Mathematical Modelling, 5th

Projects:Carbon cycle in a box modelBoat dynamics: OscillationsJ S. Miller. Physics in a Toy Boat. Am. J. Physics 26, 199 (1958) pop-pop boat

Advanced Population dynamicsClimate Oscillator

Immigrant dynamicsBifurcation TheoryPartial diffential equations

Boat dynamics: OscillationsPhysics in a Toy Boat. pop-pop boat

Climate Oscillator

Atmosphere 725(Annual increase ~3)

Surface waterDissolved inorg. 700

Dissolved org. 25(Annual increase ~ 0,3)

Surface biota3

Intermediate andDeep water

Dissolved inorg. 36,700Dissolved org. 975

(Annual increase ~ 2,5)

Short-lived biota~110

Long-lived biota ~450(Annual decrease ~1)

Litter~60

Soil 1300 - 1400(Annual decrease ~1)

Peat (Torf)~160

Fossil fuelsoil, coal, gas

5,000 - 10,000

Respiration &decomposition

~36

Primaryproduction

~40

Detritus~4

Detritus decomposition

54-50

~40 ~38

5

2 - 5

2 - 5

~15~40

~120~60~90~93Deforestation

~1

‹1

‹1

~15~1

Fig. 4-3 principal reservoirs and fluxes in the carbon cycle. Units are 1015 g(Pg) C (burdens)and PgC/yr (fluxes). (From Bolin (1986) with permission from John Wiley and Sons.)

Carbon Cycle

Turnover Time, renewal time

M content if a substance in the reservoir

S total flux out of the reservoir

MS=kMQ

single reservoir with source flux Q, sink flux S, and content M

The equation describing the rate of change of the content of a reservoir can be written as

Linear System: The adjustment process is

e-folding time

Atmosphere 725(Annual increase ~3)

Surface waterDissolved inorg. 700

Dissolved org. 25(Annual increase ~ 0,3)

Surface biota3

Intermediate andDeep water

Dissolved inorg. 36,700Dissolved org. 975

(Annual increase ~ 2,5)

Short-lived biota~110

Long-lived biota ~450(Annual decrease ~1)

Litter~60

Soil 1300 - 1400(Annual decrease ~1)

Peat (Torf)~160

Fossil fuelsoil, coal, gas

5,000 - 10,000

Respiration &decomposition

~36

Primaryproduction

~40

Detritus~4

Detritus decomposition

54-50

~40 ~38

5

2 - 5

2 - 5

~15~40

~120~60~90~93Deforestation

~1

‹1

‹1

~15~1

Fig. 4-3 principal reservoirs and fluxes in the carbon cycle. Units are 1015 g(Pg) C (burdens)and PgC/yr (fluxes). (From Bolin (1986) with permission from John Wiley and Sons.)

The flux Fij from reservoir i to reservoir j is given by

The rate of change of the amount Mi in reservoir i is thus

where n is the total number of reservoirs in the system. This system of differential equationscan be written in matrix form as

where the vector M is equal to (M1, M2,... Mn) and the elements of matrix k are linear combinationsof the coefficients kij

Master Equation,

Statistical Physics

?

Simplified model of the carbon cycle. Ms represents the sum of all forms ofdissolved carbon , , and

CO2

H 2 HCO3

HCO3

,

CO 22

Atmosphere

M A

Terrestrial System

M T

Ocean surfaceDiss C= CO2,HCO3,H2CO3

M S

Deep layers of ocean

M D

F TA

F AT

F SA F AS

F SDF DS

Non-linear System: Simplified model of the biogeochemical carbon cycle. (Adapted from Rodhe and Björkström (1979) with the permission of the Swedish Geophysical Society.)

Inorganic Carbon Cycle

Free protonBicarbonate carbonate

Non-linearity in the oceanic carbon system

Carbonate acid

hydrated

Ocean: inorganic Carbon Cycle

Simplified model of the carbon cycle. Ms represents the sum of all forms ofdissolved carbon , , and

CO2

H 2 HCO3

HCO3

,

CO 22

Atmosphere

M A

Terrestrial System

M T

Ocean surfaceDiss C= CO2,HCO3,H2CO3

M S

Deep layers of ocean

M D

F TA

F AT

F SA F AS

F SDF DS

Buffer factor results from the equilibrium between CO2(g) and dissolved carbon.

Consequence: a strong dependence of FSA on MS,

a substantial increase in CO2 in the atmosphere is balanced by a small increase of MS.

FSA kSAM S

SA

Exponent

Buffer factor

Revelle factor

Degassing Dissolution

F=k (pCO2atm – pCO2

sol) = k (pCO2atm – c DICX)

Questions

• We proceed from the assumption that mankind disturbs the carbon system by burning fossil fuels with a total quantity of 300 Pg C, which is directly introduced into the atmosphere in one swoop.

• The model shall be used to answer the following two questions:

1. How does the carbon inventory disperse in the boxes?2. Where will we find the additional carbon on a long-

term basis?