Interaction Box models

2) The upper mantle reservoir that is sampled by MORE. 

No subduction of atmospheric noble gases back into the mantle occurs. 

It has been pointed out that Xe and I are both highly incompatible and would be removed together during partial melting of the upper mantle (Musselwhite et al. 1989). Therefore, generating high Xe/ °Xe ratios in the mantle due to elevated EXe ratios requires that I was efficiently subducted back into the mantle before decay of 

The observed high Xe/Ar ratios in MORB relative to that of the atmosphere are due to somewhat more efficient degassing of Ar, presumably due to a lower partition coefficient (Allegre et al. 1986). 

While the upper mantle is 25% of the mantle, the °Ar budget appears to require that 40% of the mantle has degassed to the atmosphere. Therefore, some degassing of the lower mantle must occur, although this does not qualitatively change either the degassing calculations of the upper mantle, or the relatively unradiogenic character of the lower mantle. 

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Demerjian, K. L., and Schere, K. L., Application of a photochemical box model for O3 air quality in Houston, TX, in "Proceedings of Ozone/Oxidants Interactions with the Total Environment II." Air Pollution Control Association, Pittsburgh, 1979, pp. 329-352. 

With different pectins, one found that the activity coefficient of calcium has a value half that of magnesium this is interpreted as the basis of a dimer formation in presence of calcium. The specific interaction of calcium was described as the egg-box model first proposed for polyguluronate in which oxygen atoms coordinated to calcium [46]. Recently, the comparative behaviour of Mg and Ca with homogalacturonan was reexamined [47]. 

Grant, G.T., Morris, E.R., Rees, D.A., Smith, P.J.C., Thom, D. (1973). Biological interactions between polysaccharides and divalent cations the egg-box model. FEES Letters, 32, 195-198. 

In practice, a gray-box model is developed in steps. One early step is to decide which variables and interactions to include. This is often done by the sketching of an interaction-graph. It must then be decided if a variable should be a state or a dependent variable, and how the interactions should be formulated. In the case of metabolic reactions, the expression forms for the reactions have often been characterized in in-vitro experiments. If this has been done, there are also often in-vitro estimates of the kinetic parameters. For enzymatic networks, however, such in-vitro studies are much more rare, and it is hence typically less known which expression to choose for the reaction rates, and what a good estimate for the kinetic parameters is. In any case, the standard method of combining reaction rates, r,-, and an interaction graph into a set of differential equations is to use the stoichiometric coefficients, Sij... 

Example 1. Let us consider an example that exemplifies step 3 in the core-box modeling framework. The system to be studied consists of one substance, A, with concentration x = [A]. There are two types of interaction that affect the concentration negatively degradation and diffusion. Both processes are assumed to be irreversible and to follow simple mass action kinetics with rate constants p and P2, respectively. Further, there is a synthesis of A, which increases its concentration. This synthesis is assumed to be independent of x, and its rate is described by the constant parameter p3. Finally, it is possible to measure x, and the measurement noise is denoted d. The system is thus given in state space form by the following equations ... 

Box models are commonly used biogeochemical studies that involve the interaction of biological, chemical, and geological processes that determine sources, sinks, and fluxes of elements through different reservoirs within ecosystems. 

An alternative example of the metal-in-a-box model is provided by the monocationic [(Ph4B)Ba(thf)4][BPh4], shown in Figure 34. Here, the metal coordination sphere is completed by jr-arene interactions to a BPI14 anion, which binds in a sandwich-type interaction to the metal. Secondary, nonclassical interactions between the monocation and the noncoordinated anion provide additional stabilization. Interestingly, analogous reaction conditions for the smaller metals calcium and strontium afford the dicationic species [M(thf) ][BPli4]2 (M = Ca, n = 6 M = Sr, n = 7), that are... 

A distinct class of models that describe the chemical evolution of the Earth are the so-called box models, in which assumptions are made about the geometry of distinct reservoirs and their interactions. For example, one can assign four distinct reservoirs in the upper mantle, lower mantle, continental crust, and atmosphere and develop differential equations that incorporate radiogenic ingrowth, chemical fractionation effects, and assumptions about mass transfer between the reservoirs. Successful models reproduce the observed isotopic ratios and/or... 

FIGURE 1.1 A box model for the Earth System, showing some of the major reser voirs and the interactions (white arrows) between them. 

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