Modelling Radioactivity in the Environment, Volume 4 (Radioactivity in the Environment)

MODELLING RADIOACTIVITY IN THE ENVIRONMENT

RADIOACTIVITY IN THE ENVIRONMENT A companion series to the Journal of Environmental Radioactivity Series Editor M.S. Baxter Ampfield House Clachan Seil Argyll, Scotland, UK Volume 1: Plutonium in the Environment (A. Kudo, Editor) Volume 2: Interactions of Microorganisms with Radionuclides (F.R. Livens and M. Keith-Roach, Editors) Volume 3: Radioactive Fallout after Nuclear Explosions and Accidents (Yu.A. Izrael, Author) Volume 4: Modelling Radioactivity in the Environment (E.M. Scott, Editor)

MODELLING RADIOACTIVITY IN THE ENVIRONMENT

Editor

E. Marian Scott Department of Statistics, University of Glasgow, Glasgow, G12 8QW, Scotland, UK

2003 AMSTERDAM – BOSTON – LONDON – NEW YORK – OXFORD – PARIS SAN DIEGO – SAN FRANCISCO – SINGAPORE – SYDNEY – TOKYO

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Contents

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Section 1: Models and modelling 1. Design and evaluation of environmental radioactivity models by E. M. Scott .

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2. Modelling the dispersion of radionuclides in the atmosphere by A. Mayall . .

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3. Modelling radioactivity in the marine environment: The application of hydrodynamic circulation models for simulating oceanic dispersion of radioactivity by I. H. Harms, M. J. Karcher & H. Burchard . . . . . . . . . . . . . . . . . .

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4. Radioactivity in the terrestrial environment by G. Pröhl . . . . . . . . . . . . .

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5. Modelling radionuclide transport in the geological environment: a case study from the field of radioactive waste disposal by W. R. Alexander, P. A. Smith & I. G. McKinley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6. Radioactivity in lakes and rivers by L. Håkanson & L. Monte . . . . . . . . .

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7. Radioactivity in aquatic biota by T. G. Sazykina . . . . . . . . . . . . . . . . .

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Section 2: Modelling applications 8. Modelling the dispersion of radionuclides in estuarine environments: the Odiel river estuary by R. Periáñez & J. M. Abril . . . . . . . . . . . . . . . . . . . .

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9. Predicting transfer of radionuclides: soil–plant–animal by N. Crout, N. Beresford & A. Sanchez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10. Pathways of anthropogenic radioactivity in the Northern Oceans by I. H. Harms & M. J. Karcher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

11. Modelling the behaviour of radiocaesium in forest ecosystems by G. Shaw, R. Avila, S. Fesenko, A. Dvornik & T. Zhuchenko . . . . . . . . . . . . . . .

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12. Mathematical and numerical approaches in performance assessment for radioactive waste disposal: dealing with uncertainty by J. C. Helton . . . . . . .

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13. The Chernobyl accident and aquatic biota by I. I. Kryshev, T. G. Sazykina & A. Kryshev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Foreword E. Marian Scott

Why is a book on modelling environmental radioactivity necessary? There are many reasons why such a book is necessary, perhaps the most important being that: (a) modelling is an often misunderstood and maligned activity and we hope that this book can provide, to a broad audience, a greater understanding of modelling power but also of some of the limitations, (b) modellers and experimentalists often do not understand and mistrust each other’s work yet they are mutually dependent, in the sense that good experimental science can direct good modelling work and vice-versa; we hope that this book can dispel mistrust and engender improved understanding, (c) there is an increasing reliance on model results in environmental management, yet there is also often misuse and misrepresentation of these results. We hope that this book can help to bridge the gap between unrealistic expectations of model power and the realisation of what is possible, practicable and feasible in modelling of environmental radioactivity; and finally, (d) modelling tools, capacity and power have increased many-fold in a relatively short period of time. Much of this is due to the much-heralded computer revolution, but much is also due to better science. It is useful to consider what gap if any still remains between what is possible and what is necessary. Mathematical models are general tools that synthesise observational and experimental data, allow predictions of behaviour, and provide input to decision-making processes in their field of application. As such, they are simplifications of a complex reality and are imprecise, with many sources of uncertainty. Models are developed to approximate or mimic systems and processes of different natures (e.g. physical, environmental, social or economical), and are of varying complexity. Many processes are so complicated that physical experimentation is too time-consuming, too expensive or simply impossible. As a result, investigators require models to simulate and explore these processes. Mathematical models for radioactivity in the environment are no different from the general models. They too are needed and used to describe the behaviour of a radionuclide within a vii

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possibly complex environmental system. They too may include physico-chemical descriptions of the processes (and rates), are used to test hypotheses concerning environmental behaviour and to evaluate impact assessment in response to anthropogenic induced change (either real or hypothetical) in a quantitative and objective manner. They also form part of radiological assessments by providing predictions of concentrations of radionuclides in the foodchain and ultimately, through modelling ingestion and inhalation, to prediction of dose and biological harm. In the development of the model, the modeller must work with an imperfect and incomplete description of the physical and biological system. Mathematical models incorporate factors and parameters aimed at characterising the processes being investigated and are commonly composed of a series of linked equations (often differential). The modeller must select features and processes to be parameterised, he must chose how to derive the necessary equations and thus must synthesise sometimes conflicting evidence. In the design of the model structure, the modeller must accept that there is no correct model and that acceptable predictions are possible from quite different models. He must also accept that apparently small changes may generate large effects. These make model creation something akin to an art although founded in science and impose a limit on the confidence that a user has in the model. Further, models and modellers must cope with natural intrinsic variability and imprecise observation. They must assess and accommodate the uncertainties associated with the natural processes, decide how best to incorporate them into the models, how to deal with imprecision in measurement and how to present the complete package to decision-makers and managers.

1. About this book Just as an environmental model typically will be composed of a number of linked sub-models, representing physical, chemical or biological processes understood to varying degrees, this volume includes a series of linked chapters exemplifying the fundamental nature of environmental radioactivity models in all compartments of the environment. A classification scheme for the models would conclude that most of the models are deterministic, some are mechanistic and a few are dynamic and stochastic. The model domain of applicability ranges over the short term (days) to the long term (thousands of years) (as in waste repository modelling, Chapters 5 and 12) and from the near field (a few kilometres) to the far field (many thousands of kilometres) (as in general ocean modelling (Chapters 3 and 10)). Models have been used, for example, to predict levels of Cs-137 on pasture over several seasonal cycles following the Chernobyl power plant accident (see Chapters 4 and 9); to model the global distribution of Chernobyl fallout in lakes and rivers (Chapters 6 and 7), to model technologically enhanced natural radioactivity (discharge from phosphate processing in estuaries in Chapter 8) and to model the dispersal of a radioactive pollutant discharged into the Kara Sea from dumped nuclear waste (see Chapter 10). The model, depending on its purpose and field of application, will include many different processes, the sub-processes of which may operate at quite different time and space scales and for the modeller there will be a choice of which processes to model explicitly and how to parameterise these processes. The chapters in this book present the diversity and commonality of modelling and throw some light, we hope, on the processes

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involved in moving from problem (hence purpose) through scientific knowledge to conceptual solution to ultimately practical and useful modelling tool. In this book, the different authors were asked to describe models within a given environmental compartment (atmosphere, oceans, lakes and rivers, and the terrestrial (and geological) environments). They were asked to bring together and illustrate the model development processes in these compartments, as well as to describe the current modelling practices. Finally, they were then asked to illustrate and expand on the more methodological work by describing and exploring a number of case studies, i.e. to show the use and practicality of the model in the real world. The case studies chapters are extensive, varied and detailed, allowing a breadth of material dealing with issues including scale of model (local/global, over short/long times), complexity (1D–3D), validation and parameter estimation, sensitivity and uncertainty. Many are linked specifically to a methodological chapter and there are many planned and unplanned connections between the chapters. The applications have been drawn from some of the most significant environmental radioactivity issues of recent times. We hope that they encapsulate the best current practices and look forward to some of the new directions. In its entirety, the book summarises the current state of the radioactivity modelling art and science and presents a series of extended case studies concentrating on different environments, different objectives and different approaches.

Acknowledgements My thanks are due to a number of people. First, to Murdoch Baxter, the series editor and my friend, for continual encouragement, advice and assistance. It is doubtful if I would have embarked on this task without my experience as an associate editor of the Journal of Environmental Radioactivity under his editorship. Secondly, to all the authors for all their hard work in putting together these chapters; without them, there would be no book. All graciously agreed to commit their time and effort and although the gestation period was longer than originally envisaged, I hope that they consider the finished product to have been worthwhile and worth the wait.

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MODELLING RADIOACTIVITY IN THE ENVIRONMENT E. Marian Scott (Editor) © 2003 Elsevier Science Ltd. All rights reserved

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Chapter 1

Design and evaluation of environmental radioactivity models E. Marian Scott Dept. of Statistics, University of Glasgow, Glasgow G12 8QW, UK

1. The art of modelling 1.1. Why model the environment? In a general setting, the objectives of any environmental study that involves modelling can be broadly classified as: • Description and characterisation of the state of the environment; this may require a simple underlying model, such as a mathematical description of the distribution of the characteristic of interest (or some simple numerical summaries), or more complex models, based on physical descriptions and incorporating both space and time effects to be used for prediction. These more complex models are usually based on series of linked differential equations. • Prediction of the condition or state of the environment as a result of an activity (which may include doing nothing). The model may be used in the process of deciding which action to undertake (e.g. in an accident situation where there is only limited data available in the early stages, a model can be used to predict likely concentrations and whether they will exceed action levels or derived limits). • Detection of change and design and assessment of monitoring (within both a space and time framework); in this context models are often used in a regulatory situation, to demonstrate compliance with standards which are set to protect man, the biota and the environment. 1.2. What is a good model? A good model can be described in many ways; terminology commonly and loosely used includes; it should be simple (parsimonious), realistic, efficient, valid, reliable, accurate, precise and reproducible. However, demonstration that the model is ‘good’ is more difficult than its definition since there are many aspects of model goodness. The terms ‘good model’ or ‘better model’ imply a value judgement, based on the output of a model evaluation. The evaluation is 1

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carried out, to a greater or lesser extent, based on a number of different criteria, not necessarily of equal importance and not necessarily all easily quantified, and is a process which has a role to play in every stage of model development and construction when decisions need to be made about how to proceed. Calibration and validation are two important model procedures that are part of the evaluation process. In calibration, we attempt to find parameter estimates that provide acceptable agreement between model predictions and a training data set, while one of many definitions of validation is that we aim to demonstrate the similarity between the model predictions and an independent test data set. However, there continues to be some debate about model validity, the terminology used and its meaningfulness in environmental modelling (Oreskes et al., 1994). One further criterion of particular interest and one often used in environmental radioactivity modelling is the reliability of the model. The reliability of the model can be defined by the answers to three questions (IAEA, 1989): • How large is the uncertainty associated with the model predictions? • Is the uncertainty acceptable for decision making? • What are the main factors contributing to the uncertainties? These questions emphasise the importance of uncertainty analysis, which is picked up in a number of the chapters in this volume and which is discussed further in Section 5. Factors affecting model reliability include how the problem was specified, the conceptual framework of the model (were important processes omitted, was a process inadequately described, are the model assumptions satisfied, is the model credible?), the formulation of the computational model (errors in software and codes, do the model predictions seem reasonable?) and estimation of parameter values (do the predictions seem reasonable?) and finally how uncertain are the results? This is perhaps the most important of the model evaluation phases, since the modeller must communicate confidence in and understanding of the model output to the decision-maker or stakeholder. Model complexity is also linked to model reliability in that more and more detailed science may not always lead to a better model (at least in terms of less uncertain predictions).

2. Model building processes The model building process has three main stages: design and conceptualisation, estimation (or calibration, model fitting) and finally verification (or validation, corroboration). It is often iterative. In the design and conceptualisation process, visualisation of structure and choice of environmental processes and parameterisation systems to use are included. This is then followed by creation of a mathematical model that captures the structure and processes in a series of mathematical equations. Finally at this stage, the mathematical model must in most cases be translated into a computational model. In the estimation stage are included those processes described as calibration and model fitting. Finally in the verification stage, the modeller checks how the model behaves and how model output relates to any field and laboratory experimental results. Based on this third stage, the modeller may decide to return to the model design and make changes. Each stage depends on different information sources; expert judgement may be used to design and conceptualise the model, on-site (site- and nuclide-specific or stable analogue

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element) experiments can often provide information useful in parameter estimation and model verification. At each stage, however, decisions concerning structure and input must be made and at each stage uncertainties will be introduced. 2.1. Design and conceptualisation The art of model-building is perhaps a slightly loose way of describing the steps taken in creating a model to contribute to the solution of a problem. Model-building is a creative activity, bringing together experience, experiment and observation. As mentioned in the introduction, the conceptual uncertainties in this process reflect the different experiences, expertise and perspectives that a modeller brings to the problem. Intrinsically, within any particular problem, the modeller may identify a series or class of models based on knowledge concerning the theory and also the practice of the application. This identification process may also be assisted by observational or experimental data. The model must be fit for the purpose for which it will be used, and given the diversity of purposes, it is clear that different models may be necessary and that different purposes may require different levels of model complexity. Visualisation of the model structure and identification of the processes, variables and parameters utilises current scientific knowledge. By parameter, we simply mean an unknown quantity, e.g. the rate of transfer of a radionuclide from soil to root zone in a model describing radionuclide uptake by plants. Generic values are often used in the absence of site-specific information. However, the parameter value may be tuned for specific nuclides, for specific soil types and for specific plants. These might be described as systematic factors determining the appropriate choice of value, but there may also be a stochastic or random component that is due to natural variability in the plant root system or in individual members of a population. By variable or factor we mean one of the attributes of interest, which could be the soil density or the water content which we consider is related to the radionuclide concentration in some way. We might include the description of such a relationship by describing a physical process (a sub-model) or simply including an unknown parameter or rate-constant. This is the key stage in the design process and is carried out in tandem with the review and selection of physical processes to be included. This also determines the complexity of the model, since more processes introduce extra complexity. Obviously, the definition of the purpose of the model drives the selection of processes and the conceptualisation and scientific knowledge also determine the connections that are included. From this description, it is clear that many conceptual structures are possible and may be supported by the same scientific evidence. Part of the conceptualisation process also concerns the evaluation of data resources, scientific knowledge and the synthesis of information of many different types and sources. From a statistical perspective, the goal is to find the simplest model that satisfactorily explains the experimental results, while, from a mechanistic perspective, the goal would be to include our science base. These two perspectives may be in conflict. Therefore the conceptualisation stage is crucial. Following the definition of the conceptual model comes mathematical formulation and then ultimately the computational representation and numerical solution. At issue is the choice of mathematical methods for the solution of any equations and the checking of any code so

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generated. This stage is, however, of less interest within the remit of this particular book and we focus on conceptualisation and model choice. 2.2. Parameter estimation (calibration and model fitting) Model calibration is the process whereby any unknown parameter values are estimated. Difficulties encountered here include the relative paucity of experimental data, with more complex models often requiring extensive data sets. There may be significant mis-matching between the scale of laboratory and field data in terms of spatial and temporal relevance to the model domain. The sources of information can be many. In some situations, there will be sufficient experimental data available to estimate all the parameters; however, it should be borne in mind that later new data will be required for model testing. Such information can be supplemented by expert opinion, review of the literature or comparison with analogues. Parameter identifiability is often also an issue, whereby we may have identified a model structure with its unknown parameters which require to be estimated, but we may only have limited data such that it is not possible to estimate unambiguously all the parameters, resulting in confounding. It is clear that in this stage especially, the modeller and experimentalist are mutually dependent.

3. Model assessment and evaluation It is evident that there may be many competing conceptual models of differing levels of complexity and one issue facing the modeller is model selection. It has been said that ‘no model is correct but some are more appropriate than others’; thus how do we assess the value of the models which are available and decide which if any to use? Credibility, reliability and similarity with observations are some of the many criteria which may be applied. But beyond these, modellers and model users must also consider the uncertainties implicitly and explicitly associated with the model of choice. Uncertainty stems from a number of sources including the parameters but also the model structure. Reliance on the results from a single model in an impact assessment is likely to leave the decision-maker with a significant over-estimate of the precision of the results and hence with an unrealistically narrow view of the plausible outcomes. Regardless of terminology, with complex models, immediate difficulties become apparent in model evaluation; the data requirements of environmental models are often large and the requirement for independent training and test data sets is one which may not be realised. The modeller must demonstrate that the model is appropriate for its purpose by showing that the model will reproduce, to an acceptable standard, the real world as observed. Whose definition of acceptable should be used? The terms verification and validation are frequently used and mis-used for this process and much has been written about the processes involved in model validation (Sheng et al., 1993; Caswell, 1976). In Oreskes’ (see Oreskes et al., 1994) terminology, verification means the establishment of truth, while validation refers to the establishment of legitimacy. Oreskes criticises the use of both terms. Her main argument against the act of verifying a model is that natural systems are never closed, and thus absolute verification, in the sense of asserting the

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“truth” of a model, is impossible. Model corroboration can only be attempted once the purpose of the model has been defined and it is worth noting that a model always agrees better with the particular set of data used for its calibration (Carrera et al., 1993; Chatfield, 1993).

4. Sources of information in the modelling process There is a wide variety of models that may be adopted in any given setting. The choice of model is determined by available information, knowledge and applicability to the particular context in which the project is set. All will involve combining available environmental information of different provenance and this also introduces further challenges in that the information may have been collected under very different protocols and may have widely varying properties (Cox & Piegorsch, 1996). The different sources of information include: (a) expert judgement; information based on prior knowledge, expert judgement and the environmental context within which the problem is posed all must be combined. The elicitation of expert judgement can prove difficult, and there is considerable interest in and literature about the elicitation of prior information, (b) comparison with analogues; other nuclides or stable elements may be assessed to provide information which may be applicable to the specific project. Difficulties arise here since it may be difficult to find analogues, (c) experiments; site- or process-specific experiments may be carried out either in the laboratory or in the field. This may introduce further difficulties in that the space and time scale of the experiment may not match the model, and thus the modeller must up- or down-scale and interpolate the data. A modeller uses data to aid model design and selection, in parameter estimation and in model evaluation. An experimentalist could use the model to define and test hypotheses and to highlight experiments which should be performed to improve knowledge of the system. The suitability of the data depends on the correspondence with the system being modelled and on agreement amongst the various data sources. A number of common problems are frequently encountered including data requirements far exceeding existing sources, available data not matching the conditions for which model has been developed, which includes spatial and temporal mis-matching, and observations being subject to uncertainty due in part to natural variability. Lack of observations contributes to uncertainties in input data and to parameter uncertainties, while conflicting evidence contributes to uncertainty about the model so, taken together, the implications for modelling are inexactness and uncertainty. Nonetheless, the uncertainty implicit and explicit within the data is an important issue in model testing; it sets a limit on what is defined as acceptable in terms of model performance.

5. Issues of uncertainty Uncertainty is not simply about lack of knowledge. It covers many issues such as; uncertainty about model quantities, uncertainty about model form, uncertainty about model completeness;

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but is also about natural variation and imprecision in the data and model quantities. How uncertainty is handled is and remains a key modelling issue (Granger Morgan & Henrion, 1990). Uncertainty has important implications for modelling; more than one model may be fit for the purpose; it may result in incompleteness due to processes having been omitted in the model design, and it may simply reflect the fact that knowledge is lacking. The uncertainty in the model prediction has an important role to play in model usage and is something which cannot and should not be ignored. As modellers, we have perhaps all been a little complacent about evaluating our uncertainties and although users may now be clamouring for such assessments, there are still problems about how the assessment is used in the decision-making process. Environmental assessment needs ‘usage of prediction techniques, which yield a range of predictions and associated probabilities for those predictions to occur’ (VROM, 1985). Assessment of risks and benefits and decisions concerning the impact must take account of the possible outcomes and their likelihoods. The key environmental management question is whether the uncertainty can be managed within defined acceptable limits and whether, through a sensitivity study, we can identify steps to be taken to reduce the uncertainties and limit the possible predicted outcomes? 5.1. Sources of uncertainty Uncertainty can enter at all stages of the modelling process (VROM, 1985), namely in (a) the description of an effect. In the conceptualisation, the modeller must make decisions about how a particular effect will be modelled and this has implications for the information required and how it will be collected, (b) data collection (through sampling and monitoring). In the environmental context, it has become increasingly clear that there may be substantial ‘natural’ variability, often greater than any measurement uncertainty. The combined effect of these two sources of uncertainty may be substantial and should not be ignored in any model testing or impact auditing, (c) the predictive methods themselves. Environmental models cannot reproduce exactly what is observed; they are simplifications of complex systems. Different types of uncertainty may be introduced in the modelling work; through omission of certain processes, the model may be over-simplified; the mathematical description of the process may be subject to uncertainty (i.e. our knowledge is limited); there may be resolution uncertainty (i.e. the model may have one level of space or time resolution while the data collected may be at another), and there may be different user interpretations (Kirchner et al., 1997). Although some of these uncertainties are irreducible, it is still necessary that the overall uncertainty be evaluated. There is a fundamental indeterminacy in much of environmental modelling, in that more than one model may be fit for the stated purpose, and with the data available it may not be possible to select between the alternatives. Thus a range of predictions from the different models provides one possible measure of this aspect of uncertainty (Scott et al., 1997; Kirchner et al., 1997). Model testing and validation must take into account the uncertainties introduced in the preceeding stages (McKay, 1995). Sensitivity analysis may also show which of the factors and parameters make a substantial contribution to the model output.

