cycles Sierra, C.A. 2 / 44
Max Planck Institutefor Biogeochemistry
Outline
� Mass balance equations
� Matrix representation
� Diagnostic times
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Earth system models
� Movement and transport of energy andmatter across Earth’s main reservoirs
� Use for climate change simulations
� Help us understand connections amongEarth system processes
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Biogeochemical elements follow conservation laws
� Mass cannot be created or destroyed. The change of mass M over time is the resultof inputs minus outputs of mass.
dM
dt= Inputs−Outputs
Modelling Global Biogeochemical Cycles Sierra, C.A. 5 / 44
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Compartmental theory
Definitions:
� Compartment/reservoir/pool: an amount of material defined by certain physical,chemical, or biological characteristics and is kinetically homogeneous. Wecharacterize compartments by their mass.
� Flux: amount of material transferred from one compartment to another per unit oftime, i.e. [mass time−1]
� Rate: relative speed of change of the mass of a reservoir in units of inverse time, i.e.[time−1]
Modelling Global Biogeochemical Cycles Sierra, C.A. 6 / 44
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Steps for modeling biogeochemical cycling
� Define the boundaries and structure of the system through a conceptual diagram
� Define a mathematical model that captures the structure of the conceptual diagram
� Find the best set of parameters of the model
� Manipulate the mathematical model to observe its dynamics over time and computeother interesting diagnostics
Modelling Global Biogeochemical Cycles Sierra, C.A. 7 / 44
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Simple example: radioactive decay
Modelling Global Biogeochemical Cycles Sierra, C.A. 8 / 44
Max Planck Institutefor Biogeochemistry
Conceptual model: radioactive decay
Total number of 14C atoms (N)
Loss by radioac:ve decay
Question: How much mass remains after x units of time?
Modelling Global Biogeochemical Cycles Sierra, C.A. 9 / 44
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Mathematical model
dx
dt= Inputs−Outputs
or
dx
dt= Sources− Sinks
Modelling Global Biogeochemical Cycles Sierra, C.A. 10 / 44
Max Planck Institutefor Biogeochemistry
Conceptual model: radioactive decay
Total number of 14C atoms (N)
Loss by radioac:ve decay
dN
dt= −λ ·N
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Mathematical solution
Initial value problem (IVP)
dN
dt= −λ ·N, N(t = 0) = N0
Modelling Global Biogeochemical Cycles Sierra, C.A. 12 / 44
Max Planck Institutefor Biogeochemistry
Mathematical solution
Initial value problem
dN
dt= −λ ·N, N(t = 0) = N0
Solution:
N(t) = N0 exp(−λ · t)
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Parameterization
Libby half-life for radiocarbon is 5568 years!
t1/2 = 5568 =ln 2
λ,
Then,λ = 0.0001244876 years−1
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Prediction
How much radiocarbon would be available after 10,000 years of radioactive decay if theinitial amount of 14C atoms N0 = 100?
N(t = 10000) = 100 exp(λ · 10000)
= 28.8Modelling Global Biogeochemical Cycles Sierra, C.A. 15 / 44
Max Planck Institutefor Biogeochemistry
The one pool model
x
k x
IThe mass balance equation:
dx
dt= I −O
= I − k · x
� I: External flux from outside thesystem [mass · time−1].
� O = k · x: Flux leaving the system.
� k: rate at which mass leaves the system[time−1].
Modelling Global Biogeochemical Cycles Sierra, C.A. 16 / 44
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Analytical solution of the one pool model
x(t) =I
k−(I
k− x(0)
)e−k·t
With initial condition x(0) = 0 Pg, I = 10Pg yr−1, and k = 0.5 yr−1.
0 5 10 15 20
05
1015
20
Time (year)M
ass
(Pg)
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Effect of the initial conditions
0 5 10 15 20
010
2030
40
Time (year)
Mas
s (P
g)
x0 = 0x0 = 20x0 = 40
Steady-State:At steady-state, inputs are equal to outputs(I = O = k · x), and
x =I
k
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Turnover time
At steady-state, the ratio of the stock and the input or output flux is defined as theturnover time τ
τ =x
I=
x
k · xτ =
1
k
Turnover time is usually interpreted as the time it would take to renew all mass in thereservoir, or the time it would take to fill or empty the reservoir.
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Heterogeneous interconnected systems
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Concept to math: balance equations
dSV1
dt= F0 − F1
dSV2
dt= F1 − F2
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Example: a lake ecosystem
C cycle
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The matrix with fluxes among compartments
F =
−f1,1 f1,2 . . . f1,nf2,1 −f2,2 . . . f2,n
......
