Two-Box Models of Lakes

In Section 23.1, this procedure will be applied to just one completely mixed water body. This control volume may represent the lake as a whole or some part of it (e.g., the mixed surface layer). Section 23.2 deals with the dynamics of particles in lakes and their influence on the behavior of organic chemicals. Particles to which chemicals are sorbed may be suspended in the water column and eventually settle to the lake bottom. In addition, particles already lying at the sediment-water interface may act as source or sink for the dissolved chemical. In Section 23.3, two-box models of lakes are discussed, particularly a model consisting of the water body as one box and the sediment bed as the other. Finally, in Section 23.4, one-dimensional vertical models of lakes and oceans are discussed. 

Linear Two-Box Model with One Variable Linear Two-Box Model of a Stratified Lake Box 21.7 Linear Two-Box Model for Stratified Lake Illustrative Example 21.5 Tetrachloroethene (PCE) in Greifensee From the One-Box to the Two-Box Model Linear Two-Box Models with Two and More Variables Nonlinear Two-Box Models... 

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... 

Two-Box Model for Lake/Sediment System Box 23.3 Solution of Linear Water-Sediment Model PCBs in Lake Superior (Part 3)... 

As a second example of a two-box model we discuss the case of a stratified lake which is divided into the surface layer (epilimnion E, box 1) and the deep-water layer (hypolimnion H, box 2). The model and its parameters are shown in Fig. 21.10. It includes the following processes (numbers as in the figure) ... 

Figure21.10 Two-box model for stratified lake. The numbered processes are (1) input by inlets (rj is relative fraction of input going to the hypolimnion), (2) air-water exchange, (3) loss at the outlet, (4) loss by in situ chemical transformation (chemical, photochemical, biological), (5) flux on settling solid matter, (6) exchange across the thermocline. See text for definition of parameters. Note that the substance subscript i is omitted for brevity.

Hence, the one-box and two-box models yield the same result. There is a simple reason for that. Since the only removal processes of PCE act at the lake surface, at steady-state the surface concentration in both models (C°°for the one-box model, ClE for the two-box model) must attain the same value to compensate for the input /, tot. Furthermore, since the hypolimnion has neither source nor sink, the net exchange flux across the thermocline must be zero, and this requires C(E= C,H. 

In the epilimnion/hypolimnion two-box model the vertical concentration profile of a chemical adopts the shape of two zones with constant values separated by a thin zone with an abrupt concentration gradient. Often vertical profiles in lakes and oceans exhibit a smoother and more complex structure (see, e.g., Figs. 19.1a and 19.2). Obviously, the two-box model can be refined by separating the water body into three or more horizontal layers which are connected by vertical exchange rates. 

You have constructed a linear two-box model for tetrachloroethene (PCE) in a lake in which the only input of PCE is from the outlet of a sewage treatment plant. The atmospheric PCE concentration is assumed to be zero in your model. How will the steady-state of the model be altered if the PCE input from sewage is reduced by... 

In this section we treat the exchange at the sediment-water interface in the same manner as the air-water exchange. That is, we assume that the concentration in the sediments is a given quantity (an external force, to use the terminology of Box 21.1). In Section 23.3 we will discuss the lake/sediment system as a two-box model in which both the concentration in the water and in the sediments are model variables. 

In Chapter 21 the model of a stratified lake served as a prototype of a linear two-box model (Fig. 21.10). The necessary mathematics were developed in Boxes 21.6 and 21.7. In Illustrative Example 21.5 the fate of tetrachloroethene (PCE) in Greifensee was used to demonstrate that for the case of a two-box model it is still possible to carry out back-of-the-envelope calculations. Further examples are given in Problems 23.2 and 23.3, where the behavior of anthracene in a mixed as well as in a stratified lake is assessed. 

Figure 23.5 Processes considered for the combined sediment-water two-box model to describe the fate of PCB congeners in lakes.

In the last step (Part 3), the sedimentary compartment (the surface mixed sediment layer , SMSL) was treated as an independent box (Table 23.7). The steady-state solution of the combined sediment/water system explained another characteristic of the observed concentrations, which, as mentioned above, could not be resolved by the one-box model. As shown in Table 23.8, for both congeners the concentration measured on particles suspended in the lake is larger than on sediment particles. The two-box model explained this difference in terms of the different relative organic carbon content of epilimnetic and sedimentary particles. This model also gave a more realistic value for the response time of the combined lake/sediment system with respect to changes in external loading of PCBs. However, major differences between modeled and observed concentrations remained unexplained. 

If the lake is stratified, vertical transport is commonly the time-limiting step for complete mixing. This was the reason for applying the two-box model to the case of PCE in Greifensee (Illustrative Example 21.5). Now we go one step further. We consider a vertical water column of mean depth h with a constant vertical eddy diffusion coefficient Ez. The flux Fa/VJ of PCE escaping to the atmosphere is given by Eq. 20-la ... 

Why do we need at least a two-box model to explain the long-term memory of the PCBs in Lake Superior ... 

In Equation (48) the ratio A" He/x from an individual sample has been taken as representative for the entire lake. If several measurements are available the volume weighted mean values of A He and x could be applied. This would correspond to the two box model (Eqn. 47) with box 2 representing surface water in equilibrium with the atmosphere. At the lake surface " Hei = " Hcequ and Hei = Hcequ, implying X2 = 0. Because Heequ varies only very little with temperature Heequ is approximately constant. Then FqHe.sed = Vq/Ao A Hci/xi with A Hci and x being volume weighted mean values. 

In case the actual concentration of phenanthrene would exhibit significant spatial variations, the one-box model would not be the ideal description. Instead, it may be adequate to subdivide the lake into two or more boxes in such a way that within the defined subvolumes, phenanthrene concentration would be fairly homogeneous. So we would end up with a two- or multi-box model. In certain situations this box model approach may still not be sufficient. We may need a model which allows for a continuous concentration variation in time and in space. Such models will be discussed in Chapter 22. 

Application of steady-state solution of linear water-sediment model (Box 23.3) to two PCB congeners in Lake Superior. The steady-state is calculated from Box 21.6. 

In the above studies, lake water is regarded as perfectly mixing one box. However, lake water chemistry is heterogeneous. For instance, chemical composition of surface water differs from deep water. Two box (surface water, deep water) model was applied to the interpretation of P concentration in lake water by Brezonik (1994). 

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 ... 

The meaning and typical sizes of the coefficients Ax and A2 are discussed in Box 22.4. From Eq. 22-44 we note that for small times t, c2(t) grows as f, whereas for large times it grows as i2. The critical time, crit, defined in Eq. 3 of Box 22.4 separates the two regimes. Figure 22.10 shows a2(() curves from different experiments conducted in Swiss lakes. In Illustrative Example 22.3 the shear diffusion model is applied to the case of an accident in which a pollutant is added to the thermocline of a lake. 

Let us demonstrate the power of the one-box lake model by analyzing the fate of two different polychlorinated biphenyl congeners (PCBs) in Lake Superior (North America). Characteristic data of the lake are given in Table 23.3. 

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