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5.2. Model uncertainty and sensitivity Traditionally, sensitivity and uncertainty analyses in environmental modelling contribute quantitative descriptions of the relative importance of individual parameters and processes, highlighting areas of significant contributors to the overall uncertainties of model predictions and giving markers for areas requiring substantial improvement (perhaps through directed experimental work). Sensitivity analysis (SA) is a process whereby the effects of varying one or more input factors (or parameters) on the model output are assessed. Uncertainty analysis (UA) is a process whereby the uncertainties on ‘all’ components are propagated through the model to give a quantified uncertainty on the model output. Altogether, to increase the confidence in the model and its predictions, an understanding of how its outputs respond to changes in its inputs, be they data, model structures or parameters, is of fundamental importance. This understanding is achieved by sensitivity and uncertainty analysis. SA and UA are closely linked with UA aiming to quantify the overall uncertainty associated with the response as a result of uncertainties in the model input (Saltelli & Scott, 1997; Scott, 1996, 1997). Modellers conduct SA and UA to determine (a) if and how a model resembles the system or processes under study, (b) the factors that mostly contribute to the output variability and that require additional research to reduce the variability, (c) the model parameters (or parts of the model itself) that are not influential in determining the output, and which can therefore be eliminated from the final model. Both sensitivity and uncertainty analyses require extensive re-sampling from distributions which define the properties of the input data and simulation of model response and there is a large and growing literature concerning their use in environmental modelling (Hamby, 1995; Scott et al., 1997). Sensitivity and uncertainty analyses contribute to all stages of model development, testing and assessment, including assessment of model reliability and validity. Sensitivity analysis encourages the exploration of the interactions between the various modelled processes and helps throw light on the properties of complex computational models by, in its simplest form, perturbing one parameter at a time and studying its effect on the response. For large computer models, with perhaps hundreds of parameters, sensitivity analysis may involve perturbing all parameters simultaneously, but in such a way that the main effects of individual parameters and their interactions can be estimated. Uncertainty analysis contributes estimates of uncertainties on the final predictions by propagating through the model all quantified sources of uncertainty. Thus, sensitivity analysis identifies the key contributors to uncertainties, while uncertainty analysis quantifies the overall uncertainty, so that together they contribute to a reliability assessment for the model. 5.3. Sensitivity analysis in detail Sensitivity analysis involves determination of the amount and kind of change produced in a given system parameter by a change in another parameter. It is a tool in the investigation of factors which are important or influential in determining the system response. It is an important process useful in the model design stage (Saltelli et al., 2000). To perform a sampling

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based sensitivity analysis (or Monte Carlo analysis), it is necessary first to identify the parameters of interest, define, for each, a probability density function (p.d.f.) to reflect the belief concerning the possible values within its possible range, account for dependencies amongst the parameters and propagate the variation in parameter values through the model to generate a p.d.f. of predicted values which can then be analysed. Definition of the parameter p.d.f. can often prove difficult. Occasionally there may be experimental information available which would allow empirically based estimation of the p.d.f., but more commonly there may be little information and the p.d.f. then fundamentally reflects the modeller’s belief. The elicitation of belief-based p.d.f.s in environmental contexts is a challenging task, but is one which if successfully achieved raises the possibility of directly incorporating expert opinion into the formal analysis through Bayesian methodology (Cooke, 1994a, b; EC, 1995). Given the p.d.f., there is a number of widely used methods of sampling, including simple random sampling (SRS) (parameter values chosen at random from the p.d.f.) and Latin hypercube sampling (LHS) (the range of each parameter is partitioned into n intervals and one value is selected from each interval of each parameter). SRS is simple, reliable, easy to analyse but inefficient. LHS is basically a stratified sampling procedure and is generally more efficient than SRS (Andres, 1987, 1997; Iman & Conover, 1980). However, parametric uncertainty reflects only one (and perhaps not the dominant) source of uncertainty. The other key contributor is uncertainty about the model structure itself. 5.4. Uncertainty analysis in detail Under the broad term uncertainty is included: structural uncertainty about the model form, parameter uncertainty and scenario uncertainty (future conditions). In the development of the model, the modeller must work with an imperfect and incomplete description of the physical system so he is uncertain. He must select features and processes to be parameterised, he must synthesise sometimes conflicting evidence and he must prioritise, so he is uncertain. Any unknown parameters must be defined. The parameters may have associated with them large uncertainties, arising from two sources; natural stochastic variation and lack of knowledge on the modeller’s behalf (Hoffman & Hammonds, 1994). Model or structural uncertainty reflects the facts that an acceptable model prediction may be achieved in many different ways and that the modeller must use judgement to decide on the structure of the model. In an attempt to explicitly consider this source of uncertainty in the predictions (particularly important when the model is being used as a management decision tool), a number of problems must first be resolved. It may be more difficult to quantify the uncertainty in the model structure when compared to a single parameter. Models may differ in terms of the physical, chemical or biological processes included; they may have different state spaces; they may have been constructed using a different knowledge base; the complexity of the models may thus be different, yet all may be valid but still provide different responses. The issue of how uncertainty about model structure can be quantified and incorporated in a standard sensitivity and uncertainty analysis is an important one which is yet to be fully addressed. There is, however, growing interest in the sensitivity of predictions to model structure and in evaluating the contribution of model structural uncertainty to the overall uncertainty within the general framework of sensitivity analysis (Draper, 1995; Beven,

Design and evaluation of environmental radioactivity models

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1993; Beven & Binley, 1992; Scott, 1997). The issue of structural uncertainty is one that can be addressed using a Bayesian formulation, within which, it is possible to elicit and make use of expert beliefs. 5.4.1. Bayesian methods Bayesian methods provide a formal mathematical framework within which expert opinion and judgement can be incorporated into uncertainty analysis. One simple and practical empirical approach to assessing structural uncertainty which has been used in model benchmarking (such as IAEA VAMP and IAEA IASAP programmes) supposes that a discrete set of models, say m (Draper, 1995) exists. Let x denote the data and y the quantity to predict, then using Bayes Theorem, we can describe the distribution of the output y, given the input parameters x and model structure as p(y|x, S) =



p(Si |x)p(y|x, Si),

(1)

where Si , i = 1, . . . , m denotes the different structural alternatives or models and p(A|B) represents the conditional probability of A given B. The analysis presented in equation (1) allows, through the application of Bayes Theorem, the distribution of the predicted quantity to be defined in terms of prior beliefs p(Si |x) for the model structure given the data and the predictive distribution for y given the model structure Si . However, the prior distributions p(Si |x) must be defined. There is a substantial literature on the elicitation of priors from experts (Cooke, 1994a, b) but the approach must be carefully designed and there are a number of difficulties. The role of model validation/verification in this process is an important one, and if we are able to define an indicator (‘degree of model support provided by the observational data’), then in equation (1) above, the p(Si |x) can be empirically evaluated. In this way, a more robust and hopefully reliable set of predictions can be achieved. In addition, in the purest sense, the variability in the model predictions provides a direct measure of uncertainty based on structural uncertainty as well as user interpretation and choice of parameter values. This approach has been adopted in a number of international assessment programmes, most notable VAMP, BIOMOVS, IASAP and the IAEA Mururoa assessment (IAEA, 1995; BIOMOVS, 1993; Scott et al., 1997, 1998) and in an assessment of a waste repository (Draper et al., 1999). These studies (Hoffman & Thiessen, 1995; Hoffman et al., 1996; Scott et al., 1997, 1998) showed variation amongst the predictions and differences between the model predictions and experimental data. On further analysis, one of the main sources of variation in the results was found to be ‘modeller interpretation’ (IAEA, 1995; Kirchner et al., 1997). The final conclusions were that the uncertainties had two main sources; (a) user-related (based on interpretation of the scenarios and experience in use of the model) and (b) model-related (including here the parameterisation of specific processes, differences in model structures and complexity) (BIOMOVS, 1993). Agreements were within several orders of magnitude and it was concluded that ‘agreement within a factor of 2 was considered acceptable’. The disagreements can, at least in part, be explained by the different model structures.

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5.5. Why perform SA and UA? Sensitivity and uncertainty analyses are important tools in the modelling toolkit (Saltelli et al., 2000) and should be performed routinely. In model assessment and testing, sensitivity and uncertainty analyses provide essential information for determining the acceptability of the uncertainties and identifying influential factors. In model validation, the agreement of the model to observed or measured data can only be judged in the light of the predictive uncertainties. In model design, sensitivity analysis encourages the identification of key processes to be further investigated and gives a metric by which to compare different model structures. Taken together, sensitivity and uncertainty analysis contributes to a more reliable model and one whose abilities are better understood. A sensitivity analysis, where the uncertainty in the output is apportioned to the uncertainty in the system input parameters, provides an element of verification in that it ensures that the response of the computational model to the input is the expected one. In our introduction, we stated that the modelling process is iterative. In this iterative process, we are striving to improve the quality of the model, perhaps by reducing complexity, or by designing experiments to reduce uncertainty, and ultimately amending the model and repeating the testing process. SA and UA play crucial roles in this refinement process. Through an improved understanding of the interplay between the model factors and the relative contributions they make to the overall uncertainties, we can target specific factors and parameters for more detailed study to reduce uncertainty; we can identify factors which over the whole design space are non-influential in determining the outcome and which can perhaps be combined or omitted. In general, SA and UA can assist the modeller in deciding whether model performance is sufficient to the task.

6. Conclusions It is clear that environmental modelling is an important, non-trivial activity with many functions. Modelling is of great value but does have limitations, which sometimes may not be fully appreciated. The message is that • The evaluation of environmental models is non-trivial and is dependent on the context and purpose of the model. • There is a huge diversity in the types and complexity of models, so that different evaluation tools may be required. • Model development and data collection should be more integrated. They are mutually dependent activities with, in reality, the dependence being often broken. • Currently, much model development and evaluation is hampered by lack of data, or data of insufficient frequency or ‘wrong’ spatial scale. • The practical use of an environmental model is not simply the domain of the individual scientist; stakeholders, and regulators are both involved. • Uncertainty is ubiquitous. Management of uncertainty must operate at all levels in the model and must be explicitly incorporated and expressed as part of the model output.

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References Andres, T. H. (1987). Statistical sampling strategies. Proc. NEA Workshop on Uncertainty Analysis for Performance Assessments of Radioactive Waste Disposal Systems, Seattle. Andres, T. H. (1997). Sampling methods and sensitivity analysis for large parameter sets. Journal of Statistical Computation and Simulation, to appear. Beven, K. & Binley A. (1992). The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes, 6, 279–298. Beven, K. (1993). Prophecy, reality and uncertainty in distributed hydrological modelling. Advances in Water Research, 16, 41–51. BIOMOVS (1993). Final report, BIOMOVS Technical report 15. Stockholm: Swedish Radiation Protection Institute. Carrera, J., Mousavi, S. F.,Unusoff, E. J., Sanchez-Villa X., Galarza G. (1993). A discussion on validation of hydrogeological models. Reliability Engineering and System Safety, 201–216. Caswell, H. (1976). The validation problem. In B. C. Patton (Ed.), System Analysis and Simulation in Ecology. Academic, New York, Vol. IV, pp. 313–325. Chatfield, C. (1993). Model uncertainty, data mining and statistical inference. J.R. Statist. Soc. A, 158 (3), 419–466. Cooke, R. (1994a). Parameter fitting for uncertain models: modelling uncertainty in small models. Reliability Engineering and System Safety, 44, 89–102. Cooke, R. (1994b). Uncertainty in dispersion and deposition accident consequence modelling assessed with performance-based expert judgement. Reliability Engineering and System Safety, 45, 35–46. Cox, L. & Piegorsch, W. (1996). Combining environmental information I: Environmental monitoring, measurement and assessment. Environmetrics, 7, 299–308. Draper, D. (1995). Assessment and propagation of model uncertainty. JRSS (B), 46, 193–227. Draper, D., Pereira, A., Prado, P., Saltelli, A., Cheal, R., Eguilor, S., Mendes, B. & Tarantola, S. (1999). Scenario and parametric uncertainty in GESAMAC: a methodological study in nuclear waste disposal risk assessment. CPC, 117, 142–155. EC (1995). Review on development of methodologies for modelling with uncertainty and variability: Munvar project. European Commission report, EUR 16174 EN. Granger Morgan, M. & Henrion, M. (1990). Uncertainty: a Guide to Dealing with Uncertainty in Quantitative Risk and Policy Analysis. Cambridge Press. Hamby, D. M. (1995). A comparison of sensitivity analysis techniques. Health Physics, 68 (2), 195–204. Hoffman, F. O. & Hammonds, J. S. (1994). Propagation of uncertainty in risk assessments: the need to distinguish between uncertainty due to lack of knowledge and uncertainty due to variability. Risk Analysis, 14, 707–712. Hoffman, F. O. & Thiessen, K. M. (1995). Use of Chernobyl data to test predictions and uncertainty estimates from exposure assessment models. Proceedings of ‘Environmental impact or radioactive releases’, IAEA-SM-339/20, 325–336. Hoffman, F. O., Thiessen, K. M. & Watkins, B. (1996). Opportunities for the testing of environmental transport models using data obtained following the Chernobyl accident. Health Physics, 70, 5–7. Iman, R. L. & Conover, W. J. (1980). Small sample sensitivity analysis techniques for computer models, with an application to risk assessment. Communications in Statistical Theory and Methods, A9 (17), 1749–1842. International Atomic Energy Agency (1989). Evaluating the reliability of predictions made using environmental transfer models. Safety Series, 100. International Atomic Energy Agency (1995). Validation of models using Chernobyl fallout data from the central Bohemia region of the Czech republic. IAEA-Tecdoc 795. Kirchner, G., Peterson, S. R., Bergstrom, U., Bushell, S., Davis, P., Filistovic, V., Hinton, T. G., Krajewski, P., Riesen, T. & de Haag, P. (1997). Effect of user interpretation on uncertainty estimates: Examples from air-tomilk transfer of radiocaesium. Journal of Environmental Radioactivity, in press. McKay, M. D. (1995). Evaluating prediction uncertainty. Los Alamos National Lab Technical Report, NUREG/CR6311. Oreskes, N., Shrader-Frechette, K. & Belitz, K. (1994). Verification, validation, and confirmation of numerical models in the earth sciences. Science, 263, 641–646. Saltelli, A. & Scott, E. M. (1997). The role of sensitivity analysis in the corroboration of models and its link to model structural and parametric uncertainty. Reliability Engineering and System Safety, 57 (1), 1–4.

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Saltelli, A., Chan, K. & Scott, E. M. (2000). Sensitivity Analysis. New York: Wiley. Scott, E. M. (1996). Testing and assessment of environmental models. In Theory and applications of sensitivity analysis of model output in computer simulation, EUR report, 16331. Italy: JRC Ispra. Scott, E. M. (1997a). Uncertainty and sensitivity studies of models of environmental systems. In Proc. of Winter Simulation Conference. San Diego, 1996, Ch. 213. Scott, E. M. (1997b). Evidence and quantification of environmental model structural uncertainties. In Report of the Technical committee on Uncertainty modelling to European Safety and Reliability Association, Lisbon, June 1997. Scott, E. M., Gurbutt, P., Harms, I., Heling, R., Nielsen, S. P., Osvath, I., Preller, R., Sazykina, T., Wada, A. & Sjoeblom, K. (1997). Benchmarking of numerical models describing the dispersion of radionuclides in the Arctic Seas. Science of the Total Environment, 202, 123–134. Scott, E. M., Gurbutt, P., Harms, I., Heling, R., Kinehara, Y., Nielsen, S.P., Osvath, I., Preller, R., Sazykina, T., Wada, A. & Sjoeblom, K. L. (1998). Radiologicalimpact assessment within the IAEA Arctic Assessment Project (IASAP). Radiation Protection Dosimetry, 75, 257–261. Sheng, G., Elzas, M. S., Ren, T. I. & Cronhjorth, B. T. (1993). Model validation: a systemic and sytematic approach. Reliability Eng. and Syst. Safety, 42, 247–259. VROM (1985). Handling uncertainty in environmental impact assessment. Report no. 18, Ministerie van Volkshuisvesting Ruimtelijke Ordening en Milieubeheer.

MODELLING RADIOACTIVITY IN THE ENVIRONMENT E. Marian Scott (Editor) © 2003 Elsevier Science Ltd. All rights reserved

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Chapter 2

Modelling the dispersion of radionuclides in the atmosphere A. Mayall Environment Agency∗ , England

1. Introduction The release of radionuclides into the atmosphere has the potential to cause the radiation exposure of humans and other biota. In some cases, the radiological impact, i.e. the level and significance of the radiation exposure, can be estimated from measurements of radioactivity concentration in the environment. However, in many cases, suitable measured values will not exist (for example, when predicting the impact of future radioactive discharges) and in these cases models are needed to provide estimates of the values. These models attempt to relate a measured or estimated discharge of a radionuclide from a particular source to the resulting activity concentration of the radionuclide in air at the point of interest and its level of deposition to the ground. These models, known as atmospheric dispersion models, are mathematical representations of the transport and dilution of material in the atmosphere. Dispersion modelling is required for a number of reasons, for example: • The authorisation of discharges for regulatory purposes – the authorisation of future radioactive discharges often requires a prospective assessment, or forecast, of the attributable radiological impact. • Retrospective impact studies or ‘dose reconstruction’ – for example, in health impact studies related to past discharges, the available measured data may not be suitably extensive or may be lower than the detection limits of the measurement techniques. Therefore it may be necessary to estimate, using dispersion models, the full extent of the dispersion and deposition. • Accident consequence assessment – this can be prospective and used as input into emergency plans and for assessing the risk posed by nuclear plant, or retrospective, in which a more complete picture of a past accident is constructed. ∗ The views expressed herein are those of the author and do not necessarily reflect the policy of the Environment

Agency. 13

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• Real time impact assessment – in the event of a real accidental release of radionuclides to the atmosphere, it is desirable to be able to predict the spatial and temporal spread of radionuclides so that appropriate monitoring and countermeasures may be targeted. 2. Fundamentals of meteorology and dispersion The mixing of radionuclides in the atmosphere depends primarily on the weather systems that they encounter. Therefore an understanding of particular aspects of meteorology is essential to an understanding of atmospheric dispersion and dispersion models. 2.1. Structure of the atmosphere The atmosphere consists of a number of layers each with its own characteristics: 2.1.1. Troposphere This is the lowest layer of the atmosphere in which the majority of the weather systems occur and which contains about 80% of the total mass of the atmosphere. Its depth varies with location but on average it is about 16 km at the equator and 9 km near the poles. When compared to the diameter of the earth (about 13,000 km), the depth of the troposphere is relatively small. In general, the temperature decreases with height in the troposphere. However, this lapse rate, which on average throughout the troposphere is about 6.5◦ C per km, varies depending on a number of factors such as the level of water vapour in the air and the time of day. The heating of the earth’s surface gives rise to vertical convection, which keeps the air in the troposphere relatively well mixed. The lowest section of the troposphere (about 1 km depth on average but does vary from about 100 m up to about 2000 m) has properties that are directly influenced by the earth’s surface and is known as the boundary layer or mixing layer. The properties of the boundary layer have a major impact on the dispersion of radionuclides released within it, particularly over short to medium distances (< 100 km). 2.1.2. Stratosphere This layer lies immediately above the troposphere and the boundary between the two known as the tropopause. This layer rises to about 50 km above the earth’s surface. Its strong winds can influence the weather in the troposphere below. The temperature in the stratosphere does not fall with height but initially remains constant and then in the upper reaches rises, until at its upper boundary the temperature is approximately the same as it is at the earth’s surface (a few tens of degrees C). There is little vertical mixing of air in the stratosphere. Above the stratosphere there is the mesosphere and thermosphere, which combined extend to about 300 km above the earth’s surface, but because they contain relatively little air and no weather systems they are of no interest in the consideration of dispersion modelling. 2.2. Atmospheric turbulence The capacity of the atmosphere to dilute and disperse depends primarily on the degree of atmospheric turbulence. If there was no turbulence, the dispersion of a substance would be

Modelling the dispersion of radionuclides in the atmosphere

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a lot slower since it would rely solely on the random motion of gas molecules (molecular diffusion). Turbulence arises from the variation of wind velocity in time and space and convective (thermal) instability. Turbulent air flow is disordered, with random fluctuations of velocity that can be large and comparable to the mean velocity and for this reason it is responsible for introducing great variability in measured meteorological quantities. 2.2.1. Wind and mechanical turbulence Wind is normally defined as the horizontal velocity of air at a point, having direction and speed. Wind direction is defined as the direction from which the wind is blowing in degrees clockwise from true north. Wind fields are in reality three-dimensional because there will also be air movement in the vertical plane. In meteorological observations the mean wind is often an average of measurements taken over a 10-minute period. A gust is a fluctuation about the mean wind, typically lasting from a few seconds to a few minutes. Winds are always turbulent due to the roughness of the earth’s surface. Buildings, trees and hills for example, will enhance gustiness (or mechanical turbulence), however the mean wind will be reduced by the friction created by a rough surface. To avoid the worst distortions from turbulence, wind measurements are normally made in an area of undisturbed flow at 10 m above ground level. Wind speed is normally measured in either knots (nautical mile per hour) or in metres per second (1 knot = 0.514 m s−1 ). 2.2.2. Effect of wind speed For substances released into the atmosphere from a chimney stack at a constant rate the concentration of the substance in the receiving air will be inversely proportional to the wind speed at that point. For example, if the wind speed doubles, the substance will then be released into twice as much air per unit time, thus halving the concentration of the substance in the air. 2.2.3. Effect of height on the wind speed and direction Above a certain minimum height the wind blows steadily in a direction parallel to the isobars, the lines of equal pressure on a weather map. Wind speed is then proportional to the pressure gradient and such winds are called gradient winds. The lowest height at which these winds occur is called the gradient height and is normally about 600 m. Below this height the frictional effects of the earth’s surface slow the wind down. As the wind slows down with decreasing height the wind can then blow more directly to the point of the low pressure system and therefore the wind direction also varies with height. The difference between the gradient wind direction and the direction at the ground surface may be as much as 30◦ . 2.2.4. Surface roughness As described above the objects on the earth’s surface (roughness elements) will have a frictional effect on the wind flow. Both the height and the spacing of the roughness elements on the surface will influence the frictional effect on the wind. A single parameter, the surface roughness length (z0 ) is used to represent this effect. Typical values of the roughness length are given in Table 1. In summary mechanical turbulence is created by the roughness of the earth’s surface and the wind shear caused by the variation of wind speed and direction with height.

