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## Dynamical Climatology

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**Gerrit Lohmann**2. June 2005, 15.15 o‘clock Biogeochemical cycles Clocks 14-C Thermohaline Circulation Some homework Dynamical Climatology**S**total flux out of the reservoir M content if a substance in the reservoir 4 - 1 Turnover Time, renewal time single reservoir with source flux Q, sink flux S, and content M Q S=kM M The turnover time of carbon in biota in the ocean surface water is 3 x 1015/(4+36) x 1015yr ≈ 1 month The equation describing the rate of change of the content of a reservoir can be written as**Since ∑ Si = S, these time scales are related to the**turnover time of the reservoir, 4 - 2 If the reservoir is in a steady state (dM/dt = 0) then the sources (Q) and sinks (S) must balance. If material is removed from the reservoir by two or more separate processes, each with a flux Si, then turnover times with respect to each process can be defined as: In fluid reservoirs like the atmosphere or the ocean, the turnover time of a tracer is also related to the spatial and temporal variability of its concentration within the reservoir.**4 - 2**In fluid reservoirs like the atmosphere or the ocean, the turnover time of a tracer is also related to the spatial and temporal variability of its concentration within the reservoir. Fig. 4-2 Inverse relationship between relative stand- dard deviation of atmospheric concentration and turnover time for important trace chemicals in the troposphere. (Modified from Junge (1974) with per- mission from the Swedish Geophysical Society.)**4 - 3**Atmosphere 725 (Annual increase ~3) Deforestation ~1 ~93 ~90 ~60 ~120 ~1 Surface water Dissolved inorg. 700 Dissolved org. 25 (Annual increase ~ 0,3) Short-lived biota ~110 ~15 Long-lived biota ~450 (Annual decrease ~1) ~15 ~40 Detritus decomposition 54-50 Primary production ~40 Respiration & decomposition ~36 Litter ~60 ~40 ~38 Surface biota 3 2 - 5 ‹1 Detritus ~4 2 - 5 Intermediate and Deep water Dissolved inorg. 36,700 Dissolved org. 975 (Annual increase ~ 2,5) Soil 1300 - 1400 (Annual decrease ~1) Peat (Torf) ~160 ‹1 5 Fossil fuels oil, coal, gas 5,000 - 10,000 Land Sea 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.)**PDF of residence times be denoted by ø ( )**4 - 4 The residence time is the time spent in a reservoir by an individual atom or molecule. It is also the age of a molecule when it leaves the reservoir. PDF lake exponential decay 238 U The (average) residence time removal is biased towards young particles "short circuit" case: Sink close to the source The (average) age of atoms in a reservoir is given by [PDF is always decreasing ]**4 - 5**The adjustment process is e-folding time**4 - 6**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 equations can be written in matrix form as where the vector M is equal to (M1, M2,... Mn) and the elements of matrix k are linear combinations of the coefficients kij**where and are the eigenvectors of the matrix**k. In our case we have 4 - 6 or, in component form and in terms of the initial conditions:**4 - 6**response time turnover times of the two reservoirs**4 - 8**ATMOSPHERE (dust) LAND BIOTA 0.00009 96.9 0.02 0.01 0.14 0.10 6.0 6.0 33.6 0.10 Land (upper 60 cm of soil) 87.5 OCEAN BIOTA SURFACE OCEAN 6460 32.2 1.6 - 4.0 0.58 1.87 1.4 0.03 2812 0.39 0.60 0.69 DEEP OCEAN 323-645 0.11 MINE- RABLE P SEDIMENTS 1.29X108 The global phosphorus cycle. Values shown are in Tmol and Tmol/yr. (T=10^12) The mass of P in each reservoir and rates of exchange. Phosphate PO4(3-)**4 - 8**Table 4-1 Response of phosphorus cycle to mining output. Phosphorus amounts are given in Tg P (1Tg=1012g). In addition, a pertubation is introduced by the flux from reservoir 7 (mineable phosphorus), which is given by 12 exp(0.07t) in units of Tg P/yr Tcycle = 5300 years**4 - 9**1 2 Q T S1 S2 Example: An open two-reservoir system**Simplified model of the carbon cycle. Ms represents the sum**of all forms of dissolved carbon , , and 4 - 12 F AT Terrestrial System M T Atmosphere M A F TA F SA F AS Ocean surface Diss C= CO2,HCO3,H2CO3 M S F DS F SD Deep layers of ocean M D Non-linear System: Simplified model of the biogeochemical carbon cycle. (Adapted from Rodhe and Björkström (1979) with the permission of the Swedish Geophysical Society.)**4 - 12**where the exponent SA (the buffer, or Revelle factor) is about 9. The buffer factor results from the equilibrium between CO2(g) and the more prevalent forms of dissolved carbon. As a consequence of this strong dependence of FSA on MS, a substantial increase in CO2 in the atmosphere is balanced by a small increase of MS. atmosphere to the terrestial system**4 - 12**uptake of atmospheric CO2 by terrestrial biota with MTB being the content of carbon in terrestrial biota and D, a Michaelis constant. Mass MTB may grow without bounds. To avoid such a mathematical explosion, Williams (1987) suggested that the factor MTB in Equation (33) be replaced by**4 - 13**Fig. 4.13 Calculated and observed annual wet deposition of sulfur in mgS/m2 per year.**ATMOSPHERE**Water Conservation Equation Thermodynamic Equation Equations of Motion Turbulence Parameterization Cloud Parameterization Radiation Hydrologic Equation Thermodynamic Equation Vegetation Land Ice 4 - 14 Sensible Heat Evaporation Sensible Heat Evaporation Radiation Radiation Precipitation Precipitation Wind Stress Equations of Motion Salt Equation Runoff Thermodynamic Equation Turbulence Parameterization Sea Ice OCEAN LAND Schematic diagram showing the components of a global climate model (GCM).**4 - 14**organized fluid motion molecular diffusion continuity of tracer mass**4 - 14**Eddy correlation technique, eddy diffusivity**4 - 15**Fig 4-15 Orders of magnitude of the average vertical molecular or turbulent diffusivity (which is largest) through the atmosphere, oceans, and uppermost layer of ocean sediments.