A Simple Eutrophication Model for the Bay of Keszthely, Lake Balaton G. Jolânkai and A. Szôllôsi-Nagy Budapest, Hungary INTRO
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A simple eutrophication model for the Bay of Keszthely, Lake Balaton G. Jolânkai and A. Szôllôsi-Nagy Budapest, Hungary Abstract. Lake Balaton and especially its Bay at Keszthely is facing the danger of increasing eutrophication. Utilizing former plant nutrient discharge studies and records and also the records of lake hydrology, hydrobiology and water chemistry, a simple algae-dissolved reactive phosphorus model has been constructed to describe the eutrophication process. The model is mainly based on the specific condition that in Lake Balaton the main direct source of phosphorus is the uptake from the the sediment with the indirect source being the input phosphorus load. The crucial point of model ling is the determination of the time varying (or rather varying as a function of meteorological factors) sediment phosphorus uptake coefficient. Consequently a special parameter estimation technique was required. The dynamics of the model are nonlinear. On the other hand the measure ment equation is linear with a white noise Gaussian measurement error sequence. To identify the unknown constants and the time varying parameters, the output error method was adopted. A recursive algorithm based on the maximum likelihood approach is developed for determining the unknown parameters and noise statistics. The algorithm utilizes the sensitivity matrix equation for the parameters. Un modèle simple d'eutrophisation pour le baie de Keszthely, lac Balaton Résumé. Le lac Balaton et surtout sa baie à Keszthely est soumis à un danger d'eutrophisation. Un modèle simple algues-phosphore réactif dissous a été construit sur la base des données collectées sur l'hydrologie, l'hydrobiologie, la chimie des eaux et sur les études existantes sur les charges en nutrient du lac. Le modèle tient compte de la circonstance spéciale que le phosphore disponible pour les algues provient premièrement et directement des sédiments du fond et indirectement de la charge extérieure. Le point crucial de la modélisation est la détermination du coefficient de l'absorption du phosphore issu du fond, qui varie selon le temps (ou bien en fonction des paramètres météorologiques). On a eu besoin donc d'une technique spéciale pour l'estimation des paramètres. Le modèle est dynamique et non linéaire, tandis que l'équation de mesure (des variables d'état) est linéaire avec une série d'erreurs de bruit blanc dont la distribution est normale. L'identification des constantes inconnues et des paramètres variables dans le temps a été effectuée par la méthode de l'erreur 'output'. Un algorithme récursif a été élaboré sur la base de la méthode maximum-vraisemblance pour l'estimation des paramètres inconnus et de la statistique de bruit. L'algorithme utilise l'équation matricielle de sensibilité pour les paramètres. INTRODUCTION In spite of the fact that according to the relevant literature the mathematical model ling of the eutrophication process is in a fairly well developed stage, at least as far as the theoretical approaches are concerned, the practical application of these models raises many additional problems. These can be divided into two main groups: (1) The desired level of model complexity is restricted by the available data base. Consequently where reliable records for some of the elements of nutrient trans port or the food chain are not available then at this point the modelling becomes only a mathematical game. (2) The next crucial point is the determination of model parameters. The adaptation of literature values or the estimation of the ranges of parameter values can be decided only with the possession of widespread knowledge on hydrological, hydro- biological and chemical processes taking place in the lake (or part of the lake) in question. 137 138 G. Jolankai and A. Szôllosi-Nagy Based on their former water quality modelling experiences the authors believe, contrary to the opinion of many others, that the main model parameters can only be determined with reasonable accuracy from the use of actual field data. There are two possible ways of model parameter determination: (1) To carry out objective oriented field experiments directly designed to measure the effects of the individual parameters, and (2) parameter estimation techniques with the utilization of available measure ment data. Using hydrological, nutrient input, chemical and hydrobiological data available for the Bay of Keszthely, which is the most endangered part of Lake Balaton, and also the results of earlier experimental and research work, a simple eutrophication model will be presented below. The basic principles of the model formulation were: The theoretical model selected to describe the main processes of nutrient transport and algae growth is calibrated by up-to-date parameter estimation techniques using the available records of about seven years in length, in such a manner that at a later model ling stage the dependence of model parameters on other parameters could also be taken into consideration. The problem