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## The Standard Model

Both the ecosystem dispersion model and the chemokinetic model are described in this chapter, since the modeling paradigm used for the two submodels is the same.

The ecosystem dispersion model takes as input the time- and space-dependent air concentration of the pollutant at the predefined grid points and follows the fate of the pollutant across the different trophic levels of the ecosystems considered. Each grid point of the air dispersion model represents one local food web, through which the trace of the pollutant is followed.

Note that population dynamics are not accounted for, since there does not seem to be a general mechanism for population dynamics, valid for all kinds of interacting populations [10, 28].

The chemokinetic model takes as input the time- and space-dependend exposure of an individual to the pollutant under consideration and computes the uptake and the distribution of the chemical in the different body tissues and organs. Several different types of individuals can be simulated by specifying the corresponding parameters. The metabolism of the pollutant under consideration is modeled by Michaelis-Menten kinetics.

Both codes simulating the processes described above are based on a compartment model of the system to be simulated [19, ] (also called continuously stirred tank reactor (CSTR) model). These systems are of the following type, and one of these systems is set up for each grid point of the computational grid of the atmospheric dispersion model. The modeler divides the total system under consideration (i. e. the ecosphere and the geosphere) into different compartments. The total mass of pollutant over time in a compartment k, , is then given by a function , which has to be computed. Again, note that we have where is a variable denoting the point in space where the compartment system resides. In what follows, we skip this dependency for ease of notation. In the compartment model implemented in OLAF, compartment no. 1 is always the "air compartment", and the total mass in it is represented by the pollutant concentration calculated by the MESOPUFF II model (see Section 2.2). As a consequence, the function can be considered as an external variable which does not need to be calculated here any more. By the first law of thermodynamics, the change of mass in compartment no. k (), , is the difference between the inflow and the outflow to other compartments. The inflow from compartment no.  to compartment no. k is proportional to the mass inside compartment j, while the outflow out of compartment k to compartment no. i is proportional to the mass in compartment k. As a consequence, the following ordinary differential equation holds for :

where is the time-dependent proportionality constant for the flow from compartment j to compartment k, and is the time-dependent proportionality constant for the flow from compartment k to compartment i (). These parameters are dimensionless. The differentiation symbol denotes differentiation with respect to t. Again, we have suppressed in the notation used here any dependency on the spatial grid point considered. In what follows, we assume without loss of generality for and all , i. e. at any time there is at least one inflow into each non-air compartment considered. Define now the matrix by using, as usual, as the entry in row k and column i and define the diagonal elements by

When using the vector notation , we see that the equations

hold, where the starting condition m(0) depends on the spatial location, i. e. the grid point of the computational grid used. Note that the function  is given and does not have to be computed.

In the current version of the OLAF system, a back flow from an arbitrary compartment to the air compartment is not handled, since it has to be expected that the quantities involved flowing back are too small to be tracked numerically once they are added to the large amount of mass in the air compartment. As a consequence, it is possible to use values for some to model a release of pollutant from a compartment to the air. But, while the mass-balance of the compartments is correctly computed (i. e. a certain mass is deducted), the corresponding mass-balance of the air compartment is not taken into account. Moreover, instead of the mass of the air compartment, the concentration in it is computed by MESOPUFF. This means that instead of dimensionless scalars () values with dimension have to be used. Now let c(t) be the concentration of pollutant in the air at the time . (Again, we suppress the dependence on the spatial scale.) We then have to solve the system

which is a linear inhomogeneous system of ordinary differential equations of dimension n-1.

Next: Mass Conservation and Nonnegativity Up: Compartment Models Previous: Compartment Models

Joerg Fliege
Wed Dec 22 12:25:31 CET 1999