Modelling Global Biogeochemical Cycles Part I · Modelling Global Biogeochemical Cycles Sierra, C.A. 6 / 44. Max Planck Institute for Biogeochemistry Steps for modeling biogeochemical

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


Assigned reading

Chapter 4: Modeling Biogeochemical Cycles



Outline

� Mass balance equations

� Matrix representation

� Diagnostic times



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



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



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]



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



Simple example: radioactive decay



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?



Mathematical model

dx

dt= Inputs−Outputs

or

dx

dt= Sources− Sinks



Conceptual model: radioactive decay

Total number of 14C atoms (N)

Loss by radioac:ve decay

dN

dt= −λ ·N



Mathematical solution

Initial value problem (IVP)

dN

dt= −λ ·N, N(t = 0) = N0



Mathematical solution

Initial value problem

dN

dt= −λ ·N, N(t = 0) = N0

Solution:

N(t) = N0 exp(−λ · t)



Parameterization

Libby half-life for radiocarbon is 5568 years!

t1/2 = 5568 =ln 2

λ,

Then,λ = 0.0001244876 years−1



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


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].



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)



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



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.



Heterogeneous interconnected systems



Concept to math: balance equations

dSV1

dt= F0 − F1

dSV2

dt= F1 − F2



Example: a lake ecosystem

C cycle



The matrix with fluxes among compartments

F =

−f1,1 f1,2 . . . f1,nf2,1 −f2,2 . . . f2,n

......

. . ....

fn,1 fn,2 . . . −fn,n



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



Reaction order

� Zero order: k0

� First order: k1 · [A]

� Second order: k2 · [A] · [B]

� Third order: k3 · [A]2 · [B]



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



Nomenclature of pool models

No. Pools 1 2 3

Parallel

Series

Feedback



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



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



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



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).



Run multiple-pool models numerically

� Parallel: no exchange of matter amongpools

� Series: Progressive transfer of matteramong pools

� Feedback: Multiple exchange of matteramong pools.



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



Effect of model structure: same inputs and rates

1900 1920 1940 1960 1980 2000

020

0040

0060

0080

00

Years

Mas

s

ParallelSeriesFeedback



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.



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



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.



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




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 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 Mean Age: 60.5 yrs

I Mean transit time: 22.35 yrs

I Mean pool ages: 13, 43, 143 yrs.




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




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



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


55 Pg C yr-1

x =

6009002300

, I =

0550

dxAdt

= FAT + FAS − FTA − FSA

dxSdt

= FSD + FSA − FDS − FAS

dxTdt

= FTA − FAT



System of differential equations

Atmosphere (600 Pg C)

Surface ocean (2300 Pg C)Terrestrial biosphere (900 Pg C)


55 Pg C yr-1


55 Pg C yr-1

dxAdt

= xTkAT + xSkAS − xA(kTA + kSA)

dxSdt

= IS + xAkSA − xS(kDS + kAS)

dxTdt

= xAkTA − xTkAT



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



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


Documents

Modelling Global Biogeochemical Cycles Part I · Modelling Global Biogeochemical Cycles Sierra, C.A. 6 / 44. Max Planck Institute for Biogeochemistry Steps for modeling biogeochemical