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A. Mayall Table 1 Typical values of roughness length and n for various types of terrain (from Clarke, 1979)1 Terrain

n

z0 (cm)

Sea Sandy desert Short grass Open grassland Root crops Farmland Open suburbia Cities, woodlands

0.07 0.1 0.13 0.15 0.2 0.24–0.255 0.3 0.39

0.01 0.1 0.5 2 10 20–30 50 100

1 Reproduced by kind permission of NRPB.

2.2.5. Thermal turbulence and atmospheric stability The wind not only mixes air in the horizontal plane but also in the vertical, often through a depth of 1000 m or more. The depth of the boundary layer depends on the way the temperature changes with height, or the stability of the boundary layer. Strong winds create turbulence in the form of a series of eddies, with air being continually forced up to be replaced by air from above, this results in a well-mixed boundary layer. However, this mixing does not create the same temperature at all levels. Rising air cools and descending air warms at a rate of about 1◦ C per 100 m – this is known as the dry adiabatic lapse rate (DALR). If a ‘parcel’ of air is moved upwards by the wind it expands adiabatically, i.e. without any heat transfer to the outside, due to the reduction in atmospheric pressure with height. Since some of the heat energy within the parcel is used up in this expansion, the temperature of the air parcel decreases. If the temperature reduction within the rising parcel matches the temperature decrease in its surroundings then it will have the same density as its surroundings and its vertical motion will be neither suppressed nor enhanced. A turbulent boundary layer has this temperature lapse throughout its depth. This is known as neutral stability since a parcel of air displaced in the vertical experiences temperature and density changes in line with its surroundings and neither gains nor loses buoyancy. In light winds the temperature change with height is influenced mainly by incoming and outgoing thermal radiation. When the temperature falls with height at less than the DALR or even increases with height (a temperature inversion) the boundary layer is said to be stable. This is because air displaced upward cools with respect to its surroundings and therefore its upward motion is suppressed. Stable conditions typically occur at night, particularly when there is no cloud cover coupled with light winds. Under these conditions, in the absence of incoming solar radiation, the earth’s surface radiates heat into space and through a process of conduction at the surface the lower levels of air subsequently cool. An extreme example of stable conditions is when the temperature increases with height, which leads to complete suppression of vertical air movements which is very poor for pollutant dispersal. This is known as a temperature inversion. This type of inversion is caused by heat radiation from the earth’s surface; these usually ex-

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tend from ground level. Inversions may also occur at a substantial height above the ground (elevated inversions) and these can be caused by: • the subsidence of an air mass aloft which then becomes warmer and comes to rest on a cooler air mass, • the interface of two air masses (or fronts) of different temperature where the warmer air rises above the cooler air, and • the flow of an air mass over land at a different temperature, e.g. when a cool sea breeze blows over warm land which forces the warm air upwards. If the temperature falls with height at a rate greater than the DALR then air displaced in the vertical continues to be warmer and thus less dense than its surroundings and therefore continues to rise until constrained at a greater height – this is an unstable boundary layer. These conditions typically occur on cloudless, hot, sunny days in light winds when the earth’s surface is being heated more than usual. 2.2.6. Categorisation of atmospheric stability Pasquill (1961) related experimental data on stability and vertical dispersion to readily observed factors. He developed an empirical scheme which defined stability in terms of six categories, A to F, in which A represents the most unstable conditions, D neutral and F stable. Later a very stable category, G, was added. The approach suggested by Pasquill depends on knowing the wind speed at a height of 10 m and assessing the amount of incoming solar radiation (insolation) in qualitative terms. Table 2 sets out the relationship of wind speed and insolation to the Pasquill stability category. Strong insolation corresponds to sunny midday conditions in midsummer and slight insolation to similar conditions in midwinter. Night refers to the period from one hour before sunset to one hour after dawn. The neutral category D should be assumed, irrespective of wind speed, for overcast conditions during day or night, and for any sky conditions during the hour preceding or following night as defined above. Smith (1973) developed the Pasquill scheme further in the following ways: • the introduction of a more objective approach to the estimation of sensible heat flux in the lower atmosphere as a function of time of day, month and cloud cover, this is the amount of heat per unit area (measured in watts per square metre, W m−2 ) passing between the air and the ground surface as a result of temperature difference, often caused by solar heating during the day or radiative cooling at night, Table 2 Pasquill stability categories based on solar insolation and wind speed (from Pasquill, 1961) Surface wind speed at 10 m (m s−1 )

Insolation Strong

Moderate

Slight

Night > 3/8 low cloud

< 4/8 cloud

<2 2–3 3–5 5–6 >6

A A–B B C C

A–B B B–C C–D D

B C C D D

– E D D D

– F E D D

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A. Mayall

• the introduction of allowance for surface roughness, and • the substitution of the seven Pasquill stability categories by a continuous numerical scale or stability parameter (P ). In practical situations the value of the stability parameter, P , can be deduced from the vertical sensible heat flux, which can then be estimated from the amount of incoming solar radiation, and the 10 m wind speed. Smith provided graphs (nomograms) for the practical application of this method. Once the amount of incoming solar radiation (S) has been estimated then the sensible heat flux (H ) is given by the following approximation: H = 0.4(S − 100) W m−2 . The categorisation of the atmospheric stability has an important influence over the degree of dispersion of material released to the atmosphere. Several workers have used experimental data to derive dispersion parameters for use in models, representative of the various stability categories in the form of the standard deviations of distributions of concentrations in the crosswind (σy ) and vertical directions (σz ). Curves of these parameters against downwind distance were suggested by Gifford (1961) and these became known as the Pasquill–Gifford curves. 2.2.7. The boundary layer As described above the atmospheric boundary layer is the lower region of the atmosphere which is influenced by the effects of the earth’s surface. The boundary layer may be subdivided into three sub-layers: 2.2.7.1. Roughness layer. This is the layer nearest to the ground and represents the region in which turbulence is not fully developed. The depth of this layer is represented by the roughness length (z0 ) which can be interpreted by the eddy size of surface objects. The roughness length is approximately 1/30th of the average height of the surface protrusions. Table 1 gives representative roughness lengths for different surfaces. 2.2.7.2. Surface layer. This extends from the roughness height to about 10–200 m. In this layer, the heat flux and turbulence are assumed to be independent of height, and the rotational effect of the earth is negligible. 2.2.7.3. Transition layer (Ekman layer). This extends from the top of the surface layer to the height of the boundary layer (h) which may be up to 2000 m. The Pasquill/Smith/Gifford approach to stability classification assumes that the meteorology of the boundary layer is defined by measurements or observations made close to the earth’s surface namely the mean wind speed, surface heat flux and surface roughness. Research has shown that this is not strictly correct, particularly in very unstable conditions (Hunt et al., 1990). Therefore as we shall see, the boundary layer structure and hence turbulence are defined in the new generation of models for practical application, in terms of the boundary layer height (h) and the Monin–Obukhov length (LMO ) (m). A key difference in this approach is that the variation in boundary layer properties with height is taken into consideration. The Monin–Obukhov length is defined by the ratio of the mechanical flux to heat flux and put simply is related to the height below which mechanically generated turbulence dominates

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Table 3 Typical meteorological parameters for the Pasquill–Gifford categories and representative Monin–Obukhov length Pasquill–Gifford category

Boundary layer height, h (m)

Wind speed (m s−1 )

Monin–Obukhov length, LMO (m)1

h/LMO

A (v. unstable) B C D (neutral) E F G (v. stable)

1300 900 850 800 400 100 100

1 2 5 5 3 2 1

−2 −10 −100 ∞ 100 20 5

−650 −9 −8.5 0 4 5 20

1 Illustrative values.

over the buoyant production of turbulence and above which the reverse is true. It is defined mathematically as: LMO =

−u3∗ ρCp T , kgH

where: u∗ is the friction √ velocity (a measure of drag and mechanical turbulence) at the earth’s surface (m s−1 ) = τ0 /ρ, τ0 is the surface momentum flux or shear stress (μ du/dz) where μ is the viscosity coefficient and du/dz is the change in wind speed with height, ρ is the air density (∼ = 1.225 kg m−3 ), T is the near-surface absolute temperature (K), Cp is the specific heat of air at constant pressure (∼ = 1004.6 J kg−1 K−1 ), −2 g is the acceleration due to gravity (9.8 m s ), k is the von Karman constant – determined experimentally as 0.4, H is the surface heat flux (J s−1 m−2 or W m−2 ). The Monin–Obukhov length is strongly dependent on wind speed since the friction velocity depends on shear stress which in turn depends on the change in wind speed with height. Table 3 shows the approximate relationships between the Pasquill–Gifford stability categories and the Monin–Obukhov length. Figure 1 gives an indication of how the various meteorological parameters are related to the stability categories. 3. Source terms The sources of anthropogenic radioactivity in the atmosphere are many and varied, including: • nuclear weapons testing in the atmosphere (largely ceased in the 1960s), • routine and accidental releases from the nuclear fuel cycle, • other ‘non-nuclear’ establishments which use radioactive materials such as hospitals and research establishments, and

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Fig. 1. Non-dimensional representation of variation of Monin–Obukhov length with atmospheric stability (from CERC, 1999 – reproduced by kind permission of CERC).

• installations that release natural radionuclides such as polonium-210 and radium-226, e.g. coal-fired power stations. Releases to atmosphere may be categorised by reference to whether they are made from a single point source such as a chimney stack, or from a building vent, or an area source such as the radon gas emitted from uranium mine tailings. This chapter is concerned with releases from point sources. Releases from the various sources may be either continuous or virtually so (quasicontinuous) or non-continuous, intermittent or virtually instantaneous. The source term is a set of parameters, which includes the list of radioactive species present in the aerial effluent and their respective release rates (radioactivity released per unit time) or the total quantity of radioactivity released over a finite period. The source term may also include other relevant parameters such as the temperature of the effluent and the exit velocity of the release. The definition of the source term is an important first step in any radiological assessment. 4. Atmospheric dispersion modelling 4.1. Introduction Many mathematical models have been developed for estimating the spread of materials released into the atmosphere.

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For releases of short duration, puff models have been developed which take into account the diffusion of the material in the puff along three axes corresponding to the wind direction, and the crosswind and vertical directions. Continuous releases can be treated as a release of a series of ‘puffs’, each of which can be tracked individually, taking into account changes in meteorological conditions along its route. Another type of model, the grid model, divides the area of interest into an array of grid cells; the equation representing the diffusion of material is integrated to simulate the atmospheric dispersion. All of these models are able to use threedimensional windfields as input and can provide a detailed representation of the physical processes involved in dispersion. However they are generally considered too complex for routine practical application and have mainly found application in the estimation of longrange dispersion of radionuclides, particularly in the field of emergency response. In the case of long duration or continuous releases the diffusion of the released material in the direction of the wind will be relatively small compared to the transport by the wind itself (advection). The models adopted to predict dispersion under these conditions are termed plume models of which the best known is the semi-empirical Gaussian plume model. This is an analytical solution to the diffusion equation based on ideas introduced by Sutton (1932) and developed by, amongst others, Pasquill (1961), Smith (1973) and Gifford (1961). The model was developed in parallel with schemes for categorising atmospheric stability which are used to provide dispersion parameters for the model. Gaussian plume models are still used today because they are relatively simple to apply, and their parameters are related to readily available meteorological quantities such as wind speed and cloud cover. The basic model was originally developed for dispersion over land of neutrally buoyant releases from isolated point sources in flat terrain. The basic model has changed little over the years although it has been adapted to be applied to a more diverse range of conditions than originally intended. It should be remembered that all dispersion models are approximations and this is particularly so in the case of the analytical models such as the Gaussian plume model and the more recent generation of analytical models introduced in Section 4.3. Over the last 10–20 years a new generation of models has been developed. These ‘new generation’ models do not use the Pasquill–Gifford type schemes to categorise atmospheric stability according to discrete categories based on surface data but rely on methods which allow consideration of the height dependence of turbulence. Despite the gradual move towards these newer models, no discussion of dispersion modelling would be complete without reference to Gaussian plume models because: • they continue to be used in many applications, • they share some similarities with the new generation models, and • it helps in the understanding of some of the general dispersion modelling concepts. 4.2. Short to medium range dispersion modelling 4.2.1. The Gaussian plume dispersion model There are many versions of the Gaussian plume model in use around the world. A major reason for the variation is the difference in the way that the dispersion parameters for the model are derived. One such practical version of the model, which relies in particular on the work of Pasquill (1961), Smith (1973) and Hosker (1974), was published as National Radiological Protection Board Report R91 (Clarke, 1979). This subsequently became known as the ‘R91

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Table 4 Examples of computer implementations of Gaussian plume models Name

Country

Description

Reference

AEROPOL CAP88-PC

Estonia USA

Kaasik (1998) USEPA (2000)

NORMAL

Czech Republic

ONM9440

Austria

PC-CREAM

UK

Gaussian plume using Pasquill–Gifford categories. Gaussian plume for radiological assessment using Pasquill–Gifford categories. Approved by United States Environmental Protection Agency (USEPA). Gaussian plume using Pasquill–Gifford categories for radiological assessment. Gaussian plume model using a discrete stability classification scheme. Includes a continuous release version of ‘R91 model’ (PLUME), i.e. Gaussian plume using Pasquill–Gifford categories and Hosker formulation for σz .

Pecha et al. (1999) Pechinger & Petz (1995) Mayall et al. (1997)

model’ which became a de facto standard for use within the UK for the calculation of the dispersion of radioactive discharges. However, its use in the field of modelling non-radioactive discharges has been limited where the US EPA’s ISC (Industrial Source Complex) model (also a Gaussian model) and others, have been used. Examples of Gaussian plume computer based models in use in Europe and the USA are summarised in Table 4. The following summarises an example of a Gaussian plume model (the ‘R91 model’) which remains in regular use for routine radiological assessments in the UK at the time of writing. It is also the model suggested for use in the European Union in a methodology for assessing the consequences of routine releases of radionuclides to the environment (Simmonds et al., 1995) and implemented in the PC-CREAM software (Mayall et al., 1997). The activity concentration in air of a radionuclide, C(x, y, z) is given in the basic form of the Gaussian plume model for a point source as:    2 Qo y (z − he )2 C(x, y, z) = , exp − + 2πσy σz u10 2σy2 2σz2 where: C(x, y, z) is the activity concentration in air at the point (x, y, z) in Bq m−3 . The origin of the co-ordinate system is at ground level directly beneath the discharge point, x is the downwind distance (m), y is the cross wind distance from the centre line of the plume (m), z is the height above ground for which the concentration is calculated (m), σy is the standard deviation of the horizontal Gaussian distribution (m), σz is the standard deviation of the vertical Gaussian distribution (m), Qo is the release rate (Bq s−1 ), u10 is the mean windspeed at 10 m (m s−1 ), and he is the effective release height (m). The model assumes that the distribution of air concentration about the central axis of the plume in both the vertical and horizontal planes takes the form of a Gaussian or Normal

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curve. The concentrations obtained are applicable to releases which are short enough to be unaffected by changes in wind direction. For releases of longer duration the horizontal spread of material is heavily influenced by changes in the wind direction rather than by the dispersion and diffusion of material about the plume’s axis. For a continuous release in which the meteorological conditions (e.g. stability category) are assumed to remain constant and the windrose assumed to be uniform (i.e. a uniform distribution of material around 360◦ or 2π radians), the equation above can be modified to:   Qo (z − he )2 C(x, z) = exp − , √ 2σz2 2πx 2πσz us where C(x, z) is the mean activity concentration in air at the point (x, z) in Bq m−3 (which is the same in all directions from the release point) and us is the wind speed at the height of the plume (or effective release height) or u10 if he < 10 m. In this form of the equation the crosswind direction (y) terms are removed owing to the assumption of a uniform spread around 360◦. Wind frequency data are normally divided into a number of angular sectors. For example, in the UK, data are often quoted for 12 × 30◦ sectors. If the dispersion within a single sector is required, the result from the equation above must be multiplied by the total number of sectors. Meteorological conditions will not remain constant during a prolonged release and therefore calculations should be done for each combination of meteorological conditions (e.g. of stability category, wind speed and wind direction) and the frequency weighted sum calculated. 4.2.1.1. Reflection from the ground and from the top of the boundary layer. When material is discharged from an elevated source, the plume will disperse vertically as well as horizontally and eventually reach the ground. On reaching the ground a non-depositing plume is effectively ‘reflected’ back into the atmosphere. To take account of this reflection the mean activity concentration in air is modified by the addition of a reflected term to the basic equation: C(x, z) =

     Qo (z − he )2 (z + he )2 √ exp − + exp − . 2σz2 2σz2 2πx 2πσz us

A similar modification is made to the form of the equation which includes the horizontal diffusion terms. At large distances downwind, after multiple reflections from the ground and top of the boundary layer, or when the value of the vertical dispersion coefficient becomes greater than the depth of the boundary layer, the vertical concentration profile is assumed to become uniform throughout the boundary layer. For these conditions the equation can be simplified to the following: C(x, z) =

Qo , x2πush

where h is the height of the boundary layer (m). This represents the uniform spread of released material within the volume of a moving annulus of constant height (the boundary layer).

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4.2.1.2. Input parameters for the Gaussian plume model. In order to apply the model it is necessary to categorise the meteorological conditions as described in Section 2.2 into the Pasquill–Gifford stability categories. The stability category determines the dispersion coefficients or values of the standard deviations of the distributions of air concentration about the axis of the plume (σz and σy ). Vertical standard deviation (σz ). To facilitate numerical analysis and implementation in computer applications, Hosker (1974) fitted equations to Smith’s σz curves. The Hosker equation has the form: σz =

ax b F (z0 , x), 1 + cx d

where F (z0 , x) is a roughness correction factor given by:    1 F (z0 , x) = ln f x g 1 + j , hx    1 , F (z0 , x) = ln f x g 1 + hx j

for z0  0.1 m, for z0 < 0.1 m.

The values for the coefficients for specific roughness lengths and stability categories are given in Table 5. Horizontal standard deviation (σy ). The crosswind or horizontal spread of a plume is influenced by two factors – short-term turbulent diffusion and longer-term changes in wind Table 5 Coefficients given by Hosker (vertical dispersion coefficient (σz ) (from Clarke, 1979)1 Pasquill–Gifford stability category A B C D E F Roughness length (z0 ) (m) 0.01 0.04 0.1 0.4 1.0 4.0

a 0.112 0.13 0.112 0.098 0.0609 0.0638 f 1.56 2.02 2.72 5.16 7.37 11.7

1 Reproduced by kind permission of NRPB.

b

c 1.06 0.95 0.92 0.889 0.895 0.783

g 0.048 0.0269 0 −0.098 −0.0957 −0.128

d 5.38 × 10−4 6.52 × 10−4 9.05 × 10−4 1.35 × 10−3 1.96 × 10−3 1.36 × 10−3

h 6.25 × 10−4

7.76 × 10−4 0 18.6 4.29 × 103 4.59 × 104

0.815 0.75 0.718 0.688 0.684 0.672 j 0.45 0.37 0 −0.225 −0.6 −0.78

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Fig. 2. Horizontal standard deviation due to turbulence (σyt ) as a function of distance and Pasquill–Gifford category (from Clarke, 1979 – reproduced by kind permission of NRPB).

direction. Pasquill (1961) based his original values for σy on experiments involving very short releases or short averaging times for continuous releases. The R91 model adopted this ‘three minute’ component for releases of 30 minutes or less, which is conservative in terms of the maximum ground level concentration on the plume centreline. The short-term turbulent component to σy (σyt ) is given graphically in Fig. 2 which was derived by Gifford (1968). For releases of longer duration, account also needs to be taken of changes in wind direction which can send a series of ‘plumes’ in different directions hence smoothing longer term av-

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erage concentrations over a wider arc. This can be particularly marked during periods of low wind speed, stable conditions, when the plume may meander due to large-scale eddy motions (Etling, 1990). This component (σyw ) may be estimated from the following equation (Clarke, 1979): σyw = σθ x

or

0.065x 7t/u10 ,

where σθ is the standard deviation of the horizontal wind directions when averaged over consecutive three minute periods and sampled over the release duration and t is the release duration in hours. The combination of the two components for short-term turbulence and longer-term wind direction changes, into a single value for σy may be achieved by using the equation suggested by Moore (1976): σy =



2 . σyt2 + σyw

Wind speed and boundary layer depth. Wind speeds at different heights are used in the different versions of the model. The basic Gaussian plume model equation requires the mean wind speed measured at a height of 10 m, a height commonly used for meteorological measurements. This is because the values of σy derived empirically for use with the basic model, correspond with the use of the wind speed measured at 10 m. The absence of σy from the continuous release form of the equation means that the wind speed no longer needs to be that measured at 10 m. Studies have suggested that the best fit between predicted and measured concentrations is obtained using the wind speed at the height of the release. If the release height is lower than 10 m then the wind speed at 10 m is generally adopted. For the purpose of converting a wind speed measured at the 10 m reference height, u10 , into the wind speed at any other height u(z) the following power law may be used (Clarke, 1979): u(z) = u10 (z/10)n , where u(z) is the wind speed at height z. The shape of the profile depends on n (see Table 1) which is a function of ground roughness and atmospheric stability. Boundary layer depth is not commonly measured at many meteorological stations, Table 3 gives default values for this parameter, together with wind speeds for each stability category for the Pasquill scheme for occasions when meteorological data are not available. 4.2.1.3. Sensitivity. Hamby (2002) carried out a parameter sensitivity analysis on the longterm, sector average version of the Gaussian plume model. It was found that the model is most sensitive to the value of the wind speed although the sensitivity decreased with increasing distance. The model was also found to be particularly sensitive to σz , the importance of which increases as stability increases. It was also found that for distances within about 2 km, the sensitivity to the joint frequency distribution of wind speed, wind direction and stability category, increased with increasing release height.

