Multibox model

The remaining two chapters of Part IV set the basis for the more advanced environmental models discussed in Part V. Chapter 21 starts with the simple one-box model already discussed at the end of Chapter 12. One- and two-box models are combined with the different boundary processes discussed before. Special emphasis is put on linear models, since they can be solved analytically. Conceptually, there is only a small step from multibox models to die models that describe the spatial dimensions as continuous variables, although the step mathematically is expensive as the model equations become partial differential equations, which, unfortunately, are more complex than the simple differential equations used for the box models. Here we will not move very far, but just open a window into this fascinating world. 

Figure 19.5 In multibox models, the exchange between two fairly homogeneous regions is expressed as the exchange flux of fluid (water, air etc.), Q . Normalization by the contact area, A, yields the exchange velocity vex = QJA. This quotient can be interpreted as a bottleneck exchange velocity vex =...

Lakes and oceans are often vertically stratified. That is, two or more fairly homogeneous water layers are separated by zones of strong concentration and density gradients. In Chapter 21, two- and multibox models will be developed to describe the distribution of chemicals in such systems. In these models, volume fluxes, Qex, are introduced to describe the exchange of water and solutes between adjacent boxes (Fig. 19.5). Qex has the same dimension as, for instance, the discharge of a river, [L3TT ]. The net mass flux, LFnet, from box 1 into box 2 is given by ... 

This is the first of several chapters which deal with the construction of models of environmental systems. Rather than focusing on the physical and chemical processes themselves, we will show how these processes can be combined. The importance of modeling has been repeatedly mentioned before, for instance, in Chapter 1 and in the introduction to Part IV. The rationale of modeling in environmental sciences will be discussed in more detail in Section 21.1. Section 21.2 deals with both linear and nonlinear one-box models. They will be further developed into two-box models in Section 21.3. A systematic discussion of the properties and the behavior of linear multibox models will be given in Section 21.4. This section leads to Chapter 22, in which variation in space is described by continuous functions rather than by a series of homogeneous boxes. In a sense the continuous models can be envisioned as box models with an infinite number of boxes. 

One-box/ two-box/ multibox models Model consisting of one or several boxes. Each box is characterized by one or several state variables Example Two-box model of a lake consisting of the boxes epilimnion and hypolim- nion... 

Some of the mathematical tools, such as the linear one-box model, are both fairly simple and nonetheless sufficient for handling a great variety of situations. More complex systems require the use of multibox models. In some cases, continuous time-space models are needed. The mathematics of the latter involve partial differential equations and quickly lead beyond the scope of this book. In this chapter, a few important concepts were discussed which allow the reader both to make some approximative calculations and to critically analyze the results from computer models in which time-space processes are employed. 

The offshore advective flux for Si shown in Fig. 17.3 (30 X 10 mol d l) was calculated by difference, based on the total flux of dissolved Si supplied to the shelf system (32 X 10 mol Si d-1), the estimated deltaic burial rate (1-3 x 10 mol Si d ), and the nearshore particulate flux (0.1-0.7 x 10 mol Si d ). This advective flux is in good agreement with the results of Daley (1997), who estimated that 30 x 10 mol d of Si leave the shelf, based on seasonal field data and a multibox model for the shelf. Most of the silicate (94%) supplied to the shelf by external sources appears to be transported to the open ocean in either dissolved or particulate form. Approximately 36% of the Si leaving the outer shelf is in particulate form according to these calculations. Biogenic silica export may have contributed to the lack of closure in the Edmond et al. (1981) silicate budget for the shelf, although deltaic burial also remains as a potentially important sink. 

The two box models discussed above can be extended to a multibox model. In the limit of infinitely small boxes the multibox model corresponds to the continuous model of Jassby and Powell (1975). Using Equation (40), the vertical turbulent diffusivity as function of depth can be obtained from x if t is at steady state and can be treated as ideal tracer with source strength of lyr/yr ... 

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