In lake ecosystem models different strategies have been applied to deal with the problems described above. The major distinction is the degree of simplification of the physical, biogeochemical, and ecological components of the lake ecosystem and the mathematical formulation of the processes considered in the model. In this section, we give an overview of approaches for physical, biogeochemi-cal, and ecological submodels of lake ecosystem models.
Key elements distinguishing physical submodels of lake ecosystem models are the spatial resolution of the model and the description of mixing processes.
Already in the early 1970s, it was recognized that simple one-box mass-balance models (mainly for phosphorus contained in phosphate and phytoplankton) are extremely useful for improving the basic understanding of eutrophication processes. Such models represent the whole lake as a mixed reactor and calculate changes in average concentration(s) as a consequence of input, transformation, sedimentation, sediment release, and output. In conjunction with more complex biogeochemical and ecological submodels (see below), such models are still in use today, particularly for shallow lakes with a small degree of stratification.
Thermal stratification suppresses vertical mixing significantly during the summer, when warm, less-dense water layers lay on the top of colder and denser layers. Due to diurnal temperature variations and wind-induced mixing, a frequently mixed surface layer, the epilimnion, builds up on top of the hypolimnion, which is separated from the epilimnion by a zone with a strong temperature gradient and high stability, the metalimnion. Obviously, a simple description of such a system can be obtained with two mixed reactors representing epilimnion and hypo-limnion and with an exchange process that is strong during the winter and weak during the summer.
Closing the nutrient cycle in the lake-sediment system requires an extension of the mass balances to the sediment. This can be done by adding an additional mixed reactor describing the sediment, either to a one- or two-box lake model. This makes it possible to describe mineralization of organic particles deposited in the sediment and resulting oxygen consumption and nutrient release. More boxes can be used to describe anoxic and anaerobic mineralization processes in deeper sediment layers.
A better description of mixing in the lake can be achieved by resolving the depth of the lake continuously. Such one-dimensional (1D) lake models are able to resolve the often very strong gradients in vertical concentration profiles of dissolved oxygen, nutrients, and phytoplankton.
Particularly for narrow lakes with a large longitudinal extension and a high throughflow, it may be necessary to resolve the longitudinal dimension as well. This can be the case for reservoirs. Such 2D models are then able to distinguish different vertical profiles along the longitudinal direction of the lake.
The increasing availability of 3D mechanistic mixing models makes it more and more possible to couple such models with biogeochemical and ecological lake models. This leads to a 3D description of all processes in the lake. From the biogeochemical and ecological perspective, such models make it possible to distinguish processes and substance concentrations in the pelagic zone (open water column) from those in the littoral zone (close to the land and sediment) and to resolve concentration gradients across the lake. This can be relevant for the description of increased productivity in the neighborhood of nutrient-delivering inflows.
Options for the description of mixing processes in lakes depend on the spatial resolution of the model described in the previous subsection. Use of box models usually requires an empirical parametrization of mixing processes between the boxes. Models with resolution of the vertical dimension of the lake often also rely on empirical para-metrization of mixing processes. Such empirical parametrization of mixing processes is usually calibrated with the aid of temperature profiles in the lake. Such profiles lead to quite reliable estimates ofmixing intensity between epilimnion and hypolimnion. However, due to very small temperature gradients, they often do not provide sufficient information for a calibration of mixing processes in the hypolimnion of deep lakes. This leads to the requirement of including profiles of phosphate, nitrate, and/or dissolved oxygen into the calibration process. This can be problematic as there may be nonidentifiability problems between mixing intensity and transformation processes.
Physically based mixing models can be derived for 1D, 2D, and 3D lake models. They describe stratification and mixing caused by heat exchange over the lake surface and by momentum uptake due to wind forcing. Often turbulence due to seiche oscillation is also relevant for lakes. The main advantage of replacing empirical and semiem-pirical mixing models by such mechanistic mixing and transport models is that this decreases the need for empirical parametrization of the physical part of the ecological lake model. This improves the predictive capability of the models, at least conditionally on assumptions regarding climate forcing.
Biogeochemical submodels differ in the consideration of nutrients, in modeling of the element cycles and exchange with the sediment, and in their description of the mineralization process.
The most important elements that can limit phytoplank-ton growth are phosphorus, nitrogen, carbon, and, for diatoms, silicon. These are usually taken up in the form of phosphate, ammonium or nitrate, carbon dioxide (sometimes also bicarbonate), and silica.
Many lakes are limited by phosphorus during the summer stratification period. This leads to extremely small phosphate concentrations in the epilimnion during summer and makes the consideration of phosphate very important for lake models.
Nitrogen limitation is more difficult to describe than phosphorus limitation, because there are some phyto-plankton species that are able to fix nitrogen from dissolved molecular nitrogen. Nitrification of ammonium to nitrate also affects the oxygen budget of the lake. Nitrate is not only important as a nutrient for phyto-plankton, it is also important for anoxic mineralization of organic material (primarily in the sediment). This denitrifying mineralization can make the lake a significant sink for nitrate. Quantifying this denitrification capacity of the lake requires consideration of nitrate (and usually also ammonium) in the lake model.
The limiting effect of silica on phytoplankton growth depends on the geology of the watershed (determining silica input) and on the occurrence of diatoms (determining silica consumption). The consideration of silica can be important, if diatoms are distinguished from other functional groups of phytoplankton.