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4.2.2. Extensions to the basic model 4.2.2.1. Removal processes. The concentrations derived from the basic Gaussian model apply to the dispersion of inert, long-lived material which remains in the plume. However, a number of processes may act to reduce the total amount of activity in the plume, in particular radioactive decay, and various deposition processes. These processes can be taken into account by modifying the initial source strength so that the effective quantity of material reaching a given point is smaller than that released. Radioactive decay. In the case of radionuclides with radioactive half-lives which are short compared to the travel time to a given point, radioactive decay will reduce the activity concentration of the radionuclide as it travels downwind; the modified concentration can be obtained by substituting a modified release rate, Qo Dp , where   x Dp = exp −λp , us where λp is the radioactive decay constant of the radionuclide (s−1 ). Daughter products will grow into the plume with the decay of the parent radionuclide and the concentration of daughter products can be obtained by substituting Qo Dd for Qo , where      x x λd exp −λd − exp λp , Dd = λp − λd us us where λd is the radioactive decay constant of the daughter (s−1 ) and λp that of the parent. Wet deposition. Radioactive particulates and gases in the plume may be removed by different types of rainfall mechanism. There are two types of process by which this removal occurs: rain falling through the plume, which is termed washout (or below-cloud scavenging) and removal of activity incorporated in the rain cloud, known as rainout (or in-cloud scavenging). Rain falling through a plume removes material throughout the whole of the plume volume. The wet deposition rate is therefore a function of the total activity throughout the depth of the plume rather than the activity concentration at ground level. The wet deposition rate during rain can be calculated using a scavenging coefficient (), defined as the fraction of material within the plume removed by rain per unit time (usually in units of a fraction per second, s−1 ). It is not normally necessary to distinguish between the two types of scavenging process, which are often of the same magnitude, and therefore the washout coefficient usually incorporates the effects of both processes and for this reason the scavenging coefficient is often termed the washout coefficient. Calculations suggest that, at a rainfall rate of 1 mm h−1 , the scavenging coefficient (for below-cloud scavenging) of 1 μm particles is about 4 × 10−5 s−1 , while that for 10 μm particles it is about 4 × 10−4 s−1 (Underwood, 2001). The same reference suggests values of 3 × 10−5 to 3 × 10−4 for an in-cloud scavenging process. A default value for the scavenging/washout coefficient of 10−4 s−1 for 1 μm particles is suggested in an EU methodology (Simmonds et al., 1995).

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The total amount of radioactive material Q remaining in a plume, subject to continuous rainfall at a constant rate, is given by: dQ = −Q , dt which gives Q = Q0 exp[−t], where  is the scavenging/washout coefficient and t is the time from the start of the rain in seconds. As it travels the plume may coincide with rainfall in a complex pattern. Therefore in order to account for the intermittent nature of rainfall a number of simple approaches may be used. One such approach involves the assumption that the entire plume is rained upon for a fraction of its travel time, the fraction being equal to the fraction of time for which it rains at a given point. If a more realistic representation of the temporal and spatial distribution of rainfall is required a more complex model should be used. Details of a probabilistic model for this purpose is given by Jones (1981a). A review of the modelling of wet deposition from a short release is given in Jones (1986a). Dry deposition. The removal of radionuclides from the atmosphere to the ground may also occur in the absence of rain by diffusive dry deposition. Dry deposition is a complex process by which particulates and gases are removed from the air by impaction with the ground and vegetation. The rate at which radioactive substances are deposited depends on the nature of the airborne material, the ground surface (e.g. surface roughness and vegetation type) and the state of the atmosphere (e.g. stability and friction velocity) and can be estimated using the concept of a deposition velocity (Vg ). Deposition velocity is defined as the ratio of the amount of material deposited on the ground per unit area per unit time, to the air concentration per unit volume just above the ground (usually at 1 m height). Deposition velocity is often expressed in units of metres per second (m s−1 ). The dry deposition rate (D) or the deposition density (the time integral of the deposition rate) is given by: D = Vg C, where C is the activity concentration in air at ground level (e.g. z = 1 m) or its time integral, respectively. A number of reviews have been carried out of deposition velocity. A comprehensive review is given by Underwood (2001). Table 6 gives some examples of deposition velocity derived for different sets of conditions. The deposition behavior of gases is different to that of particulate material. For example, it normally assumed that noble or inert gases such as argon-41 and krypton-85 do not deposit on the ground either via dry or wet deposition and therefore that their deposition velocities and scavenging/washout coefficients are effectively zero. However reactive gases such as iodine and its compounds do deposit and their deposition velocities can often be greater than those associated with smaller particulates. Many studies have been carried out to determine the deposition velocities of specific reactive gases in particular situations. For example, one such study considered the deposition onto crops of carbonyl sulphide (CO35 S) gas released to atmosphere from gas cooled nuclear

Modelling the dispersion of radionuclides in the atmosphere

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Table 6 Example deposition velocities for particulates derived on a ‘best judgement’ basis1 (m s−1 ) Surface type Low crops2 Urban horizontal surfaces Urban vertical surfaces

Aerodynamic particle diameter (μm) 1 3

10

6.6 × 10−4 6.4 × 10−4 6.7 × 10−5

1.7 × 10−2 8.9 × 10−3 3.5 × 10−3

4.8 × 10−3 3.8 × 10−3 4.8 × 10−3

1 From Underwood (2001). Deposition velocities derived on a conservative basis may be up to an order of magnitude greater for the smaller particle sizes but only a few tens of percent greater for the larger particle sizes. 2 Assuming typical values of normalised aerodynamic resistance (R ) of 10 and a friction velocity (u ) of 0.2 m s−1 . a ∗

power stations in the UK. The results indicated a wide range of values for deposition velocity (2 × 10−5 m s−1 to 1.1 × 10−2 m s−1 ) which appeared to increase with increasing crop mass (Collins & Bell, 1996). Sehmel (1980) reviewed the deposition velocities of gases and particulates and the values for elemental iodine were largely in the range 1 × 10−3 to 4 × 10−2 m s−1 . The values for organic compounds of iodine such as methyl iodide are generally lower (1 × 10−6 to 1 × 10−4 m s−1 ) due to poor absorption by vegetation. Representative values for deposition velocity of 10−3 m s−1 for 1 μm particles, 10−2 m s−1 for elemental iodine and 10−5 m s−1 for organic iodine are suggested by Simmonds et al. (1995). Where the plume is being depleted by dry deposition the activity concentration in air is given by the appropriate Gaussian plume or other equation but the source strength may be modified to reflect that the amount of material reaching a given point has been depleted by dry deposition. Simmonds et al. (1995) describe a method, the source depletion model, by which such a depleted source term may be calculated. The integrals in the model require numerical solution. There are limitations associated with the source depletion model. The model assumes that turbulence is sufficient to maintain the plume’s Gaussian vertical profile as material is deposited at the ground. The results of this model should be treated with caution if a significant fraction of material is predicted to be removed from the plume in stable conditions. The Atmospheric Dispersion Modelling System (ADMS) allows for the modification of the plume’s vertical concentration profile as a result of deposition (CERC, 1999). Gravitational settling. The model described above for dry deposition is primarily for diffusive deposition and makes no allowance for the settling of particulates solely under the influence of gravity. A particle held in still air or in horizontal laminar flow will fall at a rate equal to its settling or terminal velocity (Vs ). Terminal velocity depends on a particle’s mass and diameter and can be calculated using Stokes’ law: Vs =

mg , 6πrη

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where: m g r η

is the mass of the particle (kg), is the acceleration due to gravity (9.8 m s−2 ), is the particle radius (m) and is the air viscosity coefficient (2 × 10−5 N s m−2 ).

In general gravitational settling will be a minor contributor to overall deposition if the activity median aerodynamic diameter (AMAD) of an aerosol is less than about 10 μm. However, in certain situations, for example accident scenarios, larger particulate material may be released which could deposit at velocities of up to 1 m s−1 and for these situations a different approach to modelling deposition and its effect on plume depletion and concentration distribution is necessary. A review of suitable methods has been carried out by Underwood (1985). ADMS combines the contributions from diffusive deposition and gravitational settling using the following equation (CERC, 1999): Vd =

Vs , 1 − exp(−Vs /Vg )

where: Vd is the overall deposition velocity, Vs is the terminal velocity and Vg is the deposition velocity (diffusive). 4.2.2.2. Complex effects Plume rise. Material released to the atmosphere may rise above its original release point if it has vertical momentum or is more buoyant than the ambient air. The buoyancy can occur when the material has a lower density than air, either because of its temperature or composition. Plumes can also gain or lose buoyancy because of chemical reactions or the latent heat of their constituent material. Plume rise can be an important element in dispersion modelling particularly over short distances. Although a large number of models for plume rise from isolated sources have been given in the literature, there are difficulties in using any of them to calculate the concentration in air at ground level, one of the key quantities required in radiological assessment. Amongst the more accurate of these models is the one derived by Briggs (1984) used to predict the trajectory of the plume centre-line for a stack height greater than a few tens of metres. This model is suggested for use by Simmonds et al. (1995). Building wake effects. The flow of air around single buildings or groups of buildings is extremely complex, an understanding of the flow is important to the prediction of downwind concentrations from radionuclides released into such airflows. The main effects of building structures on airflow are: • incorporation of a proportion of the discharged radionuclides into the immediate eddy in the lee of a building (entrainment),

Modelling the dispersion of radionuclides in the atmosphere

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• general downward movement of the plume centreline in the lee of the building at further distances (downdraught), • enhancement of plume spread in the downstream wake (turbulence), and • plume rise following the initial rapid dilution in the case of buoyant releases. Many nuclear fuel cycle plants release radionuclides at a height such that dispersion may be influenced by the presence of turbulent recirculating wakes. This may be an important consideration in the calculation of concentrations and hence radiation doses in the vicinity of the site. However, these near-field effects can often be ignored in the calculation of longer range dispersion. In general the initial rapid spread of released material in the lee of a building will tend to enhance ground level concentrations at very short distances but at longer distances the effect will often lead to lower ground level concentrations than might be the case in the absence of the building. Inevitably, simple models are unable to describe every feature of complex airflow around buildings. For situations where there is full entrainment of the release into the re-circulating wake region of a building or buildings, it may be assumed that the released material is thoroughly mixed within the wake region and effectively emitted across the cross-sectional area of the building. To simulate this initial rapid spread of the plume in the wake, a ‘virtual point’ upwind of the actual source may be adopted for use with the Gaussian plume equation and the vertical and crosswind standard deviations of plume spread modified to reflect the effective area which the building presents to the air stream. The virtual source model assumes that the enhanced and rapid dispersion provided by the turbulent building wake is equivalent to the dispersion at some distance downwind for an isolated point source. An example of a virtual source model is given by Barker (1982) and suggested for use by Simmonds et al. (1995). Simmonds et al. (1995) also discuss models for situations where no or little entrainment occurs from sources well above the roof. The virtual source model may be used for the calculation of downwind concentrations at distances greater than 3 building heights, i.e. further than the immediate near-wake region, for releases with no significant buoyancy. The model is simply a modification of the Gaussian plume model, where: • The effective source height is set to one third of the building height (Hb ) based on empirical evidence. • The wind speed at height 10 m is used. • The horizontal and vertical dispersion coefficients representative of the distance from the upwind ‘virtual source’ to the receptor are used instead of those from the real source. The ‘virtual’ distances for both the horizontal (xvy ) and vertical dispersion coefficients (xvz ) are found by assuming that at zero real distance (i.e. at the downwind edge of the building) the dispersion coefficients are equal to one third of the building’s cross-sectional dimensions, i.e. σz = Hb /3 and σy = Wb /3, where Wb is the building width. In general xvy is not equal to xvz and both need to be evaluated for each stability category. The above assumes that it is only the turbulent component of lateral spread that is affected by the building and that the formulation of σy can be extended to include lateral spread due to fluctuations in wind direction. In simple applications of the virtual source approach an effective release height (he ) of Hb /3 has been adopted in the Gaussian plume model whilst omitting the computation of the mod-

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ified dispersion coefficients, this is a conservative approach that will tend to overestimate downwind concentrations. As the release height increases relative to the building height there will be a reduction in the amount of released material entrained in the near wake and the virtual source model becomes less valid. However, where release heights are lower than the height at which the building loses its influence, the effective height of release will be lower because of the downwash of the plume in the main wake region. If the exit velocity of the release is small relative to the wind speed at release height, i.e. less than 1.5 times the wind speed, the turbulence around the stack itself may also cause downwash of the released material. The stack downwash correction may be calculated as follows (Briggs, 1973): hs = hs − 4di(1.5 − we /us ), where: hs hs di we us

is the stack height modified for stack down-wash (m), is the physical stack height (m), is the internal diameter of the stack at release height (m), is the exit velocity (m s−1 ), is the wind speed at stack height (m s−1 ).

The following formulae may then be applied to obtain an effective release height (he ) (Briggs, 1973): hs > Hb + 1.5 lb

no building influence: set he = hs ,

Hb + 1.5 lb > hs > Hb + 0.5 lb

set he = 2hs − (Hb + 1.5 lb),

where Hb is the building height (m), lb is the smaller of Hb and Wb , where Wb is the building width (m). ADMS, an example of a new generation model, uses the following composite approach (CERC, 1999). See also Fig. 3. • Several buildings are modelled as a single effective building for each wind direction. • The disturbed airflow for the building or group of buildings is divided into two key regions: the immediate recirculating flow region (or cavity) and the main turbulent wake region further downstream. • The degree of entrainment into the recirculating flow region is calculated region where uniform concentrations are assumed. • The downstream concentrations are calculated using a ‘two-plume’ concentration distribution – one plume emanating from the entrained material in the recirculating flow region and the remainder the unentrained material from the discharge point. For more extensive reviews of this subject the reader is referred to those published by Hosker (1984) and more recently by Robins (2001). In general it is the ‘simple’ models of the types introduced above which are used in the majority of modelling for regulatory and impact assessment purposes. However, it is possible to model the complex fluid motion around objects such as buildings using computer programs

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Fig. 3. Stages used in ADMS to calculate concentrations downwind of building wakes (from CERC, 1999 – reproduced by kind permission of CERC).

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which apply the theories of fluid dynamics. These are known as computational fluid dynamics (CFD) codes. Physical (e.g. wind tunnel) modelling may also be used. CFD and physical modelling are covered in Section 4.8. Coastal meteorology. For those sources that are situated by the coast or by large lakes, consideration may need to be given to the special conditions that are encountered in such areas. Conditions at such sites differ from those at inland sites in a number of ways which can affect atmospheric dispersion. These differences, caused by differences in temperature and roughness of the land and water surfaces, include the occurrence of sea and land breezes, the presence of an internal boundary layer whenever the airflow crosses the coastline and a more moderate climate in general. There are a large number of reports covering aspects of dispersion at coastal sites many of which are referred to by Simmonds et al. (1995), which also describes models for use in situations where a sea-breeze is blowing and one which takes into account the changes in surface roughness and atmospheric stability around the water–land interface. Complex terrain. It has long been accepted that the semi-empirical Gaussian solution of the advection-diffusion equation, despite being relatively simple to apply, has some significant limitations, which are discussed in the following sub-section. One such limitation relates to the inadequacies of a model which assumes a straight line trajectory and constant wind speed throughout the model’s output domain, for estimating dispersion in areas of hilly terrain where Table 7 Criteria for neglecting complex terrain effects (from Jones, 1986b) Condition Neutral/unstable: Gradient of surrounding terrain Ridge upwind of source

Isolated hill upwind of source Hill or ridge downwind of source Stable: Gradient of surrounding terrain Obstacle upwind of source

Obstacle downwind of source

Criterion < 1 in 10 Either he > 1.5H or x > 20H (neutral) x > 10H (v. unstable) Either he > 1.5H or x > 7H Either he > H + σz (x) or σz (x) > H < 1 in 100 (or < 1 in 1000 in v. stable) Either he > H or x > 40H (slightly stable) x > 100H (v. stable) he > H + σz (x) or σz (x) > H

Notes: 1. Reproduced by kind permission of NRPB. 2. Wind speed must be at least 1 m s−1 . 3. he = the release height. 4. H = the obstacle (hill, ridge etc.) height. 5. x = the distance between obstacle and source. 6. σz (x) = the vertical dispersion coefficient.

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flow patterns can be extremely complex and far from linear. For example, Castro et al. (1988) studied the effects of different shaped hills upwind of sources. They found that for winds blowing perpendicular to the hill, the ‘terrain amplification factor’ (i.e. the ratio of the maximum ground level concentration with the hill to that without) could be 4–8 for sources about half the height of the hill. This effect decreased as both the source height and wind bearing (away from perpendicular) increased. Criteria for the conditions when it is reasonable to neglect the effects of complex terrain on the airflow, based on a change of 30% in wind speed between flat and the complex terrain, are given in Jones (1986b) and in Table 7. In general situations that do not meet these criteria can only be simulated by more advanced numerical models. However, the new generation of practical analytical models (e.g. ADMS, AERMOD) and some of the older generation models (e.g. ISC) do include some capability of modelling flow over complex topography. The ADMS model includes a complex terrain module based upon a PC based model known as FLOWSTAR which is generally of use when the hills have gradients of between 1 in 10 and 1 in 3 (Carruthers et al., 1989; CERC, 1999). The model uses analytical solutions for the equations of motion and requires input grids of terrain height and roughness length data. The model takes into account the perturbations of wind speed and of turbulence as air streams pass over or around hills. 4.2.3. Some limitations of the Gaussian plume model The Gaussian plume model was developed from experimental work carried out in the conditions of flat terrain, uniform surface roughness and heat flux. Therefore in theory the use of the basic model is limited to those conditions. In general the Gaussian plume model assumes: • Constant airflow and turbulence characteristics throughout the area of interest. The basic model assumes a straight line trajectory and that the meteorological conditions prevalent at the discharge point at the time of release remain constant throughout the journey of the plume. The assumptions are that wind speed and direction are uniform throughout the region of interest, that the surface roughness remains constant and that the terrain over which the plume travels remains flat. • Areas of complex topography do not affect the airflow. When calculating short-range dispersion from nuclear sites particularly from those situated by the coast the immediate surrounding terrain may often be flat. However, in many situations particularly when estimating long range dispersion or short range dispersion for sites situated in areas of complex terrain, the model may be inadequate. See also Section 4.2.2.2. • Non-calm conditions. The basic Gaussian plume equation contains the reciprocal of the wind speed and therefore a zero wind speed cannot be used. Its use in conditions of a nonzero but very low wind speed is also not advised, since the wind is extremely variable in these conditions and a well-defined plume is unlikely to exist (see Section 4.6). 4.3. New generation dispersion models The Gaussian plume approach coupled with the Pasquill–Smith–Gifford approach to stability categorisation and plume spread parameterisation is one which developed largely during the

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1960s and 1970s. From the late 1980s a new approach to the description of the atmospheric boundary layer began to be implemented which was influenced by work which demonstrated that the boundary layer could be represented by dimensionless parameters such as h/LMO , where h is the boundary layer depth and LMO is the Monin–Obukhov length introduced in Section 2.2. This ratio can be regarded as a measure of the stability of the atmosphere in lieu of the Pasquill–Gifford categories. Examples of this type of ‘new generation’ model are the Atmospheric Dispersion Modelling System (ADMS), AERMOD (American Meteorological Society/Environmental Protection Agency Regulatory Model) and OML (Operationelle Meteorologiske Luftkvalitetsmodeller). 4.3.1. ADMS (UK) This personal computer (PC) based model for short-range dispersion modelling (< 30 km) was developed during the 1990s by Cambridge Environmental Research Consultants (CERC), the UK Meteorological Office and National Power, and was sponsored by a number of UK organisations including NRPB, Health and Safety Executive and the Environment Agency (Carruthers et al., 1994). Its use in the UK and in other countries has been growing steadily, particularly in the field of the assessment of dispersion of non-radioactive emissions. The key differences of ADMS when compared to the ‘old generation’ Gaussian plume model are summarised below: • A non-Gaussian (skewed) plume or puff in the vertical plane in unstable/convective conditions. • The plume rise model allows for the penetration of inversion layers. • The stability may be calculated from a range of different meteorological data e.g. surface sensible heat flux, the reciprocal of the Monin–Obukhov length or cloud cover, time of day, time of year and temperature. • It can model gravitational settling and its effects on plume concentration profile. • It can estimate short-term fluctuations in air concentrations due to turbulence on time scales less than one hour based on empirical data. • It includes detailed modelling of the atmospheric boundary layer. Although it includes the capability to model the dispersion of radionuclide releases and external dose from airborne radioactivity, its use in radiological assessments has up to the time of writing been limited (see Section 4.4). 4.3.2. AERMOD (USA) The US Environmental Protection Agency’s standard dispersion model since the early 1980s is based on the basic Gaussian plume model (the ‘ISC’ model). The ISC model is probably the most widely used worldwide of practical air dispersion models. However, it is expected that this will be replaced by the ‘new generation’ AERMOD. AERMOD, like ADMS, uses a model which accounts for a range of meteorological factors. For example, it uses the Monin–Obukhov length to account for the effects of atmospheric stability. In some respects its basic dispersion model uses a similar approach to ADMS – they are derived from the same body of meteorological research and in general, account for plume dispersion in similar ways although some details differ. However, there are more significant differences in how the two new generation models handle the various additional features such