. . ....
fn,1 fn,2 . . . −fn,n
Modelling Global Biogeochemical Cycles Sierra, C.A. 23 / 44
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Mathematical formulation for the fluxes
The law of mass action:Reaction Rate = k · [A]α · [B]β
The rate of the reaction is proportional to a power of the concentrations of all substancestaking part in the reaction
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Reaction order
� Zero order: k0
� First order: k1 · [A]
� Second order: k2 · [A] · [B]
� Third order: k3 · [A]2 · [B]
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Max Planck Institutefor Biogeochemistry
First order models are very common
No. Pools 1 2 3
Parallel
Series
Feedback
I
↵C1 C2
dC1
dt= I − k1C1
dC2
dt= αk1C1 − k2C2
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Nomenclature of pool models
No. Pools 1 2 3
Parallel
Series
Feedback
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Max Planck Institutefor Biogeochemistry
General matrix representation
Any model using first-order reaction terms with constant coefficients of the form
dx1dt
= I1 +∑
α1,jkjxj − k1x1
dxidt
= Ii +∑
αi,jkjxj − kixi
dxndt
= In +∑
αn,jkjxj − knxn
can be expressed as a system of the form
dx
dt= I + A · x
Modelling Global Biogeochemical Cycles Sierra, C.A. 28 / 44
Max Planck Institutefor Biogeochemistry
Analytical solution
The systemdx
dt= I + A · x,
with initial conditions x(t = 0) = x0, has solution
x(t) = eA·(t−t0)x0 +
(∫ t
t0
eA·(t−τ)dτ
)I.
At steady-state:
xss = −A−1 · I
Modelling Global Biogeochemical Cycles Sierra, C.A. 29 / 44
Max Planck Institutefor Biogeochemistry
Turnover time for heterogenous systems at steady-state
The ratio of all stocks over all inputs is defined as the turnover time of the system
τ =
∑xss∑I
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Max Planck Institutefor Biogeochemistry
Numerical solutions
Calculate the solution of the IVP taking advantage of the know values of the derivativesand the initial value. For the IVP
x′(t) = f(t, x(t)), x(t0) = x0,
an approximation to the solution is
x(t+ h) ≈ x(t) + hf(t, x(t)).
This leads to a recursion of the form
xn+1 = xn + hf(tn, xn).
Modelling Global Biogeochemical Cycles Sierra, C.A. 31 / 44
Max Planck Institutefor Biogeochemistry
Run multiple-pool models numerically
� Parallel: no exchange of matter amongpools
� Series: Progressive transfer of matteramong pools
� Feedback: Multiple exchange of matteramong pools.
Modelling Global Biogeochemical Cycles Sierra, C.A. 32 / 44
Max Planck Institutefor Biogeochemistry
Mathematical form
� Parallel:
dx(t)
dt= I
γ1γ2
1 − γ1 − γ2
+
−k1 0 00 −k2 00 0 −k3
x1x2x3
� Series:
dx(t)
dt= I
100
+
−k1 0 0a21 −k2 00 a32 −k3
x1x2x3
� Feedback:
dx(t)
dt= I
100
+
−k1 a12 0a21 −k2 a230 a32 −k3
x1x2x3
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Effect of model structure: same inputs and rates
1900 1920 1940 1960 1980 2000
020
0040
0060
0080
00
Years
Mas
s
ParallelSeriesFeedback
Modelling Global Biogeochemical Cycles Sierra, C.A. 34 / 44
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Model properties: Ages and transit timeste: particle enters reservoir
t: now
Age: t-te
Input flux Output flux
� System age is a random variable that describes the age of particles or molecules within a system sincethe time of entry.
� Pool age is a random variable that describes the age of particles or molecules within a pool since thetime of entry.
� Transit time is a random variable that describes the ages of the particles at the time they leave theboundaries of a system; i.e., the ages of the particles in the output flux.
Modelling Global Biogeochemical Cycles Sierra, C.A. 35 / 44
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Age and transit time are not always equal
Bolin & Rodhe (1973, Tellus 25: 58)
� Mean transit time > mean age: humanpopulation
� Mean transit time = mean age: Uranium,Radiocarbon
� Mean transit time < mean age: Oceanwater
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Formulas for transit times and ages
For the autonomous system x(t) = Ax(t) + I
� System ageI f(a) = zT eaA x∗
‖x∗‖
I E[a] = −1T A−1 x∗
‖x∗‖ = ‖B−1 x∗‖‖x∗‖
� Pool ageI f(a) = (X∗)−1 eaA II E[a] = −(X∗)−1 A−1 x∗
� Transit timeI f(τ) = zT eτ A I
‖I‖I E[τ ] = −1T A−1 I
‖I‖ = ‖x∗‖‖I‖
Metzler & Sierra (2018). Mathematical Geosciences 50: 1.
