mathematical modelling, 5th projects: carbon cycle in a box model boat dynamics: oscillations j s....
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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
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
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
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?