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as building effects and complex terrain, and in how the meteorological input data are derived from the raw data. 4.3.3. OML (Denmark) OML is described as a ‘modern Gaussian plume model’, based on boundary layer scaling instead of Pasquill–Gifford stability classification (Olesen, 1995). Unlike ADMS and AERMOD it cannot account for the effects of complex terrain. 4.4. Intercomparison of new and old generation models Since the development of the new generation models there has been considerable interest in how the old and new compare and there has been a number of intercomparison studies carried out. Two related studies (Hall et al., 2000a, b) sought to compare the new and old generation models for regulatory applications and to review existing intercomparison studies. The models examined were AERMOD and ADMS – representative of the ‘new generation’ of models, and ISC – the USEPA model chosen as representative of ‘old generation’ Gaussian models. The intercomparisons included many different dispersion conditions and endpoints and revealed differences that were significant. One output of particular relevance to the radiological assessment of continuous releases is the annual mean concentration. Figures 4 and 5 show the ratios of model output for the maximum annual mean concentration and for its distance from the source, for dispersion over flat terrain. It can be seen that differences in the predictions of maximum concentration for an isolated 40 m stack are small – within 10–20% but the ISC Gaussian model predicted that this occurred at twice the distance predicted by the other two models. The differences in maximum concentration were more pronounced for higher stacks with the ISC model predicting lower values than the other models. If the effects of buildings were taken into account, ADMS predicted significantly higher concentrations than either of the other two models by more than a factor of 2. The study also considered how differences in surface roughness, thermal buoyancy and complex terrain would affect the predictions. In the case of complex terrain it is often assumed that hill gradients of less than 10% have little effect on dispersion in most stabilities. The study looked at an area of hilly terrain with a maximum gradient of 14% and examined six cases where the upwind position of a 40 m stack relative to the terrain was varied. This comparison indicated that there was little difference between using the ADMS model with or without the use of its complex terrain module when it came to predicting the maximum annual mean concentration and where it occurred. Of the conclusions of the study the following are particularly noteworthy: • Overall the models showed significant differences and high levels of variability against one another so that it is difficult to discern any consistent relationships between them. • The models agreed most closely in neutral conditions for the lower stack heights (e.g. 40 m). • Dispersion in the new generation models was in general quite sensitive to values of the boundary layer height and the Monin–Obukhov length scale. • The pre-processors included in the new generation models for the conversion of raw meteorological data into input data for the models were a cause of significant differences in

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Fig. 4. Ratios of pairs of model predictions for a non-buoyant release of maximum annual mean concentration in flat terrain (from Hall et al., 2000b).

Fig. 5. Ratios of pairs of model predictions for a non-buoyant release of distance to maximum annual mean concentration in flat terrain (from Hall et al., 2000b).

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the estimates of boundary layer height and the Monin–Obukhov length scale which are the critical parameters for those models. • It appeared that the new generation models were still in a state of scientific development and should be used with caution and a high degree of understanding. The authors concluded that further development of the new generation models should be encouraged by an open attitude to their contents. Despite the application of an improved understanding of dispersion processes, up to the time of writing ADMS had not been widely or routinely applied in radiological assessment in the UK. This could be for a number of reasons including: • The substantial investment in Gaussian-based models, which are often embedded within larger computer software systems and the practical problems associated with interfacing the output from ADMS with these systems. • Insignificant differences in the results from the two types of model for assessments of continuous releases from lower stack heights when neutral conditions dominate. • The reasonable accuracy, particularly in the calculation of long-term averages, provided by the Gaussian plume models (see Section 4.5). 4.5. Validation Ever since the advent of mathematical models for the prediction of the spread of materials released to the atmosphere there have been validation studies carried out for the purpose of determining the accuracy of the models. Crawford (1978) reported the uncertainties derived from validation studies associated with the Gaussian plume model. Amongst the main conclusions of this work was that the annual average for a specific point within 10 km of the source in flat terrain could be predicted in general within a factor of 2. However if the concentration for a specific one hour period was required the accuracy fell to around a factor of 10. Little and Miller (1979) carried out another review of the validation data. Table 8 gives a summary of some of the key results from this work. Table 8 A summary of results from a review of validation studies of the Gaussian plume model1 Conditions

Ratio of predicted to observed concentration

Continuous gamma dose rates up to 6.8 km from source Short-term SF6 releases from a 36 m stack (all stability categories except A) Kr-85 measurements at 30–140 km (annual averages)

0.56–3.0 0.1–10 (100%) 0.33–3.0 (89%) 0.25–1.0 (77%) 0.1–10.0 (95%) 0.003–100 (very close to source) 0.5–2 (2–15 km)

Complex terrain, plume centreline (number of experiments)

1 From Little & Miller (1979).

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Table 9 Summary of Performance Scores1 (from Hanna et al., 1999) Rank

First Second Third

Number of occasions ISC3

AERMOD

ADMS

5 2 17

6 11 7

19 5 0

1 For 24 comparisons.

Table 10 Median performance measures across all comparisons (from Hanna et al., 1999)

Ratio of maximum predicted to maximum observed Percentage within a factor of 2

ISC3

AERMOD

ADMS

6.7 33

0.77 46

0.8 53

More recently there have been further validation studies carried out which include the new generation of dispersion models. Carruthers et al. (1996) describe a study which used LIDAR measurements. The study largely focussed on tall stacks (140–260 m) with thermally buoyant releases, but included one site with a passive release from a low (12 m) stack. One of the main conclusions from this work was that the model outputs (from ADMS and the Gaussian plume model as represented by R91) were significantly dependent on the model’s determination of the atmospheric stability. For example, in some cases the R91 model indicated neutral stability based on ground measurements whilst ADMS indicated unstable conditions based on data for upper levels of the boundary layer. Another study by Hanna et al. (1999) sought to compare the performance of the new generation ADMS and AERMOD models and the Gaussian based ISC3. The study used five field data sets from the USA from a range of source height, building and terrain configurations. A summary of the model performance against all the field data sets is given in Tables 9 and 10. The data above indicate that ADMS produced the highest scores in general although no model was consistently good or bad across all comparisons. Another intercomparison, this time of ADMS with the Gaussian plume model as represented by R91, involving field data (measurements of Kr-85) has recently been carried out around the spent nuclear fuel reprocessing plant at Sellafield in the UK (Hill et al., 1999). Sellafield is a large complex site with a multitude of buildings of different shapes and sizes. Krypton-85 is released under authorisation from two high stacks (> 100 m) from routine operations. Seventy 24-hour field measurements were taken at three locations around the site of which about 80% were taken during neutral conditions. Table 11 gives a summary of the results. These data suggest that it made little difference which model was chosen to predict off-site concentrations at Sellafield for the conditions under consideration. However, the majority of the intercomparisons in this study were for neutral conditions for which the differences in results between the two models would be expected to be small.

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Table 11 Performance of ADMS and R91 in the prediction of field data around the Sellafield site (from Hill et al., 1999) Model

Within factor of 2 (%)

Within factor of 5 (%)

Within factor of 10 (%)

R91 ADMS

34 35

62 56

75 73

There is growing international co-operation in the area of model harmonisation and evaluation. In 1991, a European initiative was launched for increased co-operation and standardisation of atmospheric dispersion models for regulatory purposes. This was begun in recognition of the introduction at that time of the new generation models and the need for these to be developed and evaluated against standard field data sets in a consistent manner (Olesen, 2001). Fitness for purpose. Jones (1989) discusses the accuracy and uncertainties of Gaussianbased dispersion models. Given that annual average concentrations close to a site can generally be predicted within a factor of 2 of the observed value, it was concluded that such models are generally adequate for the assessment of compliance with radiation dose limits. However, the adequacy of simple models in the field of accident consequence assessment, including realtime modelling, is less clear and depends partly on the specific type of output required. Overall it appears from the many validation studies and reviews that the models based on the simple Gaussian plume approach are reasonably accurate, particularly when predicting long-term average concentrations at short to medium distances in relatively flat terrain. This class of model is also capable of predicting a maximum short-term concentration reasonably well but that it is less accurate when predicting when and where it will occur. Some studies suggest that the new generation models are generally more accurate, particularly for nonneutrally stable conditions. 4.6. Low wind speed dispersion The Gaussian plume model and the new generation models cannot be used when the wind speed is effectively zero as the basic equations include the reciprocal of the wind speed. The plume dispersion models used for the majority of predictions are also inappropriate when the wind speed is very low since the assumptions no longer represent the physical processes involved. For example the models generally assume that the diffusion in the direction of the wind is small when compared to the transport or advection by the wind. This assumption is not correct at low wind speed, particularly in unstable conditions (Jones, 1997). This may not be of particular significance in the assessment of long term average concentrations from continuous releases of radionuclides where the frequency of low wind speeds is relatively low. However, when considering the dispersion of short term releases, for example in the case of a nuclear accident, the entire release could occur in low wind speed conditions. From a practical viewpoint Smith defined low wind speed as that below which normal meteorological instruments begin to perform inadequately (Smith, 1992). Cup anemometers

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do not start to record wind speed until it is greater than about 2–3 m s−1 and wind vanes have a starting speed of about 0.5–1 m s−1 . The UK Meteorological Office classes situations where there is no movement of the anemometer but some movement of the wind vane as a speed of 1 m s−1 , and situations where there is occasional movement of the anemometer with some movement of the wind vane as 1.5 m s−1 . If neither the anemometer nor the vane show any movement the wind is classed as ‘calm’. Smith (1992) analysed the frequency of occurrence of light winds in the UK. The majority of light winds are seen at night with clear skies or broken cloud particularly when the surrounding area has large obstacles such as trees and buildings. Light winds or calms in the day time tend to occur in the centre of high pressure systems. Smith examined data for 6 meteorological stations in the UK and found that the frequency of hours when the wind speed at 10 m was less than 2 m s−1 ranged from 10–30%. The frequency of calms ranged from less than 1% to 7%. The frequency of light winds tended to be greater in hilly areas. Smith (1992) also found that there was a 0.1 probability that given one hour of wind speed 1 m s−1 or less, this speed would persist for at least another 6 hours. 4.6.1. Dispersion models for use in low wind speed conditions The modelling of dispersion under low wind speed conditions is of particular relevance for unstable and stable conditions. In unstable conditions the mean wind speed may effectively be zero but the fluctuations about the mean wind speed may not be, since these are driven by convective motion. In stable conditions the mean wind speed and the fluctuations about the mean may both be very small. In extremely stable cases it is possible for the wind near the ground to stagnate completely, even though there might be wind at higher levels. Substances released into the stagnant air can only be diluted through the effects of molecular diffusion and the momentum and buoyancy of the release itself. A review of low wind speed modelling is given by Jones (1997). Model for low (non-zero) mean wind velocity. Hanna (1992) suggested that the Gaussian plume model can be applied to light winds if σy is replaced by the product σθ x, where σθ is the standard deviation of the wind direction and x is the travel distance. Studies carried out by Hanna on actual plume widths suggested that such widths were consistent with the assumption that the cross-wind spread in low wind stable conditions is given by: σθ = A/U, where A is a constant with a value of about 0.5 m s−1 , when the angle θ is given in radians. Hunt (1997) has suggested a model for use in unstable conditions. Model for zero mean wind velocity. Smith (1989) has developed a model for use with neutrally buoyant releases in unstable conditions where the mean wind velocity over a short period is effectively zero. However, the air is not completely still since the presence of convective thermals result in vertical and horizontal turbulent motion. The model aims to calculate the probability that material will be carried to the point of interest by considering the probability distributions of horizontal and vertical wind speed.

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4.7. Long range dispersion modelling 4.7.1. Simple models The models described in Section 4.2 are plume models of various types and are primarily for use with distances over which it may be assumed that the mean wind direction and other meteorological parameters do not change. This is typically a few tens of kilometres and up to about 100 km. When modelling long range dispersion the assumption of constant meteorological conditions becomes less valid due to the increased likelihood of changes in meteorological conditions. For example, the mean duration of a particular stability category is only a few hours. Changes in stability before the material has spread uniformly throughout the boundary layer affect dispersion in two ways. Firstly, the depth of the boundary layer changes allowing material to disperse throughout the new boundary layer. Secondly, the change of stability alters the rate at which the plume increases in size. In addition, the increase in wind speed with height will affect the plume as it grows to fill the boundary layer (Jones, 1981b). Jones (1981b) suggests that changes in wind direction may only exert a small effect on the annual average concentration at long distances in a given sector for two reasons. Firstly, directional changes resulting in the transfer of material between sectors will tend to balance out. Secondly, concentrations at great distances are largely determined by neutral conditions in which the trajectories are likely to be approximately straight. Jones (1981b) suggests therefore that for releases in the UK, long range dispersion can be approximated by assuming that they occur in neutral stability with a boundary layer depth of 800 m, but allowing for a range of wind speeds. 4.7.2. Complex models The semi-empirical Gaussian model is an analytical solution to the diffusion equation, there are many other models that have been developed which attempt to provide a numerical solution to the diffusion equation. The numerical models are complex and use three-dimensional wind fields over large model domains. Numerical models have found particular application for emergency response purposes in the prediction of long-range dispersion in the aftermath of accidental releases of radionuclides. The Chernobyl accident in 1986 prompted a widespread re-assessment of the ability to model the dispersion and deposition of radionuclides and other pollutants over long distances. Models for long distance transport can therefore serve a number of purposes: • realtime emergency response – providing input into decision making on monitoring and countermeasures; • if reliable measurements are available the models can be used to back-calculate an estimation of the source term; and • once the acute stage of the accident is over, models can be used to interpolate and extrapolate from measured data to build a more extensive picture of the contamination. Models may be categorised according to how they treat the three-dimensional atmospheric system. Models that make use of a grid, which is fixed with respect to the earth, are known as Eulerian models. Grid models are finite difference approximations to the equation of motion, diffusion, mass conservation and removal.

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The second type of reference system is called Lagrangian – this is where the co-ordinate system is defined with respect to the flow of air rather than to the ground. The majority of the long range models developed for emergency response are of this type and therefore examples of this type are described in more detail below. Many examples of Eulerian and Lagrangian models are included in the online Model Documentation System administered by the European Environment Agency. One example of a Lagrangian model is the UK Meteorological Office’s Nuclear Accident Model (NAME) (Maryon, 1996). NAME models diffusion processes that occur over short timescales and changes in the shape of the plume which occur over longer periods. The model uses analysed (hindcast) and forecast meteorology from the Meteorological Office’s Numerical Weather Prediction model (NWP). NAME simulates dispersion by tracking the 3dimensional trajectories of air parcels, and computes air concentrations and related quantities by ‘Monte Carlo’ or ‘random walk’ methods. Large numbers of virtual particles are released into the ‘model atmosphere’, as represented by the winds and other meteorology stored on the NWP grids, so as to represent the release at the site of interest. The release may be a single ‘puff’ or a series of ‘puffs’ that might continue over a week or more as at Chernobyl. Each particle represents an air parcel carrying a fraction of the total release. The activity concentrations of the radionuclides are calculated by summing the quantity of radioactivity in the grid volumes for the surface and upper levels in the atmosphere. The quantities are also summed over successive periods to give time-integrated or time-averaged values. The surface values are of particular interest as these are used to compute the ground level concentrations. It is assumed that material is well-mixed throughout the boundary layer and therefore is of uniform concentration. An important component of the NAME system is the European Rainfall Radar (ERR): a rolling 10-day archive of hourly rainfall rates derived from weather radar and other sources. This is linked to the model and enables an accurate indication of likely rainfall ‘hot spots’. 4.7.3. Uncertainty and real-time modification of model output The output from dispersion models is subject to uncertainty and care must be taken in its interpretation. The precision of the model output can give users a false sense of confidence in the results and it should always be remembered that precision does not necessarily equate to accuracy. Model uncertainty may be reduced by the input of measured data from the field during the accident. These measured values may be used to ‘calibrate’ the model in real time so as to improve predictions at other points or to refine the estimate of the source term, which may not be accurately known until well after the event. To develop and automate appropriate techniques is a difficult undertaking, but some progress in this direction has been made with NAME. Measured data and the model output can be combined in a least-squares technique to estimate the source term and its time profile. Robeau & Oishi (1991) describe an approach for estimating the source term in an accident from radiological measurements made in the field. In a real situation model adjustment will depend upon an adequate flow of measurements.

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4.7.4. Real time decision support Many countries have developed emergency response decision support systems which include models of the type described above. One such example is RODOS (Real Time Online Decision Support) which is being developed by a consortium of European institutes and funded largely by the European Commission. One of the aims of this project is to develop a comprehensive decision support system that is applicable across the whole of Europe. One of the RODOS sub-systems includes different dispersion models for the different model domains. For example, the near-field model includes a puff model (RIMPUFF) whilst the medium (mesoscale) and long-range dispersion is performed by a Eulerian model (MATCH). The model system is interfaced with a weather forecasting capability provided online by numerical weather prediction models (Mikkelsen & Desiato, 1993). 4.7.5. Validation of long range models A joint IAEA/WMO/CEC project was set up in the late 1980s to review and compare models of atmospheric transport and deposition of radionuclides over long distances using meaasured data obtained following the Chernobyl accident. The model results were compared with measured values of caesium-137 and iodine-131 air concentration and deposition from around Europe over a two week period following the accident. The study was known as the Atmospheric Transport Model Evaluation Study (ATMES) and involved over 20 different models. For the purposes of the study the same meteorological, rainfall and source term data were used. About two thirds of the models evaluated were of the Lagrangian puff type and the remainder of the Eulerian grid type with considerable variation in complexity. Some models were implemented on PC (which at the time were no more powerful than ‘386’ based machines) and others on CRAY supercomputers. The model output and measurement dataset were subjected to various statistical analyses. In general, Lagrangian models of varying complexity appeared to outperform the Eulerian models, although the latter performed better when a 5 day forecast wind field was used rather than an analysed (hindcast) wind field. Many of the models were able to predict the arrival times of the radioactive cloud at two sites in Germany, for which good measured data were available, within a few hours. Analysis of performance in predicting total deposition showed that 6 models were able to describe the deposition field adequately with more than 40% of results within a factor of two. The use of the 5 day forecast data led to a deterioration in model results although in a real time situation forecasts would be for shorter periods and would be continually updated to allow modification of model runs. A significant conclusion from the ATMES study was that the increased complexity of a model did not necessarily improve its performance. Another example of a large-scale validation study, which followed on from the ATMES study, was the European Tracer Experiment (ETEX) (Van Dop et al., 1998). The objectives of ETEX were: • to test the capability of organisations involved in emergency response to produce realtime predictions of dispersion; • to evaluate the validity of their predictions; and • to assemble a database which allows the further evaluation of long-range atmospheric dispersion models in general.

