1. Mathematical Modelling, 5th Projects: Carbon cycle in a box model Boat dynamics: Oscillations J S. Miller. Physics in a Toy Boat. Am. J. Physics 26, 199 (1958) pop-pop boat Advanced Population dynamics Climate Oscillator Immigrant dynamics Bifurcation Theory Partial diffential equations

2. Boat dynamics: Oscillations Physics in a Toy Boat. pop-pop boat

3. Climate Oscillator

4. Atmosphere 725 (Annual increase ~3) Surface water Dissolved inorg. 700 Dissolved org. 25 (Annual increase ~ 0,3) Surface biota 3 Intermediate and Deep water Dissolved inorg. 36,700 Dissolved 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 fuels oil, coal, gas 5,000 - 10,000 Respiration & decomposition ~36 Primary production ~40 Detritus ~4 Detritus decomposition 54-50 ~40 ~38 5 2 - 5 ~15 ~40 ~120~60~90~93 Deforestation ~1 ‹1 ~15 ~1 Fig. 4-3 principal reservoirs and fluxes in the carbon cycle. Units are 10 15 g(Pg) C (burdens) and PgC/yr (fluxes). (From Bolin (1986) with permission from John Wiley and Sons.) Carbon Cycle

5. Turnover Time, renewal time Mcontent if a substance in the reservoir Stotal flux out of the reservoir M S=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

6. Linear System: The adjustment process is e-folding time

7. Atmosphere 725 (Annual increase ~3) Surface water Dissolved inorg. 700 Dissolved org. 25 (Annual increase ~ 0,3) Surface biota 3 Intermediate and Deep water Dissolved inorg. 36,700 Dissolved 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 fuels oil, coal, gas 5,000 - 10,000 Respiration & decomposition ~36 Primary production ~40 Detritus ~4 Detritus decomposition 54-50 ~40 ~38 5 2 - 5 ~15 ~40 ~120~60~90~93 Deforestation ~1 ‹1 ~15 ~1 Fig. 4-3 principal reservoirs and fluxes in the carbon cycle. Units are 10 15 g(Pg) C (burdens) and PgC/yr (fluxes). (From Bolin (1986) with permission from John Wiley and Sons.)

8. The flux F ij from reservoir i to reservoir j is given by The rate of change of the amount M i i n reservoir i is thus where n is the total number of reservoirs in the system. This system of differential equations can be written in matrix form as where the vector M is equal to (M 1, M 2,... M n ) and the elements of matrix k are linear combinations of the coefficients k ij Master Equation, Statistical Physics

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10. Simplified model of the carbon cycle. M s represents the sum of all forms of dissolved carbon,, and Atmosphere M A Terrestrial System M T Ocean surface Diss C= CO 2,HCO 3,H 2 CO 3 M S Deep layers of ocean M D F TA F AT F SA F AS F SD F 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.)

11. Inorganic Carbon Cycle Free proton Bicarbonatecarbonate Non-linearity in the oceanic carbon system Carbonate acid hydrated

13. Ocean: inorganic Carbon Cycle

14. Simplified model of the carbon cycle. M s represents the sum of all forms of dissolved carbon,, and Atmosphere M A Terrestrial System M T Ocean surface Diss C= CO 2,HCO 3,H 2 CO 3 M S Deep layers of ocean M D F TA F AT F SA F AS F SD F DS Buffer factor results from the equilibrium between CO 2 (g) and dissolved carbon. Consequence: a strong dependence of F SA on M S, a substantial increase in CO 2 in the atmosphere is balanced by a small increase of M S. Exponent  Buffer factor Revelle factor DegassingDissolution F=k (pCO 2 atm – pCO 2 sol ) = k (pCO 2 atm – c DIC X )

15. 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?