Dynamic modelling of metals – Time scales and target loads

Over the past decade steady-state methods have been developed to assess critical loads of metals avoiding long-term risks in view of food quality and eco-toxicological effects on organisms in soils and surface waters. However, dynamic models are needed to estimate the times involved in attaining a certain chemical state in response to input (deposition, fertilizers or manure) scenarios. Starting from a mass balance, a universal dynamic model was developed by defining appropriate dimensionless quantities, which depend only on the metal under consideration. For any given metal, the model (differential equation) is characterised by the interplay of four (dimensionless) variables: the initial condition, i.e. the concentration at the start of the simulation, the input (driving force), time, and the concentration of the metal at any given point in time. Depending on the question asked, one of these quantities is fixed and the functional relationship between the other three provides the answer.The model allows to investigate the time development of the soil chemical status under a constant future input of the metal to predict (i) the future metal concentration as a function of time (scenario analysis), (ii) the time when a prescribed chemical state (e.g., a critical concentration or steady state) is reached (delay times), and (iii) which future input (reduction) is needed to reach a prescribed chemical state within a prescribed time period (target loads). The general solutions are illustrated with concrete examples, using (average) data from the Netherlands for four metals: cadmium, lead, copper and zinc. The modelling approach set out in this paper illustrates the potential use of dynamic models in the support of policies aimed at reducing emissions of metals by providing an understanding of the structural properties of the model, independent of site-specific parameters. It thus allows assessing temporal behaviour and time scales before embarking on detailed modelling for individual sites.