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The experiment consisted of two releases of perfluorocarbon tracers from Brittany in western France during the period October–December 1994. The resulting concentrations were measured at 168 locations spread over Europe over the subsequent 3 day period. About 30 modelling groups were required to predict the dispersion over Europe in realtime over a 60 hour period. When the chemical analyses of the samples were completed, the statistical evaluation of model predictions against measured tracer concentration values took place. One of the main conclusions was that the best that could be expected from current models is that cloud arrival time could be predicted within 3–6 hours and the maximum ground level air concentration predicted within a factor of 3. More recently a system for gathering and comparing model results in realtime has been developed (RTMOD – Real Time Model Evaluation) (Galmarini et al., 2001). This provides the possibility of obtaining the results of a number of models in the event of an emergency. These data can then be used to produce an ensemble dispersion forecast using a number of statistical parameters such as: • Models envelope – the total area in which at least one model predicts a ground level concentration above a certain threshold, • Confidence in contamination level – the percentage of the models that predict values above a threshold in a particular area of the output domain, • Confidence in time of arrival – the percentage of models that, for a given time, have predicted that the cloud has passed a particular location, and • Maximum concentration trend – the maximum concentration at a specific location and time period for any of the models. Such information may then be used by decision-makers. 4.7.6. Global box models Some radionuclides, owing to their long radioactive half-lives and their behaviour in the environment, may become globally dispersed following their release to atmosphere. The radionuclides which are particularly important in this context are krypton-85, tritium, iodine-129 and carbon-14. Global circulation models are compartmental (or box) models in which compartments represent particular environmental sectors. Each compartment in a global circulation model may represent a large part of an environmental medium, in which the radionuclide is assumed to be uniformly mixed. The models developed by NRPB include not only those sectors related to the earth’s troposphere but also other parts of the environment into which the relevant radionuclide may transfer (Simmonds et al., 1995). Global circulation models are often used in addition to local or regional scale models to predict the impact on a global scale from the continuing circulation of the key relevant radionuclides in the global environment. For this reason they are mainly used in the estimation of collective doses. 4.8. Wind tunnel and computational fluid dynamics models The near-field dispersion of a released material can be heavily influenced by site-specific factors such as buildings and their wakes. The air flow in these areas can be very complex

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such that the assumptions inherent within the practical models such as the Gaussian plume and the new generation analytical models can be very inaccurate. In some cases the only solution is to adopt physical or numerical modelling techniques One example of a physical modelling technique is the use of wind tunnels. Such an approach requires the construction of a scale model of the site of interest and its environs and the use of wind and source emission to the same scale as the model. Concentration measurements and photography are used to characterize the resulting dispersion for a particular set of conditions and the results converted to full-scale. The scales used in such experiments may range from 1/2000 to 1/100. Typical applications of such models include: • The assessment of dispersion in urban environments, • The behavior of heavier than air (dense) gases, • The behaviour of fire plumes. In addition to wind tunnels, water tanks are also used to model dispersion using fluorescent dyes to mimic the pollutant emission and the use of video recording techniques. Another approach to solving dispersion problems in complex situations is to use computational fluid dynamics (CFD) computer codes. These use numerical solutions of the equations of motion. For some types of release, for example of toxic gases, the consideration of complex airflow is potentially important since the quantity of interest is often a very short term average or instantaneous concentration. A review of the application of CFD codes to near-field dispersion is given in Hall (1997). Hall concluded that it is feasible to use such methods for practical near-field dispersion applications. However, the costs of CFD are high compared to using simple analytical models and the results very sensitive to the way in which the codes are used. The use of CFD codes is a very specialised area and it can take months for a user to become effective. 4.9. Guidelines on the use of dispersion models In 1995 a Working Group of the UK’s Royal Meteorological Society issued a policy statement covering guidelines for the use of models (Royal Meteorological Society, 1995). The intention was to promote the use of best practice in the use of models and emphasizing the quality assurance principle of ‘fitness for purpose’. The guidelines cover the following key aspects: • • • • • • • • • •

Statement of context and objectives, Justification of choice of modelling procedure, Use of software, Input data, Results presentation and conclusions, Explicit quantification, Sensitivity analysis, Uncertainty and variability, Quality assurance of models, Auditability.

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5. External radiation exposure from airborne radionuclides The presence of beta-gamma emitters within a cloud or plume of radionuclides can lead to the external radiation exposure of people on the ground. The exposure can arise from being immersed within the plume itself or from a plume as it travels overhead. The following describes approaches for estimating the radiation exposure for both gamma and beta radiation. 5.1. Gamma radiation If a person is immersed within a plume of radionuclides then the estimation of the absorbed dose in air from a plume emitting gamma photons is most simply achieved by use of a semiinfinite cloud model. The assumption in this model is that the activity concentration in air is uniform over the volume of the plume from which photons can reach the point at which the dose is delivered. The absorbed dose rate in air for a particular radionuclide can be expressed as: Dγ = k1 C

n 

Ij Ej ,

j =1

where: Dγ C Ej Ij n k1

is the absorbed dose rate in air (Gy y−1), is the activity concentration of the radionuclide in air (Bq m−3 ), is the initial energy of the photon (MeV), is the fraction of photons of initial energy Ej emitted per disintegration, is the number of photons of particular energies emitted per disintegration, is 2.0 × 10−6 (Gy y−1 per MeV m−3 s−1 ).

Where the concentration distribution in the plume is sufficiently non-uniform to invalidate this approach, for example at short distances when the plume is overhead and has a welldefined structure, and the photon energies are greater than about 20 keV, a finite cloud model should be used. A semi-infinite cloud model will tend to overestimate the external dose at short distances when the plume is overhead. The more complex finite cloud model involves dividing the plume into a large number of small volume sources and summing the dose contributions from all of these volume sources at the point of interest. A finite cloud model is included within the PC-CREAM software tool (Mayall et al., 1997). At larger distances, when the dimensions of the plume are greater, the predictions of both models are similar. MacDonald et al. (1988) carried out a study using environmental measurements; chemical tracer and wind tunnel simulations, with the aim of improving the estimates derived from traditional modelling approaches. The study focussed on releases of argon-41 from a nuclear power station in the south-west of England. The main features of plume behavior observed were the influence of plume rise and the effects of building entrainment at short distances. At low wind speeds the discharges were subject to significant plume rise, broadly consistent with theoretical prediction, but this reduced as the wind speed increased. There was no sign of full entrainment of argon-41 in the recirculating wake of the buildings despite the release

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points being on the sides and upwind face of the reactor buildings. The plume generally appeared to retain a discrete Gaussian character although the plume appeared wider than would otherwise be the case. A comparison of dose estimates indicated that the dose derived from measurements was approximately half that derived in an assessment using a modelling approach excluding the effects of plume rise. Plumes may be subject to considerable meander during low wind speeds in stable conditions (Etling, 1990). For these situations Overcamp (1991) proposed an approach based on a fluctuating plume model which estimates the absorbed dose rate for an off-axis plume at various distances from the receptor. The results using this approach for very stable (category G) conditions were similar to the results obtained using a traditional Gaussian model and a horizontal dispersion coefficient (σy ) modified to account for wind direction fluctuations (σyw ) and lower than those obtained without a modified σy by factors of 2–3. In order to calculate the radiation dose to humans the absorbed dose rate in air should be converted into an effective dose or organ equivalent dose. The ratio of the effective dose to the absorbed dose rate in air is energy dependent and ranges from 0.013 for 15 keV up to 0.92 for 10 MeV photons, with the ratio in the range 0.6–0.8 for the majority of photon energies (60 keV to 2 MeV) (Simmonds et al., 1995). The doses evaluated using the above approach are appropriate to individuals standing outside where there is no dose reduction due to building shielding. Where buildings are present doses to people will be lower. The dose reduction will depend on factors such as the time spent indoors and the type of buildings. Theoretical studies of the shielding provided by buildings indicate that the dose rates inside the building would be between about 0.01 and 0.9 of the dose rates in the absence of the building, depending upon the nature of the construction materials and the type of building (Brown, 1988). A typical value of 0.2, representative of a house, is suggested for general assessments in EU countries although this could be considerably lower for large multi-storey buildings (Brown & Jones, 1993). Doses to people standing outdoors in an urban area could also be affected by the shielding provided by the surrounding buildings; a factor of 0.7 may be used for these situations (Brown & Jones, 1993). 5.2. Beta radiation The range in air of electrons emitted by the radionuclides of interest is in general small (several metres at most) compared to the dimensions of the plume and an infinite cloud model can be used to estimate the absorbed dose rate. The absorbed dose rate in air for a particular radionuclide can be expressed as: Dβ = k2 C(x, o)

m 

Ij E j ,

j =1

where: Dβ is the absorbed dose rate in air (Gy y−1), C(x, o) is the activity concentration of the radionuclide in air (Bq m−3 ), E j is the mean energy of the particle (MeV),

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Ij is the fraction of electrons of mean energy E j emitted per disintegration, m is the number of β particles and conversion electrons of particular energies per disintegration, k2 is 4 × 10−6 (Gy y−1 per MeV m−3 s−1 ). In beta decay the particles are emitted with a spectrum of energies which is characterised partially by the maximum energy Eβ max . The mean energy of the beta particle, E j , is to a good approximation, equal to one third of the maximum energy. The range of beta particles in tissue rarely exceeds a few millimetres and therefore the principal organ of exposure is the skin. The radiosensitive cells nearest the skin surface are located at the basal layer of the epidermis at a depth of about 70 μm. The dose rate in skin is evaluated from the absorbed dose rate in air, allowing for exponential absorption of the beta particle flux in the 70 μm layer, and is given as Hβ = 0.5e−μd Dβ wr , where: Hβ is the equivalent dose rate in skin (Sv y−1 ), wr is the radiation weighting factor for beta radiation = 1, μ is the absorption coefficient in tissue and taken as inversely proportional to the range in tissue corresponding to the mean energy of the electron considered (m−1 ), d is the thickness of the epidermis and taken as 70 μm. The infinite cloud model assumes that beta radiation is incident on the target tissue from all directions, a factor of 0.5 is therefore included to allow for the shielding provided by the body itself. The shielding of beta particles by clothing may also be an important factor when assessing doses. For a typical maximum beta energy of 0.5 MeV the dose reduction due to typical thick winter clothing is about 0.005 and for typical summer clothing about 0.07 (National Radiological Protection Board, 1997). For more details on the assessment of doses to the skin see National Radiological Protection Board (1997).

6. The future This Chapter aimed to set out the current application of dispersion models in the field of radiological assessment of radionuclides released to atmosphere. The application of atmospheric dispersion modelling is set to continue, driven in part by regulatory requirements and by a general need to improve understanding of the impacts of all forms of air pollution at local, regional and global scales. With the increasing trend for openness and transparency in the application of scientific methods in decision making, public confidence is likely in future to play an increasingly prominent role in model acceptability. It is therefore important that continued development is carried out in a harmonised and transparent manner with significant emphasis on validation. In the regulatory context some countries may continue to require the use of specifically approved dispersion models. In others, such as the UK, a less prescriptive regime may continue

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where the choice of model may need to be justified for each application. The advent of the new generation of models means that there is no longer one type of practical dispersion model for short to medium distance applications. Ever-increasing computer power coupled with better meteorological data may enable further improvements to be made in the accuracy of predictions of long-term averages and possibly in short-term concentrations, including when and where maximum levels will occur. The inherent variability and randomness of atmospheric turbulence means that continued improvements in model accuracy will ultimately be limited. In Section 4.7.5 the practice of ensemble forecasting in the field of real-time emergency consequence assessment is described. This is a promising trend and the similar use of more than one model for the prediction of dispersion from routine discharges is now possible and may be the way forward in, for example, regulatory decision making. The potential for different models to produce different answers, which reflect the very real scientific uncertainty, can then be made more explicit. The agreement of results from models, which apply different approaches, can engender greater confidence while significant differences should be explained. One way of combining results from more than one model is to use ‘fuzzy mathematics’. For example, it is feasible to treat best estimate predictions from a number of models as ‘fuzzy numbers’ and to calculate the ‘fuzzy’ mean prediction across a group of models (Fisher, to be published). Many applications of dispersion models use a deterministic approach, whereby explicit consideration of parameter uncertainty is neglected and a single answer, without any indication of uncertainty, results. This is often acceptable if it is assumed that parameter values have been chosen so as to maximise the result when, for example, performing assessments for checking compliance with limits. However, there is an increasing need to carry out ‘realistic’ assessments, particularly where decisions are required on, for example, the optimisation of radiological protection which involve decisions about the allocation of financial and other resources. The uncertainty associated with such predictions can therefore become more important and the probabilistic treatment of dispersion model results, frequently used in the field of accident consequence assessment, may in future be desirable in other forms of assessment. Uncertainty analysis is further discussed in Chapters 1 and 12. Acknowledgements The author would like to thank Dr R. J. Timmis and Dr B. E. A. Fisher of the Environment Agency for useful discussions. References Barker, C. D. (1982). The application of the virtual source entrainment model in nuclear safety calculations. Central Electricity Generating Board, report TPRD/B/5238/N82. Briggs, G. A. (1973). Diffusion estimates for small emissions. In Atmospheric Turbulence and Diffusion Laboratory Report. NOAA/ATDL ATDL-106. Oak Ridge National Laboratory. Briggs, G. A. (1984). Plume rise and buoyancy effects. In D. Randerson (Ed.), Atmospheric Sciences and Power Production. US Technical Information Centre DOE/TIC-27601. Brown, J. (1988). The effectiveness of sheltering as a countermeasure in the event of an accident. Radiological Protection Bulletin, 97. Chilton: NRPB.

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Brown, J. & Jones, J. A. (1993). Location factors for modification of external radiation doses. Radiological Protection Bulletin, 144. Chilton: NRPB. Carruthers, D. J., Hunt, J. C. R. & Holroyd, R. J. (1989). Airflow and dispersion over complex terrain. In H. van Dop (Ed.), Proceedings of the 17th NATO-CCMS International Technical Meeting on Air Pollution Modelling and its Application (pp. 515–528). New York: Plenum Press. Carruthers, D. J., Holroyd, R. J., Hunt, J. C. R., Weng, W. S., Robins, A. G., Apsley, D. D., Thomson, D. J. & Smith, F. B. (1994). UK-ADMS: a new approach to modelling dispersion in the earth’s atmospheric boundary layer. Journal of Wind Engineering and Industrial Aerodynamics, 52, 139–153. Carruthers, D. J., Edmunds, H. A., Bennett, M., Woods, P. T., Milton, M. J. T., Robinson, R., Underwood, B. Y. & Franklin, C. J. (1996). Validation of the UK-ADMS dispersion model and assessment of its performance relative to R91 and ISC using archived LIDAR data. DoE/HMIP Report No. DoE/HMIP/RR/95/022. Castro, I. P., Snyder, W. H. & Lawson, R. E. (1988). Wind direction effects on dispersion from sources downwind of steep hills. Atmospheric Environment, 22, 2229–2238. CERC (1999). ADMS 3 User Guide. CERC. Clarke, R. H. (1979). The first report of a Working Group on Atmospheric Dispersion: A model for short and medium range dispersion of radionuclides released to the atmosphere. Harwell. NRPB-R91. London: HMSO. Collins, C. D. & Bell, J. N. B. (1996). Experimental studies on the deposition to crops of radioactive gases released from gas cooled reactors. Part 1. Carbonyl sulphide-35. Journal of Environmental Radioactivity, 30, 2, 99–115. Crawford, T. V. (1978). Atmospheric transport of radionuclides. In Proceedings of a Workshop on Evaluation of Models used for the Environmental Assessment of Radionuclide Releases. Oak Ridge National Laboratory, CONF770901. Etling, D. (1990). On plume meandering under stable stratification. Atmospheric Environment, 24A, 1979–1985. Fisher, B. E. A. (to be published). Treating predictions from more than one model. In Atmospheric Dispersion Modelling Liaison Committee Annual Report 2001–2002. Chilton: NRPB. Galmarini, S., Bianconi, R., Bellasio, R. & Graziani, G. (2001). Forecasting the consequences of accidental releases of radionuclides in the atmosphere from ensemble dispersion modelling. Journal of Environmental Radioactivity, 57, 203–219. Gifford, F. A (1961). Use of routine meteorological observations for estimating atmospheric dispersion. Nuclear Safety, 2, 47–57. Gifford, F. A. (1968). Diffusion in the lower layers of the atmosphere. In D. Slade (Ed.), Meteorology and Atomic Energy (p. 65). TID-24190. US Atomic Energy Commission. Hall, D. J., Spanton, A. M., Dunkerley, F., Bennett, M. & Griffiths, R. F. (2000a). A review of dispersion intercomparison studies using ISC, R91, AERMOD and ADMS. R&D Technical Report P353. Bristol: Environment Agency. Hall, D. J., Spanton, A. M., Dunkerley, F., Bennett, M. & Griffiths, R. F. (2000b). An intercomparison of the AERMOD, ADMS and ISC dispersion models for regulatory applications. R&D Technical Report P362. Bristol: Environment Agency. Hall, R. C. (1997). Application of computational fluid dynamics to near-field atmospheric dispersion. In Atmospheric Dispersion Modelling Liaison Committee Annual Report 1995–1996 (Annex B). Chilton: NRPB-R292. Hamby, D. M. (2002). The Gaussian atmospheric transport model and its sensitivity to the joint frequency distribution and parametric variability. Health Physics, 82, 64–73. Hanna, S. R., Egan, B. A., Purdum, J. & Wagler, J. (1999). Evaluation of the ADMS, AERMOD and ISC3 models with the Optex, Duke Forest, Kincaid, Indianapolis and Lovett field datasets. In Proceedings of the sixth international conference on harmonisation within atmospheric dispersion modelling for regulatory purposes. Rouen, France. Hanna, S. R. (1992). Lateral dispersion in light wind stable conditions. Il Nuovo Cimento, 13, 889–902. Hill, R. A., Teasdale, I., Chambers, N., Puxley, C. & Parker, T. (1999). Comparison between field measurements of krypton-85 around the BNFL Sellafield reprocessing plant and the predictions of the NRPB-R91 and UK-ADMS atmospheric dispersion models. In Proceedings of the sixth international conference on harmonisation within atmospheric dispersion modelling for regulatory purposes. Rouen, France. Hosker, R. P. (1974). Estimates of dry deposition and plume depletion over forests and grasslands. In Proceedings of the Symposium on Physical behaviour of Radioactive Contaminants in the Atmosphere (p. 291). Vienna: IAEA. Hosker, R. P. (1984). Flow and diffusion near obstacles. In D. Randerson (Ed.), Atmospheric Sciences and Power Production. US Technical Information Centre DOE/TIC-27601.

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Hunt, J. C. R., Holroyd, R. J., Carruthers, D. J., Robins, A. G., Apsley, D. D., Smith, F. B. & Thomson, D. J. (1990). Developments in modelling air pollution for regulatory purposes. In H. van Dop and D. G. Stein (Eds.), Proceedings of the 18th NATO-CCMS International Technical Meeting on Air Pollution Modelling and its Application (pp. 17–59). New York: Plenum Press. Hunt, J. C. R. (1997). Atmospheric diffusion from a steady source in a turbulent airflow at low mean speeds. In Atmospheric Dispersion Modelling Liaison Committee Annual Report 1995–1996 (Appendix to Annex A). Chilton: NRPB-R292. Jones, J. A. (1981a). The second report of a Working Group on Atmospheric Dispersion – A procedure to include deposition in the model for short and medium range atmospheric dispersion of radionuclides. Chilton: NRPBR122. Jones, J. A. (1981b). The third report of a Working Group on Atmospheric Dispersion – The estimation of long range dispersion and deposition of continuous releases of radionuclides to atmosphere. Chilton: NRPB-R123. Jones, J. A. (1986a). The sixth report of a Working Group on Atmospheric Dispersion – Modelling wet deposition from a short release. Chilton: NRPB-R198. Jones, J. A. (1986b). The seventh report of a Working Group on Atmospheric Dispersion – The uncertainty in dispersion estimates obtained from the Working Group models. Chilton: NRPB-R199. Jones, J. A. (1989). What is required of dispersion models and do they meet the requirements? In H. van Dop (Ed.), Proceedings of the 17th NATO-CCMS International Technical Meeting on Air Pollution Modelling and its Application (pp. 515–528). New York: Plenum Press. Jones, J. A. (1997). Atmospheric dispersion at low wind speed. In Atmospheric Dispersion Modelling Liaison Committee Annual Report 1995–1996 (Annex A). Chilton: NRPB-R292. Kaasik, M. (1998). Validation of models AEROFOUR and AEROPOL using model validation kit. In Proceedings of 5th International Conference on Harmonisation within Atmospheric Dispersion Modelling for Regulatory Purposes. Mol: France. Little, C. A. & Miller, C. W. (1979). The uncertainty associated with selected environmental transport models. Oak Ridge National Laboratory Report, ORNL-5528. MacDonald, H. F., Foster, P. M., Robins, A. G. & Thompson, I. M. G. (1988). Improved estimates of external gamma dose rates in the environs of Hinkley Point power station. Central Electricity Generating Board, RD/B/6027/R88. Maryon, R. (1996). The NAME dispersion model. Numerical Weather Prediction Gazette, 3rd Quarter 1996. Bracknell: UK Meteorological Office. Mayall, A., Cabianca, T., Attwood, C., Fayers, C. A., Smith, J. G., Penfold, J., Steadman, D., Martin, G., Morris, T. P. & Simmonds, J. R. (1997). Installing and using the PC system for assessing the radiological impact of routine releases. EUR 17791 EN/NRPB-SR296. Luxembourg: European Commission. Mikkelsen, T. & Desiato, F. (1993). Atmospheric dispersion models and pre-processing of meteorological data for real time application. Radiation Protection Dosimetry, 50, 205–218. Moore, D. J. (1976). Calculation of ground level concentration for different sampling periods and source locations. In Atmospheric Pollution (p. 516). Amsterdam: Elsevier. National Radiological Protection Board (1997). Assessment of skin doses. In Documents of the NRPB, 8 (3). Chilton: NRPB. Olesen, H. R. (1995). Regulatory dispersion modelling in Denmark. Workshop on Operational Short-range Atmospheric Dispersion Models for Environmental Impact Assessment in Europe. International Journal of Environment and Pollution, 5, 412–417. Olesen, H. R. (2001). Ten years of harmonisation activities: past, present and future. In Proceedings of the seventh international conference on harmonisation within atmospheric dispersion modelling for regulatory purposes. Belgirate, Italy. Overcamp, T. J. (1991). Modelling gamma absorbed dose due to meandering plumes. Health Physics, 61, 111–115. Pasquill, F. (1961). The estimation of the dispersion of windborne material. Meteorological Magazine, 90, 1063, 33–49. Pecha, P., Nedoma, P. & Pechova E. (1999). Modelling of radionuclide transport due to atmospheric releases used in the various stages of nuclear power plant design. In Proceedings of the sixth international conference on harmonisation within atmospheric dispersion modelling for regulatory purposes. Rouen, France. Pechinger, U. & Petz, E. (1995). Model evaluation of the Austrian Gaussian plume model ON M 9440: comparison with the Kincaid data set. International Journal of Environment and Pollution, 5, 338–349.