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Max Planck Institutefor Biogeochemistry
Three-pool model example
70 20 10
1600
100
k = 1/16 yr-1
100 500 1000
k = 1/4 yr-1k = 1/25 yr-1 k = 1/100 yr-1
100 500 1000
k = 1/4 yr-1k = 1/25 yr-1 k = 1/100 yr-1
70 3020
20
5
570 30
� One pool model:dx/dt = 100 − (1/16)x
� Three-pool model in parallel:
dx(t)dt
=
702010
+ −(1/4) 0 00 −(1/25) 00 0 −(1/100)
x1x2x3
� Three-pool model in feedback:
dx(t)dt
=
70300
+ −(1/4) 20/(55 · 25) 020/(90 · 4) −(1/25) 1/100
0 5/(55 · 25) −(1/100)
x1x2x3
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Three-pool model example
70 20 10
1600
100
k = 1/16 yr-1
100 500 1000
k = 1/4 yr-1k = 1/25 yr-1 k = 1/100 yr-1
100 500 1000
k = 1/4 yr-1k = 1/25 yr-1 k = 1/100 yr-1
70 3020
20
5
570 30
� One pool model:
I Turnover time: 16 yrs
I Mean Age: 16 yrs
I Mean transit time: 16 yrs
� Three-pool model in parallel:
I Turnover time: 16 yrs
I Mean Age: 63.8 yrs
I Mean transit time: 17.8 yrs
I Mean pool ages: 4, 25, 100 yrs.
� Three-pool model in feedback:
I Turnover time: 16 yrs
I Mean Age: 60.5 yrs
I Mean transit time: 22.35 yrs
I Mean pool ages: 13, 43, 143 yrs.
Modelling Global Biogeochemical Cycles Sierra, C.A. 38 / 44
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Three-pool model example
70 20 10
1600
100
k = 1/16 yr-1
100 500 1000
k = 1/4 yr-1k = 1/25 yr-1 k = 1/100 yr-1
100 500 1000
k = 1/4 yr-1k = 1/25 yr-1 k = 1/100 yr-1
70 3020
20
5
570 30
0 20 40 60 80 100
0.01
0.03
0.05
Age
Den
sity
func
tion
0 20 40 60 80 100
0.00
0.10
Transit time
Den
sity
func
tion Parallel structure
Feedback structure
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Max Planck Institutefor Biogeochemistry
Three-pool model example
Pool age distributions: Parallel and feedback model structures
0 20 40 60 80 100
0.00
0.10
0.20
Age
Pro
babi
lity
dens
ity
0 20 40 60 80 100
0.00
0.02
0.04
Age
Pro
babi
lity
dens
ity
0 20 40 60 80 100
0.00
40.
007
0.01
0
Age
Pro
babi
lity
dens
ity
0 20 40 60 80 100
0.00
0.10
0.20
Age
Pro
babi
lity
dens
ity
0 20 40 60 80 100
0.00
50.
015
0.02
5
Age
Pro
babi
lity
dens
ity
0 20 40 60 80 100
0.00
00.
002
0.00
4
AgeP
roba
bilit
y de
nsity
Modelling Global Biogeochemical Cycles Sierra, C.A. 40 / 44
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A simple carbon cycle model
Atmosphere (600 Pg C)
Surface ocean (2300 Pg C)Terrestrial biosphere (900 Pg C)
70 Pg C yr-1120 Pg C yr-1
55 Pg C yr-1
70 Pg C yr-1120 Pg C yr-1
55 Pg C yr-1
x =
6009002300
, I =
0550
dxAdt
= FAT + FAS − FTA − FSA
dxSdt
= FSD + FSA − FDS − FAS
dxTdt
= FTA − FAT
Modelling Global Biogeochemical Cycles Sierra, C.A. 41 / 44
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System of differential equations
Atmosphere (600 Pg C)
Surface ocean (2300 Pg C)Terrestrial biosphere (900 Pg C)
70 Pg C yr-1120 Pg C yr-1
55 Pg C yr-1
70 Pg C yr-1120 Pg C yr-1
55 Pg C yr-1
dxAdt
= xTkAT + xSkAS − xA(kTA + kSA)
dxSdt
= IS + xAkSA − xS(kDS + kAS)
dxTdt
= xAkTA − xTkAT
Modelling Global Biogeochemical Cycles Sierra, C.A. 42 / 44
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System of differential equations
dxAdt
= xTkAT + xSkAS − xA(kTA + kSA)
dxSdt
= IS + xAkSA − xS(kDS + kAS)
dxTdt
= xAkTA − xTkAT
dxAdtdxSdtdxTdt
=
0Is0
+
−(kTA + kSA) kAS kATkSA −(kDSkAS) 0kTA 0 −kAT
·
xAxSxT
dx
dt= I + A · x
Modelling Global Biogeochemical Cycles Sierra, C.A. 43 / 44
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Summary
� We use compartmental/reservoir/pool models to represent biogeochemical cycling
� Compartmental models are build using information about stocks and fluxes amongreservoirs
� Models can be expressed in matrix form, explicitly representing connections amongpools
� Major model diagnostics are turnover time, system age, and transit time
� For a single homogeneous pool at steady-state, all diagnostic times are equal
� Ages and transit time change according to the connectivity of the system
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