1. The basic difference with rivers is that the horizontal dimension(s) are comparable to depth. Thus: The velocity of the water is small (negligible) The differences along the vertical dimension are relevant The dynamics of higher levels of the trophic chain (phytoplankton, zooplankton, fishes) are relevant Slower processes (e.g. seasonal changes) must be considered The underlying hydraulic model is obviously different A lake (often artificial) can be studied as a sequence of horizontally perfectly mixed boxes.

2. The basic consequence of depth (in temperate climate) is water stratification. spring wind  heating faster than mixing summer same temp. thus mixing warm water is less dense, thus floats, and needs lots of wind to mix

3. Stratified lakes present three distinct zones. thermocline depth (m) temperature (°C) 0103020 0 10 30 20 region of rapid temperature change hypolimnium epilimnium sediments

4. A typical yearly pattern in temperate countries (dimictic lakes  holomictic lakes)

6. Loss of water quality, fish death

7. Periphyton is a mixture of algae, cyanobacteria, heterotrophic microbes attached to submerged surfaces

8.  The growth rate of a population of phytoplankton in a natural environment:  is a complicated function of the species of phytoplankton present  involves different reactions to solar radiation, temperature, and the balance between nutrient availability and phytoplankton requirements  Due to the lack of information to specify the growth kinetics for individual algal species all models characterizes the population as a whole by the total biomass of the phytoplankton present (measured in terms of chlorophyll concentration)

9. PhytPhyt NO 3 PO 4 NH 3 O C:N:P LightLight Growth rate: G max = maximum specific growth rate constant at 20°C, 0.5 – 4.0 day -1 X T = temperature growth multiplier, dimensionless X L = light growth multiplier, dimensionless X N = nutrient growth multiplier, dimensionless

10. where:  G = temperature correction factor for growth (1.0 – 1.1) T = water temperature, °C Temperature multiplier Nutrient multiplier Defines a limiting factor K MN semisaturation constant

11. X L depends on the light l(z) available for photosynthesis at depth z. It may be written using Michaelis-Menten formulation or Steele (1965) formulation where l s represents an optimal (maximum) light intensity. But The incident light on a water surface varies during the day and the season The light intensity naturally decreases with depth The presence of phytoplankton further increases light attenuation (self- shadowing)

12. The light intensity dependence on depth l(z) can be expressed by the Beer- Lambert law where l 0 is the incident radiation on the surface and the function can be written as a polynomial function of phytoplankton concentration.

13. The relation between phytoplankton and zooplankton is a typical predator- prey system. Algal blooms occur in spring.

14. Death rate: k 1R = endogenous respiration rate constant, day -1  1R = temperature correction factor, dimensionless k 1D = mortality rate constant, day -1 k 1G = grazing rate constant, day -1, or m 3 /g Z -day if Z ( t ) specified Z(t) = zooplankton biomass time function, g Z /m 3 (defaults to 1.0) Settling rate: v S = settling velocity, m/day A S = surface area, m 2 V = segment volume, m 3

15. PhytoplanktonPhytoplankton PO 4 Org. P Detr. P  Phytoplankton P  Detrital P Growth Death Settling Death Dissolution SettlingMineralizationGrowthDeath MineralizationDissolution  Inorganic P  Dissolved organic P

16. Difficult to calibrate for a specific situation. Ex. Phosphorus cycle parameters

17. It is necessary to compute all the internal processes for each volume in a grid and model the dispersion exchanges with the surrounding volumes (thousands of state variables)

18.  Reduce loads (less use of detergents or fertilizers, better treatment,…)  Artificial mixing  Selective discharges  Artificial aeration

19. Water quality WQ is, in principle, a function of all the water components c 1,…, c n in each location z 1,z 2,z 3 and at any time instant t. In practice, we are not able to define the form of such a function. We thus define suitable indicators WQ i for each components based on some kind of aggregation in time and/or space. where Stat(  ) indicates some statistics over the spatial region Z and the time interval T. Examples: -Summer average oxygen concentration in the hypolimnium -Yearly average of phytoplankton concentration in the upper 10 m -Number of times in a year in which the nitrogen concentration in the upper 1 m exceeds N.

20. Additionally, we can assume that the overall water quality is some aggregation of the WQ i. A common definition of the aggregation is an index formulated as a weighted sum: where the weights  i express some (subjective) measure of the importance of each factor in the assessment of water quality. The planning/control problem can be written: or where u are the decisions and Cost ( u ) their cost.

21. As already noted (see slides on DPSIR), when used for planning/control, the model works in conditions different from those used for calibration/validation. The accuracy of the model cannot be proved. The model can thus be used to test the effect of an input (decision, parameter, boundary condition) on the output, to understand: -the sense of the interaction (positive, negative) -the entity of the interaction (larger, smaller than other input variables). To perform such Sensitivity/Uncertainty analysis, Monte Carlo simulation is normally used: 1.Generate random numbers for model inputs 2.Run the model with the randomized inputs 3.Store the random input values and the corresponding model outputs 4.Repeat a (high) number of times the steps (1-3) 5.Calculate correlations between model outputs and random inputs

22. - Define the input to be tested - Select a suitable distribution of values (normal, lognormal,…) - Generate a set of random values - Run the model - Store the results for the selected output - Analyse the output distribution - Compute the correlation I/O

23. YASAIwYASAIw is a free open-source framework for Monte Carlo simulation in Excel. Three main functions: 1) To generate random model input values from a normal distribution: = GENNORMAL(mean, stdev) mean = nominal value for model input; stdev = e.g. 5% of mean for sensitivity analysis 2) To save the random input values and use them in a sensitivity analysis: = SIMOUTPUT(x, name, code) x = cell address of the random input name = unique name for the input code = 1 for input “assumption” 3) To save the model output values and use them in a sensitivity analysis: = VBAOUTPUT(x, name, code) x = cell address of the model output name = unique name for the output code = 2 for output “forecast”

24. YASAIw GENNORMAL functions

25. YASAIw SIMOUTPUT functions

26. Links to the model output sheets

28. Sensitivity average periphyton chlorophyll-a to the most sensitive model inputs Spearman's rank correlation coefficient (Spearman's rho), is a nonparametric measure of statistical dependence between two variables. It assesses how well the relationship between two variables can be described using a monotonic function.

29. The solution of the planning/control problems requires the use of models to determine the link between decisions and water quality. However, a full quality model may not necessary, since we are just interested in computing the link between decision and the defined water quality. Develop a SURROGATE MODEL, i.e. a (simplified) model able to reproduce the required function, at least for a certain range of values of u, which means to substitute the original function WQ( u ) with an approximation WQ= f s ( u ).

30. The overall procedure is thus: Define the range of u Define a structure for f s Simulate the original model Calibrate the parameters of f s Validate f s Use f s in the optimization procedure Common forms for f s : -A linear/polynomial function -A response surface -A neural network -….. It must be simple

31. The surrogate model: It’s NOT a copy of the system It’s based on the original model input and output and thus does not reproduce biochemical and physical phenomena It works correctly (the approximation is acceptable) only for the set of other input used for the original model simulation (design of experiments) It works correctly only within the range of u for what it has been calibrated It’s faster to execute and thus can be repeated a high number of times within the optimization procedure

32. Determination of the «best» number and position of artificial aerators for an Australian artificial reservoir.