The environmental protection of water bodies in Europe is based on the Water Framework Directive, which combines the so called Emission Limits Value and the Water Quality Objective (QO) approaches. The first one sets limits to particular type of emissions, for example the Nitrate Directive, while the second establishes Quality Standards for Biological, Chemical and Hydromorphological Quality Elements, in order to ensure the functioning of freshwater and marine ecosystem and the sustainable use of water bodies. To this regard, mathematical models are valuable tools for reconciliating these approaches, since they allow one to establish a causal link between emission levels and the Quality Standards ("direct problem") and vice-versa ("inverse problem").
In general, Quality Elements are variables or proper combination of variables which define the "status" of a water body. For example, the "chemical status" can be defined by a set of concentrations of chemicals which are potentially harmful for the ecosystem and humans, or the biological status may be based on Quality Elements which include the density of phytoplankton, the presence/absence of Submerged Aquatic Vegetation, the presence/absence of sensitive species etc. In many instances, the Quality Standards can then be expressed as threshold values, below or above which the functioning of the ecosystem is compromised and/or the risk for human health is not acceptable. If this is the case, management policies should be aimed at improving the state of the system and meet those Standards in the near future. In order to be carried in a cost-effective manner, such interventions should be based on a quantitative understanding of the relationships between the Pressures on the system and its State. This task could be very complex in large water bodies, where transport processes play a major role in creating marked gradients and pollution sources may be spatially distributed and/or not well identified. From the scientific point of view, the problem can be stated as follows: a mathematical model should enable one to "map" the spatial distribution of inputs (emissions) into the spatial distribution of the requested output, namely the "indicator" or "metric", which is subjected to a given constraint, the Quality Standard (QS), within the computational dominion. Such analysis may reveal that the QS are not respected only in a given fraction of the water body and, in the most favorable circumstances, identify the pollution sources which cause the problem. In such a case, a selective intervention, aimed at lowering the emission levels of those sources, would probably be more cost effective than the general reduction of the emission levels in the whole area. The spatial distribution of emission sources may also affect the pollution level and, in some instances, a proper redistribution of those sources in a given area, which leaves unchanged the total load, could have positive effect on the pollution level.
In this paper, we are going to investigate the above problems in the simplest possible setting, in order to provide a clear interpretation of the results in relation to the most relevant parameters. The paper is organized as follows: in the "methods" section, we present the basic equations and provide insights for solving the problem in the general case as well as in the specific one here presented. The analytical solutions are presented and discussed in the next two sessions and some concluding remarks are then summarized in the conclusive section
