The temporal change of the concentration, CX, of a substance or organism, X, in a vertically resolved water column is given by the following differential equation:
Here t is time, z is the vertical coordinate in the lake, vX is the sum of the advective velocity of the water column and the sedimentation velocity of substance or organism X, Kz is the coefficient of vertical turbulent diffusion, and rX is the total (net) transformation rate of substance or organism X. The total transformation rate of a substance is the sum of contributions by different processes. The contribution of each process is calculated as the product of the process rate with a substance-specific stoichiometric coefficient. This means that the net transformation rate of substance X is given by rX = Vi;X Pi [21
i where the sum extends over all transformation processes. ViX is the stoichiometric coefficient of the process i with respect to substance X and pi is the process rate of the process i.
In order to discuss formulations of transformation processes used in the literature, we use a simple, didactic lake model. This lake model contains five state variables: dissolved oxygen, nutrients, phytoplankton, zooplankton, and dead particulate organic material. Compared to Figure 1, this model aggregates the two functional groups of zooplankton and it omits fish and dissolved organic material. This leads to an aggregation of transformation processes also. We will give formulations of the following processes:
1. Growth of phytoplankton by primary production (gro, ALG)
2. Growth of zooplankton by grazing of phytoplankton (gro, ZOO)
3. Respiration of phytoplankton (resp, ALG)
4. Respiration of zooplankton (resp, ZOO)
5. Death of phytoplankton including grazing by zooplankton (death, ALG)
6. Death of zooplankton including predation by fish (death, ZOO)
7. Oxic mineralization of particulate organic material to nutrients including the hydrolysis step to dissolved organic material (miner)
We will now discuss typical formulations of the transformation rates of these processes used in the literature. Transformation rates of substances and organisms are then given by eqn .
Table 1 gives an overview of the structure of typical formulations of these seven process rates. Growth, respiration, and death rates are usually proportional to the concentration of the organism affected by the process. The rate formulation then multiplies this concentration by a specific transformation rate at standard conditions and several modification factors that describe the effect of important influence factors.
Table 2 shows options for the formulation of modification factors used in Table 1 for describing the dependence of process rates on important influence factors.
If several nutrients are limiting, several nutrient limitation terms can be multiplied or the minimum of the limiting factors can be used (Liebig's law).
This short overview should give an idea of how transformation process rates can be formulated. Some models use different or more complicated process formulations or they further divide processes into subprocesses. For example, the growth process of phytoplankton can more realistically be described by a nutrient uptake process into the cell and a growth process on nutrients contained in the cell.
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