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Robeau, D. & Oishi, S. (1991). Estimation of the source term in an accident from radiological measurements. In J. Sinnaeve (Ed.), Improvement of long-distance atmospheric transfer models. EUR 12549 EN. Luxembourg: European Commission. Robins, A. G. (2001). Review of flow and dispersion in the vicinity of groups of buildings. In Atmospheric Dispersion Modelling Liaison Committee Annual Report 1998–1999 (Annex B). Chilton: NRPB-R322. Royal Meteorological Society (1995). Policy Statement. Atmospheric Dispersion Modelling: Guidelines on the justification of choice and use of models, and the communication and reporting of results. RMS/DoE. Sehmel, G. A. (1980). Particles and gas dry deposition: a review. Atmospheric Environment, 14, 983–1011. Simmonds, J. R., Lawson, G. & Mayall, A. (1995). Methodology for assessing the radiological consequences of routine releases of radionuclides to the environment. EUR 15760 EN. Luxembourg: European Commission. Smith, F. B. (1973). A scheme for estimating the vertical dispersion of a plume from a source near ground level. In Proceedings of the 3rd Meeting of an expert panel on air pollution modelling. Brussels: NATO-CCMS Report 14. Smith, F. B. (1989). Short range diffusion in convective light wind conditions. In H. van Dop (Ed.), Proceedings of the 17th NATO-CCMS International Technical Meeting on Air Pollution Modelling and its Application (pp. 515–528). New York: Plenum Press. Smith, F. B. (1992). Low wind-speed meteorology. Meteorological Magazine, 121, 141–151. Sutton, O. G. (1932). The theory of eddy diffusion in the atmosphere. In Proceedings of the Royal Society, 135A, 143–165. Underwood, B. Y. (1985). The modelling of gravitational settling in the assessment of accidental releases of radioactivity. EUR 10097 EN. Luxembourg: European Commission. Underwood, B. Y. (2001). Review of deposition velocity and washout coefficient. In Atmospheric Dispersion Modelling Liaison Committee Annual Report 1998–1999 (Annex A). Chilton: NRPB-R322. United States Environmental Protection Agency (USEPA) (2000). Updated user’s guide for CAP88-PC version 2.0. EPA 402-R-00-004. Van Dop, H., Addis, R., Fraser, G., Girardi, F., Graziani, G., Inoue, Y., Kelly, N., Klug, W., Kulmala, A., Nodop, K. & Pretel, J. (1998). ETEX: A European tracer experiment; observations, dispersion modelling and emergency response. Atmospheric Environment, 32, 4089–4094.

MODELLING RADIOACTIVITY IN THE ENVIRONMENT E. Marian Scott (Editor) © 2003 Elsevier Science Ltd. All rights reserved

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Chapter 3

Modelling radioactivity in the marine environment: The application of hydrodynamic circulation models for simulating oceanic dispersion of radioactivity Ingo H. Harms a , Michael J. Karcher b , Hans Burchard c a Institute for Oceanography, Centre for Marine and Climate Research, University of Hamburg, Germany b Alfred-Wegener-Institute for Polar and Marine Research (AWI), Bremerhaven, Germany c Institute for Baltic Sea Research, Warnemuende (IOW), Rostock, Germany

1. Introduction Numerical models are increasingly used in environmental management for consulting purposes, as an assessment tool, or simply as a help to evaluate the environmental status of specific regions or situations. Frequently, models are the only tools to forecast the consequences of accidental release and to support decision making concerning remedial measures. Model based forecasts can help to detect and observe areas of high contamination but may also serve for further dose risk assessments. Dispersion scenarios may also help to monitor areas that are affected by permanent releases from reprocessing plants or other nuclear facilities. But it is not only environmental management or, more specifically, radio-ecology that benefits from numerical dispersion studies. Oceanographers and modellers are highly interested in verifying their numerical model results. Besides classical hydrographic parameters, anthropogenic radioactivity can be a valuable oceanic tracer which allows for a thorough validation of simulated circulation and dispersion scenarios. Well suited for this purpose are reprocessing plant releases because the release function is usually known and the fate of radionuclides is well monitored. This chapter reviews the application of numerical hydrodynamic models to the transport of anthropogenic radioactivity in the ocean. There is of course a large number of such applications. However, we do not claim to present a complete review. Our intention is to summarise different approaches and model configurations, to evaluate strengths and weaknesses and to give guidelines to users. 55

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1.1. Basic configurations Numerical models consist of mathematical equations which are solved at discrete time steps on a regular or irregular grid that covers the model domain, i.e. the region of interest. A model study thus starts with the selection of a model domain, the definition of the model grid and the determination of the grid size. The regional focus and the spatial extent is the most obvious difference between various model studies. Following this regional aspect, hydrodynamic-model based dispersion studies on radioactivity can be separated into three groups: • near field or local scale applications that deal with estuaries, fjords, bays or straits. Examples are the English Channel (Periánez & Reguera, 1999; Periánez, 2000), the Irish Sea (Periánez, 1998, 1999), the Muroroa Atoll (Deleersnijder et al., 1997; Tartinville et al., 1997) or Novaya Semlya bays (Harms & Povinec, 1999). Typical size of the model domain: 1–100 km, typical grid size: 0.01–5 kilometres; • medium range or regional scale applications that deal with shelf areas or semi-enclosed seas like the North Sea (Pohlmann et al., 1987; Schönfeld, 1995; Breton & Salomon, 1995; Salomon et al., 1995), the Black Sea (Margvelashvily et al., 1997; 1998; Staneva et al., 1999) or the Barents Sea and Kara Sea (Harms, 1997; Baxter et al., 1998; Nies et al., 1998). Typical size of the model domain: 100–1000 km, typical grid size: 5–50 kilometres; • far field or global / basin scale applications that deal with large ocean basins or whole oceans like the North Atlantic / Arctic Ocean (Preller & Posey, 1997; Nies et al., 1998; Preller & Cheng, 1999) or the Pacific Ocean (Ribbe & Tomczak, 1990). Also global ocean models belong to this category. Typical size of the model domain: 1000 km – global scale, typical grid size: 10–500 kilometres. The selection of a model domain and grid is first of all a question of the spatial range which should be covered by the model. However, the domain selection can also be influenced by the type of source that has to be considered. Atmospheric sources like fallout from bomb testing or from accidents can be very diffuse or scattered and might affect surface waters on large spatial scales. This would require large model domains and grids. Marine sources, however, discharge directly into the marine environment. They mostly represent well known point sources such as outlets of reprocessing plants, leaking nuclear waste units or sunken nuclear ship reactors allowing also for locally confined applications. Hydrodynamic model studies on dispersion of marine radioactivity involve basically two model types: a hydrodynamic circulation model (HD model) and a transport model. The HD model calculates the three-dimensional flow field whereas the transport model calculates the dispersion of a tracer according to the previously computed flow. Both models can be linked in a way that the two components run synchronously with a common time step. This coupling mode is called the on-line mode. But it is also possible to run both models separately. Transport models are often applied as stand-alone models that use previously calculated time series of flow fields from hydrodynamic models. This type of coupling is referred to as the off-line mode. In dispersion studies, often the HD model component remains in the background while the focus is on the transport model component. However, the HD model component is of equal importance, because the computed flow fields have a decisive influence on transport and dispersion. We will therefore focus this chapter not only on transport modelling but highlight also aspects of the hydrodynamic modelling.

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2. Hydrodynamic modelling It is a complex deduction from detailed equations of motion in fluid dynamics down to computer codes for the prediction of oceanic processes. The basic structure of a numerical hydrodynamic model, starting from first principles is outlined briefly here. Due to the limited space we will not be able to treat the relevant topics in much detail. Readers with a more fundamental interest are referred to text books such as Haidvogel & Beckmann (1999) and the citations therein. 2.1. Model physics Apart from details of air-sea exchange and flow-sediment interactions, most relevant processes in the ocean are well-described by the Navier–Stokes equations and viscous equations for potential temperature and salinity. This system of equations for the velocity components u, v and w, the potential temperature T , the salinity S, the pressure p and the potential density ρ, which are the state variables, is mathematically closed and thus well-defined. These equations describe the whole range of spatial and temporal scales from smallest dissipative eddies (orders of millimetres and milliseconds) up to the global deep water circulation known as the conveyor belt (orders of 103 years and 104 km). Since all these scales are intercorrelated through non-linear dynamics, investigations focussed on a certain scale (e.g. tides in a shelf sea) would require the prescription of all larger scales through initial and boundary conditions and the resolution of all smaller scales. Since the latter is and will even in the far future not be feasible due to limited computer resources, the smaller scales have to be parameterised. This is usually done by separating each state variable e into a mean part e¯ (the part of major interest) and a fluctuating part e˙ (due to turbulence) and then averaging the complete equations. Due to non-linearities, higher order terms appear, and the equations have to be closed by empirical assumptions. Further simplifications are the hydrostatic assumption (pressure depends only on the density structure) and the Boussinesq assumption (infinite speed of sound, leading to incompressibility of the water). For a more detailed analysis, see e.g. Sander (1998). After these manipulations, a closed set of equations (1)–(6) is obtained, the so-called hydrostatic primitive equations (see e.g. Haidvogel & Beckmann, 1999). For simplicity, the equations are formulated in Cartesian coordinates x (northward), y (eastward) and z (upward) with the respective velocity components u, v and w. The first set, (1) and (2), form the momentum equations for the prediction of the horizontal velocity in x- and y-direction: ∂t u + ∂x (u2 ) + ∂y (uv) + ∂z (uw) − ∂z (Kv ∂z u) + Fu − f v  ζ ρ(ζ ) ∂x ζ + ∂x b dz , = −g ρ0 z ∂t , v + ∂x (vu) + ∂y (v 2 ) + ∂z (vw) − ∂z (Kv ∂z v) + Fv + f u  ζ ρ(ζ ) = −g ∂y ζ + ∂y b dz . ρ0 z

(1)

(2)

Starting on the left hand side, ∂t denotes the partial differential operator ∂/∂t, which expresses the derivation of u or v in time t. The same notation is used with the spatial derivatives ∂x , ∂y

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and ∂z . The expression ∂x (u2 ) + ∂y (uv) + ∂z (uw) resp. ∂x (vu) + ∂y (v 2 ) + ∂z (vw), denotes the advection of momentum, and is often referred to as the ‘non-linear term’. The following two terms refer to sub-grid scale diffusion of momentum which needs to be parametrized and seperated into three dimensions. The spatial differentiation requires consideration of the different nature of diffusion in ‘along-isopycnal’1 and ‘cross-isopycnal’ directions. A simple approach is to approximate along- and cross-isopycnal directions by the horizontal resp. vertical direction in the model grid which is justified if the density levels are aligned horizontally and density gradients occur only in the vertical direction. Cross isopycnal diffusion of momentum then refers to vertical mixing and is parametrized by ∂z (Kv ∂z u) and ∂z (Kv ∂z v), respectively. Kv is the effective cross-isopycnal mixing coefficient due to vertical turbulence, sometimes referred to as vertical ‘eddy viscosity’ coefficient. Fu and Fv represent meso-scale mixing parameterisations to account for the horizontal ‘along-isopycnal’ diffusion of momentum. These parametrizations should depend on the spatial resolution of the model grid, because small grid sizes resolve more turbulent processes than large grid sizes and hence need less parameterised diffusion. f u resp. f v describes the acceleration due to the earth’s rotation with f = 2 sin() being the Coriolis parameter,  = 7.289 × 10−5 s−1 the angular velocity of the earth, and  the latitude. ζ The right hand side expression z ∂x b dz describes the ‘internal’ pressure due to horizontal density differences in x-direction. This term includes the ‘buoyancy’ b = −g(ρ − ρ0 /ρ0 ) with ρ being the potential density, ρ0 the reference density and g the gravitational acceleration. ) The other right hand side expression g ρ(ζ ρ0 ∂x ζ describes the ‘external’ pressure due to xdirectional gradients in the sea surface elevation ζ . The equation of continuity, ∂x u + ∂y v + ∂z w = 0

(3)

allows prediction of the vertical velocity component w from the divergence of the horizontal flow field ∂x u + ∂y v. Two prognostic equations describe the transport of temperature T and salinity S due to horizontal and vertical advection and diffusion: ∂t T + ∂x (uT ) + ∂y (vT ) + ∂z (wT ) − ∂z (Av ∂z T ) + FT = QT ,

(4)

∂t S + ∂x (uS) + ∂y (vS) + ∂z (wS) − ∂z (Av ∂z S) + FS = Qs.

(5)

The physical meaning of the individual terms is analogous to the momentum equations (1) and (2). In the temperature equation (4) for example, ∂t T is the time derivation of T and ∂x (uT ) + ∂y (vT ) + ∂z (wT ) represents the three-dimensional advection of T . The vertical diffusion of T is given by ∂z (Av ∂z T ), with Av being the mixing coefficient due to vertical turbulence, sometimes called the vertical ‘eddy diffusivity’ coefficient. Horizontal diffusion parameterisations (i.e. horizontal meso-scale mixing of T ) are represented by FT . A realisation of this term is described in Section 3.1. The external sources on the right hand side, QT and QS , denote fluxes at the sea surface such as heat exchange with the atmosphere or salt fluxes due to precipitation, evaporation or ice formation. 1 Isopycnals = lines of constant density.

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The density ρ is calculated by means of an empirical equation of state as function of potential temperature, salinity and hydrostatic pressure p0 : ρ = ρ(T , S, p0 ).

(6)

This set of equations is nearly identical for almost all ocean models. Physical differences of such models arise through different parameterisations of turbulent mixing coefficients Kv , Av and the isopycnal mixing terms F . Other, more specific differences are related to parameterisations of processes such as sea ice dynamics, sediment and suspended matter dynamics and coupling to the atmosphere through cross surface fluxes (e.g. QT ). 2.2. Model numerics In order to prepare equations (1)–(6) for approximation by digital computers, they have to be discretised in space and time. The discretisations in time, in the horizontal and in the vertical direction can basically be carried out independently. This gives a large number of degrees of freedom for discretising the physical equations and explains the large number of existing numerical ocean circulation models. The strategy for the discretisation is not only a technical matter, since the optimal discretisation strongly depends on the physical processes and scales under investigation. Therefore, the basic tendency is to construct numerical models as general as technically feasible and economically justified. Unlike in atmospheric sciences, where spectral models based on series of basic functions are frequently used, in ocean models, numerical discretisations are commonly carried out in the form of finite differences (structured grids) or finite elements (unstructured grids). Widely used types of discretisation in space and time are presented briefly in the following. 2.2.1. Vertical discretisation For vertical discretisation, the model domain is divided into a number of non-intersecting layers over which the equations are vertically integrated subsequently. These layers need to be continuous in time and space. They may be of arbitrary shape and they can be variable in time. They can intersect with the surface and the bottom, and their thickness can locally be zero (virtual or massless layers). Special realisations of such vertical discretisations are geopotential coordinates (z-levels), which do intersect with the bottom topography (Fig. 1). These, however, may cause numerical problems due to the step-like approximation of bottom bathymetry (for a discussion, see Beckmann & Döscher, 1997). Geopotential coordinates can also intersect with the sloping surface, see e.g. the model by Casulli & Cheng (1992). However, usually the surface is constrained to vary in the uppermost layer only. Another possible discretisation adjusts the layers in such a way that the top of the upper layer is the sea surface and the base of the lowest layer is the sea bottom. Such models are referred to as general vertical coordinate models; see e.g. Burchard & Petersen (1997) and the references therein. Two realisations of general vertical coordinates are shown in Fig. 2. The most well-known member of this model class is the sigma coordinate model where each layer covers a constant fraction of the water depth. The most striking disadvantage of such

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Fig. 1. Example for a vertical discretisation in a z-level type model.

Fig. 2. Cross-section through the North Sea from Scarborough in England to Esbjerg in Denmark. Upper panel: σ -transformation; lower panel: mixed-layer transformation (Burchard & Petersen, 1997). Note that for mixed-layer transformation the layer thickness near the surface is less dependent on the water depth.

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Fig. 3. Schematic vertical cross section of layer thickness distribution for an isopycnal coordinate scheme (Oberhuber, 1993).

general vertical coordinates is the pressure gradient error which occurs when sloping coordinate surfaces intersect with isopycnal surfaces. Remedies for such problems are a higher-order discretisation of the internal pressure gradient (last terms in equations (1) and (2)), high horizontal resolution or subtraction of a reference density distribution (see Blumberg & Mellor, 1987). One way to completely solve the above mentioned pressure gradient error is the introduction of so-called isopycnal coordinates (Fig. 3), in which all layers have the same potential density, see e.g. Bleck & Smith (1990) or Oberhuber (1993). Isopycnal models have the disadvantage, however, that well-mixed or unstably stratified areas cannot be described properly. Thus they are commonly coupled to integral mixed layer models which prognostically calculate the mixed layer depth. Since all three model types described above (geopotential, general vertical and isopycnal coordinates) have specific problems, attempts have been made to construct hybrid models which interpolate between these types of models. Examples are the s-coordinate model by Gerdes (1993), interpolating between geopotential and general vertical coordinates, and the double σ -coordinate model by Beckers (1991). A hybrid model combining all three types of models has recently been realized as well (see Halliwell et al., 1998). 2.2.2. Horizontal discretisation For the horizontal discretisation, the model domain is covered with a mesh which divides it into prisms. The vertical edges are aligned with the vertical coordinate and each prism is divided into a number of layers as a consequence of the vertical discretisation. Structured model grids are obtained by transforming Cartesian (rectangular) grids (Fig. 4) into spherical or curvi-linear grids. The former class is commonly used for large scale applications. The latter allows for fitting coordinate lines to coastlines or other bathymetric features or for focusing the spatial resolution to a region of major interest (Fig. 5). Unstructured grids allow for much more flexibility in fitting the grid to the geometry of the domain under investigation (Fig. 6).

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Fig. 4. Realisation of a horizontal equidistant finite difference grid (9.4 km), for a regional model of the Kara Sea (Harms & Karcher, 1999).

Generally, as concerns grid construction, a higher flexibility has to be paid for with higher computational costs per grid point. Since the necessary number of grid points often decreases significantly with grid flexibility, the decision on certain grid designs is a trade-off. Basically two methods are used for the horizontal discretisation, finite differences and finite elements. For finite differences, the differential expressions are approximated by difference expressions of specified order of accuracy. In a second order scheme for example, the discretisation error decreases with the square of the grid spacing. Usually, a finite volume approach is combined with the finite difference method, thus guaranteeing conservation of mass and property for the discrete solution which is a prerequisite for accurate dispersion studies. For finite difference methods, discrete values of the state variables are located at given places in the discrete grid. For ocean models, staggered grids have been found to be advantageous. Arakawa & Lamb (1977) classified various approaches for staggered, structured grids. In the Arakawa C-grid, the velocity components are centred on the surfaces which are orthogonal to their respective coordinates, a method which allows for accurate approximation for advective and diffusive transports. However, for coarse spatial resolution, the approximation of rotational terms is poor, such that for large-scale, coarseresolution models so-called B- or E-grids are preferred where the horizontal velocity components are collocated. Finite element methods are based on Galerkin approximations where the state variables for each element are approximated with polynomial or other basic functions. Such methods are more accurate but also computationally rather expensive per grid element since they are usually used for unstructured grids. 2.2.3. Temporal discretisation In contrast to the spatial discretisation, where usually the highest affordable resolution is used, the temporal discretisation is as coarse as possible in order to save computational time. Unless

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Fig. 5. Orthogonal conformed curvilinear grid for the global ocean with strong zooming in the Nordic Seas (Mikolajewicz, pers. com.).

implicit schemes are used for time stepping, two stability criteria for limiting the time step t have to be obeyed to obtain a stable solution of the equations. The most restrictive criterion refers to the propagation speed of long gravity waves which depends on the water depth D. An example of this criterion, valid for a discretisation on an Arakawa C-grid including rotation is given by: t <

1 ∇x √ . 2 gD

(7)

Another limiting criterion refers to the horizontal advection velocities umax , vmax which are usually much smaller than the propagation speed of gravity waves: t <

x . max{umax , vmax }

(8)

The background of the two criteria is the requirement, that neither the gravity waves nor fluid particles are allowed to travel further than one grid box x during one time step t. Since

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Fig. 6. Unstructured finite element grid for the lagoon of Venice, Italy. Note that the channels are much finer resolved than broader shallow areas (Umgiesser, pers. com.).

criterion (7) is much more limiting than (8), a so-called mode splitting according to Madala & Piacsek (1977) may be carried out. It decouples the vertical integral of the momentum equations (1) and (2) and the continuity equation (3), which are responsible for the long gravity waves (i.e. the external mode), from the equations of motion for each depth level (i.e. the internal mode). These two modes can be discretised explicitly with different time steps, both according to the respective time step limitations. However, the interaction between the two modes has to be considered carefully in order not to loose model consistency. A second method to avoid stability problems is to solve the external mode implicitly such that a violation of the time step limitation (7) does not lead to numerical model instabilities. In this case, the internal and the external mode are calculated with the same time step. A linear system of equations for the new two-dimensional field of surface elevations results from this procedure which is either solved directly or iteratively (Backhaus, 1985) or by means of subsequent application of one-dimensional solvers (Alternating-Direction Implicit, see e.g. Abbott & Minns, 1997).