setting Lake Balaton, and especially its Bay at Keszthely, is facing the problem of eutro phication. To control this process first some basic questions should be answered: (1) What are the causes of this process, that is (a) what parameter controls this process, in other words what is the limiting factor in the eutrophication process, and (b) what are the sources of this pollution? (2) What are the main processes of eutrophication the modelling of which could be reasonably well performed using the available data base? With regard to the first question,most authors (see references) agree that the eutro phication process of Lake Balaton is phosphorus limited. The sources and extent of phosphorus mass flux into the lake have been discussed elsewhere (Jolankai, 1977). The main phosphorus source of the lake and especially of the Bay at Keszthely is its tributary the River Zala (see Fig. 1). Since the lake receives about 40 per cent of the pollution via the Zala River that discharges into Keszthely Bay it is obvious that (1) Keszthely Bay is in the worst stage of degradation (highest eutrophic level) and (2) the eutrophication control of this Bay is a task of primary importance. Consequently the modelling of the eutrophication process in Keszthely Bay can be considered as the first necessary step. (Other parts of the lake still have water of fairly good quality.) Concerning the second question, a true description of the eutrophication process would include the modelling of the highly complex relationships of the aquatic ecosystem. For this purpose several models of different complexity levels are offered by the relevant literature starting with simple phosphorus-algae models and ending with multi parameter ecological models that include almost all aquatic life forms and water quality parameters (Park et al, 1974; Orlob and Chen, 1972; J^rgensen, 1975). However, from the practical point of view a reasonably applicable model should be based on the available data base and should at the same time focus on the main pro cesses. Though the results of several individual measurements and field experiments carried out in Keszthely Bay are available, sufficiently continuous and lengthy records A simple eutrophication model for the Bay of Keszthely 139 FIGURE 1. Lake Balaton and the Bay of Keszthely. are fairly rare. However, records on algal biomass and various phosphorus forms are available. The characteristic features of the eutrophication process in Keszthely Bay are briefly described below, based on the very exhaustive work of Olâh et al. (1977) on the phosphorus transport processes of Lake Balaton. (1) Contrary to most of the other lakes cited in the relevant literature, dissolved reactive phosphorus (P04-P) in Lake Balaton has a 'bypassing' way of transport from its original source (inputs) towards the utilizing algae, namely : about 72 per cent of the received amount of P04-P load immediately precipitates in water that has a high CaCO3 content and gets adsorbed in the bottom sediment. Consequently the main direct source of phosphorus is the exchange process through the sediment—water interface. (2) The concentration of P04-P in the water has not changed significantly in the past and maintains an equilibrium level of about 0.003 mg/1. However, in the last few years higher concentrations have been observed in Keszthely Bay and the equilibrium concentration is much higher. In recent years, most likely due to the greater availability of phosphorus, algal growth has significantly increased and frequent algal blooms have been observed in Keszthely Bay. The only obvious cause of these apparent changes is the increasing phosphorus load. The main task is then to describe the interrelationships between phosphorus load and algal growth by an appropriate mathematical model, the use of which could enable planners to define the probable effects of eutrophication control measures. THE NONLINEAR DYNAMIC LAKE MODEL Based on the foregoing considerations in respect of the phosphorus transport mechanism of the lake the simplest possible model that could describe the real situation with reasonable accuracy is a phosphorus-algae model that takes the sediment-water phosphorus exchange into consideration. However, contrary to the common practice of modelling the total phosphorus mechanism, the mechanism of the dissolved reactive phosphorus (P04-P) regime should be modelled, since according to experimental results in Hungary (Dobolyi and Ôrdôg, 1978), this is the only phosphorus form available for algae. At the present level of data availability the modelling procedure should end with the algae since only scattered zooplankton data are available. It also follows from the above statement that only a lumped algal loss coefficient can be used at this end of the modelling approach. 140 G. Jolànkai and A. Szôllosi-Nagy FIGURE 2. The model relationships. The model includes the following processes (Fig. 2): (1) Phosphorus precipitation and settling (since in this specific situation the precipitation of P04-P is a fairly constant and immediate process, the model considers this phenomenon with a preliminary subdivision of the phosphorus input load values).