Sometimes also carbon is limiting primary production in lakes. Most phytoplankton species need CO2 as carbon source. The dependence of the growth rate on the CO2 concentration varies among species, therefore the depletion of CO2 can lead to a change in species composition. For example, the dominance of cyanobacteria in hyper-eutrophic lakes is sometimes caused by CO2 limitation because cyanobacteria have a very efficient carbon concentration mechanism (CCM). Some species can also use HCO^ as carbon source.
Modeling of element cycles and exchange with the sediment
Simple models (with respect to element cycles) treat sedimentation of organic particles as loss from the modeled part of the system and use sediment oxygen demand, phosphate release, and ammonium release as model parameters. With a correct choice of these model parameters, this can lead to reasonable results. However, this decoupling of processes which is very strongly coupled in reality (through mineralization in the sediment) allows an inexperienced model user to work with unreasonable model parameters that violate mass conservation.
For this reason, a more detailed level of description is to model the mass balance of nutrients in the sediment explicitly. This can be done by describing the pools of particulate organic matter, dissolved oxygen, and nutrients in the sediment and calculating sediment oxygen demand and nutrient release by a simple kinetic process.
More detailed sediment models use one or more sediment layers (with different redox conditions) to achieve a more realistic description of the sediment. Particulate organic matter enters the top sediment layer through sedimentation. The mineralization of organic matter to inorganic nutrients in the sediment can be modeled using the same process description as in the water column. The inorganic nutrients produced by mineralization are released into the porewater of the sediment and diffuse into the water column, depending on the concentration difference between the sediment porewater and the water column.
Among the models which model mineralization explicitly, most do not distinguish hydrolysis of particulate organic matter into dissolved organic matter from mineralization of dissolved organic matter but combine both steps into a single 'mineralization' process.
Many models only describe oxic mineralization. Some add denitrifying mineralization, some add further steps with reduction of manganese oxide, iron hydroxide, or sulfate and finally methanogenesis.
Most models parametrize the mineralization process directly without explicitly describing the bacterial community performing the process. This limits the transferability of these models as different bacterial populations in different environments lead to different mineralization rates.
Ecological submodels can be divided into trophic-level models, functional group models, dominant-species models, and adaptive property models. Combinations of some of these model types are also possible.
Trophic-level models use the trophic levels of the food web shown in Figure 1 as state variables without further division. However, some of the trophic levels may be merged and some may be omitted. If higher trophic levels are omitted, their effect on lower levels is considered by increased death rates at the lower levels.
Phytoplankton or periphyton must be considered in each ecological lake model as it is responsible for primary production of biomass out of inorganic nutrients. Phytoplankton consists of hundreds of different species with widely varying properties such as maximum growth rate, edibility, and dependence on light, nutrients, and temperature. In trophic-level models, all these different species are modeled by a single state variable. It seems astonishing that this can work. However, the limitation of primary production by nutrients in many lakes makes production less dependent on formulation and quantification of process kinetics. In such situations, input of nutrients determines production. This may be the explanation why such simple models work astonishingly well.
If zooplankton is considered explicitly it is often modeled as a single state variable or as two state variables representing herbivorous and carnivorous or omnivorous zooplankton. Again, these classes aggregate many different species.
In most ecological lake models, fish are not explicitly modeled. The predation pressure of fish is then quantified by increasing the death rate of zooplankton. To account for changes in predation pressure, a seasonal dependence of such a death rate contribution can be considered.
Functional group models differ from trophic-level models by disaggregating the species within one or several trophic levels into groups with similar properties. Although these groups usually have the same function (given by the trophic level), they are called functional groups.
Criteria for the division of species into functional groups can be (1) ecological properties, such as growth rate, edibility (e.g., size), silica requirement, ability to fix nitrogen, or mobility; (2) taxonomic groups; (3) easily measurable properties, such as maximum extension, volume, etc., which are assumed to correlate with ecological properties. Often different functional groups have similar process formulations, but differ in model parameter values.
Dominant-species models make a slightly different approach to model the variability of properties of different species at a given trophic level. Instead of dividing the species at the trophic level exhaustively into functional groups, the key species are modeled individually. If these species can be cultivated in the laboratory, such models have the advantage that essential behavioral parameters such as maximum growth rate or parameters related to nutrient limitation can be measured experimentally. This can decrease the number of calibration parameters of the model significantly. On the other hand, these models have the disadvantage of introducing a large number of additional state variables and still not being able to describe the sum of all species correctly.
A slightly modified version of this type of model uses the properties of a key indicator species to represent functional groups. This leads to a model that combines advantages of the dominant-species model with those of the functional group approach. However, the advantage ofusing measured properties in the dominant-species approach becomes less strong when applying these to functional groups.
Biological species are adaptive. They are able to change their properties, for example, their size, edibility, chlorophyll, or phosphorus content in order to adapt to changing environmental conditions. This is only rarely accounted for in the ecosystem models discussed above (adaptation of phosphorus content is considered in some models). Recently, there have been attempts to include adaptation into aquatic ecosystem models.
One approach is called 'structural dynamic models'. In this approach, selected properties of species are dynamically changed according to the global criterion of maximization of'exergy'. Kinetic parameter values are adapted during the simulation according to exergy maximization.
A second approach identifies 'rapid evolution' as the cause for adaptation and (usually) uses an empirical para-metrization of this process.
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