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There is no general agreement on explicit or implicit treatment of the long gravity waves. For investigations of the propagation of gravity (e.g. tidal) waves, explicit schemes are believed to be more accurate, since implicit schemes tend to damp those out. This is, on the other hand, often a desirable property of implicit schemes in order to stabilise the numerical model simulations. 2.3. Some model realisations In this section, three typical ocean circulation models will be briefly described. All three models are based on hydrostatic primitive equations and have a free surface. Their most significant difference is the vertical discretisation. The chosen models are so called ‘community models’ which can be downloaded via internet from their web sites, together with user manuals and test cases. Users may also receive support from a specific user group. One model applies geopotential coordinates (MOM), one model sigma-coordinates (POM) and one model is based on isopycnal coordinates (MICOM). We selected these three models among all others because they are widely used and have been well-maintained by their developers for at least a decade. 2.3.1. Modular Ocean Model (MOM) This model has been developed over the last three decades at the Geophysical Fluid Mechanics Laboratory in Princeton, New Jersey. The model can be downloaded together with a global ocean test case at the web site http://www.gfdl.noaa.gov/MOM/MOM.html. The most recent model generation, called MOM3, has a free surface and is based on geopotential coordinates. A mode splitting technique is used for calculating the external and the internal mode with different time steps. For horizontal discretisation, a B-grid is used which expresses the design of MOM for large spatial scales. Various models for vertical and horizontal exchange are included. MOM has been applied to basically all regions in the world ocean and specifically to the global ocean. 2.3.2. Princeton Ocean Model (POM) POM has been developed since the late 1970’s in the Program in Atmospheric and Ocean Sciences in Princeton, New Jersey (Blumberg & Mellor, 1987). The model code can be downloaded from the web at http://www.aos.princeton.edu/WWWPUBLIC/htdocs.pom/. POM uses sigma coordinates, but recently a version with general vertical coordinates has been developed as well. Like MOM, POM is based on mode splitting in order to treat the external and the internal mode with different time steps. In the horizontal plane, a flexible, curvi-linear C-grid can be fitted to coastlines and topographic features. For vertical turbulent exchange, the Mellor & Yamada (1982) two-equation model is used. A hierarchy of lateral open boundary conditions is included as well, allowing for use of POM in coastal regions and other limited areas. 2.3.3. Miami Isopycnal Ocean Model (MICOM) This isopycnal coordinate model has been developed during the last 15 years at the Rosenstiel School of Marine and Atmospheric Science in Miami, Florida. The code can be downloaded from the web at http://panoramix.rsmas.miami.edu/micom/. Isopycnal coordinates are closest to nature because advective and diffusive transports occur mainly along isopycnals. The problem of the internal pressure gradient error is minimised, so that MICOM is well suited e.g.

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for overflow scenarios. The thickness of the surface layer is determined by a bulk turbulence model of Kraus & Turner (1967) type. In the horizontal, MICOM is, like POM, discretised on a C-grid, with curvi-linear coordinates. MICOM has been applied to medium and large scale studies. However, it is not suitable for shelf sea studies where well mixed shear flow dominates the dynamics. For such scenarios, HYCOM, a hybrid version of MICOM (see the MICOM web page) could be an interesting option. For more information and details on other ocean circulation models, the reader is referred to the recent review book by Haidvogel & Beckmann (1999).

3. Transport modelling of dissolved radioactivity Like HD models, numerical transport models operate step-wise in time, on a specified grid and in a selected model domain. Within each grid cell and each time step, a transport equation is solved which calculates the advection and the diffusion of a substance or a tracer. The advection term is determined by the circulation, i.e. the three-dimensional flow field, whereas the diffusion represents the sub-grid scale turbulent mixing. In general form, the time dependent total radionuclide concentration in the water R, in Becquerel per cubic metre (Bq/m3 ), is then given by: ∂t R + ∂x (uR) + ∂y (vR) + ∂z (wR) − ∂z (Av ∂z R) + FR = QR − λR.

(9)

This transport equation is formulated in the same way as equation (4) or (5): ∂t R is the time derivation, ∂x (uR) + ∂v (vR) + ∂z (wR) represents horizontal and vertical advection of R and ∂z (Av ∂z R) gives the vertical turbulent diffusion with Av being the vertical ‘eddy diffusivity’ coefficient. Horizontal mixing parametrizations (i.e. the horizontal diffusion) are denoted by FR . QR is a source term that has to be added in case a specific release should be simulated. If the half life decay of the considered radionuclide is much smaller than the simulation period (e.g. simulated 129 I dispersion over two years), radioactive decay can be omitted. In other cases (e.g. simulated Sellafield 137 Cs dispersion over the past 30 years), the decay term λR has to be considered. There are two basic approaches for discretising equation (9): the Eulerian approach that calculates the exchange of radionuclide concentrations between adjacent grid boxes in time and space and the Lagrangian approach that applies a particle tracking method which follows a trajectory in space and time. 3.1. The advection-diffusion equation (the Eulerian approach) The Eulerian approach is the most common approach in numerical dispersion modelling. It is frequently used in HD models for the dispersion of dynamically active variables like temperature and salinity. Very often, the transport algorithm for temperature or salinity (4 or 5) is simply copied and assigned to a ‘third’ passive tracer like e.g. radioactivity. Transferring such a conservative approach to radioactivity means that radionuclides drift in a soluble form in the water column.

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The horizontal diffusion in x- and y-direction, FR , can be written similar to the vertical diffusion term as FR = ∂x (Ah ∂x R) + ∂y (Ah ∂y R) with Ah being the horizontal mixing or ‘eddy diffusivity’ coefficient. As outlined before (c.f. Section 2.1), the value of Ah should depend on the horizontal grid size which means that large grid sizes require large diffusion coefficients and vice versa. Ah coefficients are usually kept as small as possible to avoid too strong smoothing. Very often horizontal diffusion is even omitted because uncontrolled numerical diffusion may be introduced through the numerical approximation of the advection scheme (see below). Both, horizontal and vertical mixing coefficients can be constant and homogeneous but may also vary in time and space. It is a well accepted approach, to use for radionuclides or other passive tracers the same horizontal and vertical ‘eddy diffusivities’ as for heat and salt (c.f. equations (4) and (5)). Vertical ‘eddy diffusivities’ may also be related to the parameterisations of turbulent mixing of momentum (e.g. the ‘eddy viscosities’) as outlined for equations (1) and (2). Applying the advection-diffusion equation in a numerical model by means of the Eulerian approach requires the discretisation of equation (9) in time and space (i.e. the numerical approximation as described for HD models above). This is a critical point because the numerical approximation of the advection term may lead to problems concerning mass conservation, numerical (artificial) diffusion, ‘over-shootings’ or the general robustness of the solution (Griffies et al., 2000). Together with the flow field, the numerical advection scheme determines the quality of the dispersion study. Two well known simple advection schemes exist: centred schemes that are based on spatial central differences and ‘up-stream’ schemes that ‘look’ for the direction from which a property has to be advected (i.e. ‘up-stream’). We take these two classical schemes as examples to describe their influence in radiological dispersion studies. Up-stream schemes are often called dissipative schemes because they create high amounts of numerical or artificial diffusion (Griffies et al., 2000). With these schemes, sharp fronts, or filaments of contaminated water will be smoothed. Up-stream schemes are very robust algorithms that avoid ‘over-shootings’ even if strong instantaneous releases into ‘clean’ environments (i.e. in worst case scenarios) are performed. The scheme is numerically efficient and thus an appropriate tool for multiple assessment studies and order-of-magnitude estimation e.g. after accidents. Critical applications for up-stream schemes would be e.g. long term releases in large model domains with low concentrations and high spatial variability in the flow field. Centred schemes are not completely free of numerical diffusion but it is much smaller than with up-stream schemes. However, centred schemes suffer from dispersive errors that often result in over-shoots and artificial extrema (Griffies et al., 2000). This might lead to an unrealistically patchy structure or to wrong peak concentrations. Since both, centred and up-stream schemes suffer from the drawbacks pointed out above, higher-order schemes such as the ULTIMATE-QUICKEST scheme (Leonard et al., 1995) have been developed. Such schemes are generally upstream-biased, with little numerical diffusion and dispersion. Fully monotone (non-dispersive) higher-order schemes can only be achieved by non-linear combination with simple upwind schemes. Among such methods are the TVD-scheme (total variation diminishing, LeVeque, 1990) and the FCT-scheme (flux corrected transport) described and tested by Gerdes et al. (1991). A description and overview of

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different advection schemes and their strengths and weaknesses in ocean and tracer modelling can be found e.g. in Webb et al. (1998), Hecht et al. (1995) or Griffies et al. (2000). If the numerical advection scheme is chosen and equation (9) is discretised on the model grid, initial and boundary conditions have to be specified for the model domain. At lateral open boundaries for example, the in- and outflow conditions for R have to be prescribed. A simple but common approach is a so-called zero gradient condition ∂Rt /∂n = 0, with n being the direction normal to the boundary. This condition would allow for an export of radionuclide concentrations out of the model domain. Horizontal and vertical boundary conditions may be used to prescribe other sources or sinks such as fluxes at the sea surface (fallout) or fluxes at the sea bed (sediment interaction). Other source or sink terms would be the reduction of radioactive concentrations due to particle scavenging. These non-conservative processes are discussed later. Starting a dispersion simulation also requires initial conditions. Each grid point has to be initialised with an initial concentration of radionuclides. This can be a background activity but also ‘zero’ activity if a release into a non contaminated environment has to be performed. Examples of the Eulerian approach and the application of advection-diffusion algorithms in transport modelling are given in Chapter 10. 3.2. The particle tracking method (the Lagrangian approach) The transport equation for a particle tracking model is identical to equation (9). In the Lagrangian notation, however, the property R (i.e. the radionuclide concentration) is represented by a statistically significant number of particles or tracers and the transport of R is achieved by moving each particle from location x0 to a new position x1 (see e.g. Maier-Reimer & Sündermann, 1982). When diffusion is neglected, a particle thus propagates along a trajectory which is determined by the velocity v and the time step t. The new position x1 of a tracer after time step t is thus given by x1 = x0 + vt. Three-dimensional diffusion is commonly discretised by the so-called random-walk or ‘Monte Carlo method’ which introduces some ‘inaccuracy’ by moving particles randomly away from the original trajectory within a range, depending on the local diffusivity Ah and Av . One advantage of a particle tracking method in comparison to the ‘Eulerian approach’ is the absence of numerical tracer diffusion. If no explicit diffusion is introduced and the flow field is stationary, all particles with the same starting conditions in space and time will follow the same pathway that is determined only by advection. Thus, the diffusion is entirely physical and can be controlled quantitatively. A better control of the diffusive fluxes allows a better reproduction of possibly strong gradients or fronts in tracer concentration. Another argument in the decision about which method to use is the computational speed. Using the ‘Eulerian approach’, the advection-diffusion equation is calculated at each grid point. The numerical effort of this approach depends on the number of grid points. Models based on the ‘Lagrangian approach’ calculate the transport of each particle in the model domain, thus they exclude areas where no tracers occur. The numerical effort of particle tracking therefore depends only on the number of particles to be traced. If the number of particles remains moderate, particle tracking can be faster than the advection-diffusion approach. This may hold for instantaneous releases, when the number

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of released particles is comparatively small. However, the use of this approach is computationally expensive for the simulation of a continuously emitting source. If initially each grid point in the model domain is addressed with a tracer concentration and if horizontal gradients tend to be rather weak, the advection-diffusion approach is superior to particle tracking, at least in terms of computational speed. Although both approaches are appropriate for all spatial scales, particle tracking studies are mostly performed on small scales whereas on large or global scales advection-diffusion transport models are favoured. A limitation for both approaches is the stability criterion that limits the time step and restricts also HD models in their computational speed. The stability criterion requires that a calculated variable should not be advected faster than one grid cell per time step (see also Section 2.2). The handling of boundary conditions is somewhat easier with particle tracking models. As long as the particles stay in the inner model domain, these approaches do not require open boundary conditions. If a particle leaves the domain it is simply excluded from further calculation. For further comparison, we would like to refer the reader to three regions where both, advection-diffusion and particle tracking approaches were applied: • The LaHague discharge to the English Channel was investigated by Periánez & Reguera (1999) and Periánez (2000) for conservative and non-conservative radionuclide dispersion, using the advection-diffusion approach. Periánez & Elliott (2001) performed a particle tracking method for non-conservative radionuclides in the same area. In particular the latter publication highlights the differences between both approaches. • Conservative radionuclide dispersion in the North Sea was performed for 137 Cs by Pohlmann et al. (1987) on the basis of advection-diffusion equations whereas Schönfeld (1995) used a particle tracking method to trace 99 Tc in the North Sea. A conservative advection-diffusion approach was also applied by Breton & Salomon (1995) and Salomon et al. (1995) for the southern North Sea, for 125 Sb and 99 Tc. • In Chapter 10, this volume, both methods are used to investigate the fate of conservative radionuclides released at dump sites in the Kara Sea. Dispersion patterns from both approaches are depicted in Fig. 7a, b and Fig. 8. 3.3. Coupling HD and transport models It has been outlined in the introduction that transport equations can be used in the on-line mode, coupled synchronously to an HD model or in the off-line mode as stand-alone models. Transport modelling in the off-line mode excludes the calculation of the HD model equations (Fig. 9a). Instead, previously determined hydrodynamic flow fields from an HD model are used. The advantage of using a stand-alone transport model is obvious: the advection speed that limits the time step in transport equations is much smaller than the equivalent speed of surface gravity waves that limits the time step in HD models. This allows for larger time steps in transport models which makes the computation faster. Additionally, the computational effort in a transport model is much smaller than with HD models which results in shorter simulation times. A discussion on this topic can be found e.g. in Periánez & Elliott (2001). Another advantage when using off-line mode coupling is that the domain, the grid size, and also the simulation period of the transport model are independent from the HD model.

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Fig. 7. Particle tracking simulations in the Kara Sea. Main pathways and transit times of a particle cloud, released instantaneously in spring in Abrasimov Bay: (a) at the surface and (b) in 25 m depth.

A stand-alone transport model is able to use results from different HD models or may use the same flow field for different sources, radionuclides or parameter settings. It is also possible to use flow fields not directly from HD models but from statistical approaches, or to combine various sources like wind induced trajectories and tidal residual flows (Salomon et al., 1994). Examples of this kind of coupling can be found e.g. in Breton & Salomon (1995) or Salomon et al. (1995). However, usually input flow fields from a separate HD model are stored at specific time intervals much larger than the HD model time step. Time interpolation of flow fields cuts off high frequency variability of the circulation and leads to non-physical smoothing. Synoptic

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Fig. 8. Simulated concentrations of 137 Cs, following a continuous 1 TBq/y release in Abrasimov Bay. The picture shows concentrations at the surface over 6 years with continuous release (Nies et al., 1998).

events like e.g. wind induced flow variability or storm surges are averaged out. This affects the dispersion results in particular in coastal areas and shelf seas considerably. In the on-line mode, the transport equations are part of the HD model such that hydrodynamics and transport can be calculated with the same time step (Fig. 9b). The computational effort is higher than with the off-line mode but can be reduced if a time splitting routine is applied (see Section 2.2). Running the transport of tracers on-line with the hydrodynamics avoids unnecessary averaging and ensures that the transport equations receive the flow data with the highest possible time resolution. It is obvious that both coupling approaches have their strengths and weaknesses. Independently from off-line or on-line coupling, flow fields should generally enter the transport model in time intervals that allow the reproduction of the dominant part of the existing flow variabil-

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(a)

(b) Fig. 9. Simplified sketch showing the schematic coupling of transport models to HD-models in the off-line mode (a) and in the on-line mode (b).

ity. The time resolution depends mainly on the objective of the dispersion scenario and the model area: A tidal simulation of tracer dispersion in a coastal bay, for example, has to resolve the tidal cycle, if the tidal flow influences the tracer dispersion there. This may require much higher temporal resolution than a large scale application, covering low frequency flow in a deep ocean basin. 4. Forcing strategies in dispersion studies Depending on the task which a hydrodynamic circulation model is bound to perform, different kinds of external driving forces have to be applied. The external driving forces initiate the circulation which in turn is responsible for the advection and diffusion of the tracers. These driving forces are applied to the model via the setting of boundary conditions at the sea surface, at lateral boundaries or as a tide generating force on the entire volume. Starting from the initial conditions, which also have to be specified at the beginning of model integration, the application of constant or space and/or time dependent boundary conditions creates a time and space dependent flow field. The details of the application of the external forcing via the boundary conditions is of course part of the experimental design which is governed by the questions to be answered by the model experiment. 4.1. Types of boundary conditions Lateral boundary conditions prescribe the in- and outflow through open boundaries of the model domain. This can be inflow by river runoff or other flows through straits. The flows

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can be prescribed as flux conditions (flow rates in m3 /s) or as sea level variations. Often non-reflecting boundary conditions have to be used in order to allow waves to propagate out of the model domain without reflections (see e.g. Orlanski, 1976). Besides the volume flux, also other characteristics like temperature, salinity or tracer parameters may be prescribed by lateral boundary conditions. At the sea surface, the flux of momentum due to the wind drag, heat- and salt- or freshwater fluxes are either prescribed directly or determined indirectly via bulk formula from other parameters like air temperature or wind velocity. In the case of coupling to a sea ice model, fluxes of heat, salt and momentum are exchanged between the two models at their interface. In this case, external boundary conditions, prescribing the forcing, are applied at the top of the sea ice model. If tidal effects should be included in regional or small scale models, sea surface elevations have to be prescribed at the open boundaries according to tidal tables or larger scale tidal models. For large scale deep ocean applications, tides may be forced by body forces according to the astronomical parameters. Tides are mostly taken into account in shelf areas, in order to give more realistic flows in tracer dispersion studies, while in the deep open ocean applications tides only have minor effects on the dispersion of tracers. If tides are important, their omission in model experiments leads to an underestimation of mixing, mostly in horizontal directions. But omitting tides can also be critical because additional advection due to residual currents might be neglected. Since in certain areas residual currents may easily be of the same order as wind or density driven currents, it is important to consider beforehand the relevance of tidal effects for the specific area in question. It is possible, however, to parameterise the effects of tides on the flow fields (Kauker & Oberhuber, 1997). This intriguing alternative is often chosen because the explicit simulation of tides requires short time steps in the HD model in order to resolve the tidal cycle properly. This may lead to unacceptable computational effort if the experiment is designed for very long integration periods. Some specific model experiments are performed to investigate the effects of a single forcing type alone, e.g. tidal forcing or wind forcing, on the dispersion processes. More common, however, is the goal to perform the model experiments as realistically as possible. These realistic studies typically include as many forcing types as necessary in order to reproduce the real ocean flows as accurately as possible on the desired spatial and temporal scales. 4.2. Atmospheric forcing effects An important aspect in terms of model forcing is the temporal and spatial resolution of the meteorological input data. On very small spatial scales below the atmospheric meso-scale, the atmospheric forcing which acts at the sea surface is determined from direct observations. For experiments on larger spatial scale the use of data from meteorological observational data sets or the products of meteorological forecast models is more common. In atmospheric forecast analyses, observed data are incorporated. For large scale dispersion studies it has been common in the past to use monthly mean atmospheric fields, sometimes in the form of multi-year climatological means (e.g. Nies et al., 1998; Preller & Cheng, 1999). This practice has changed in recent years, since products of atmospheric model reanalysis have become widely available (e.g. Kalnay et al., 1996; Gibson et

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Fig. 10. Decadal means of the surface velocity for the periods 1979–88 and 1989–98. Results from a 21 year hindcast experiment with a coupled ice-ocean model (see text for reference).

al., 1997). These offer the opportunity to force the models with consistent sets of atmospheric data like wind stress, atmospheric temperatures, precipitation and cloudiness on spatial scales of the order of 100 km for every 6 hours. While the use of monthly mean wind stress data leads to an underestimation of mixing, the use of climatological seasonal cycles excludes the investigation of interannual variability. This may create unrealistic pathways of tracer dispersion in areas where the flow patterns are changing critically as a response to interannual variability of the external forcing. It may be less problematic, however, in the case of flow patterns which are geographically stable on the interannual scale. This situation can be assumed for the Nordic Seas and for the mid-depth central Arctic Ocean where the topographic steering of the flow keeps the major current patterns in place. An example of this behaviour is given in Fig. 10 which depicts model results achieved with a coupled ice-ocean model. It is driven with daily atmospheric data from the ECMWF reanalysis (Gibson et al., 1997). The experiment has been performed for the period 1979 to 1999. For details on the experiment see Karcher et al. (2001). Water of Atlantic origin which moves northward in the Nordic Seas along the Norwegian coastline branches into three parts. Two branches enter the Arctic Ocean, one via the Barents Sea, the second one through Fram Strait and a third branch re-circulates southward in the southern Fram Strait. Here it joins with the

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Fig. 11. Vertically integrated volume fluxes in the northern Nordic Sea branching area of Atlantic Water for each of the 3 branches: the Barents Sea branch (red), the Fram Strait branch (blue) and the re-circulation branch including the polar outflow with the East Greenland Current (EGC) (green). The time series are results from a 21 year hindcast experiment (1979–99) (see text for reference). The upper panel shows monthly means, the lower panel the yearly means.

East Greenland Current (EGC) which carries the southward outflow of Polar Water from the Arctic Ocean. Even if these currents do not change their location significantly, there is still interannual variability in the partition of the flow volume between these branches (Fig. 11): years with a strong inflow of Atlantic Water via the Barents Sea branch tend to have a reduced inflow via the Fram Strait branch. The re-circulation plus EGC branch behave synchronously to the Barents Sea branch in the 1980s and to the Fram Strait branch in the 1990s. The variable partitioning of the Atlantic Water flow can be expected to have implications for the pathways of tracers which are advected by the Atlantic Water, like radionuclides from the West European reprocessing facilities. The warm and salty water of Atlantic origin, circulating in the mid depths of the central Arctic Ocean, is called the Atlantic Water Layer (e.g. Rudels et al., 1994). It is known to have warmed in the last decade (Grotefendt et al., 1998). According to model results, the warming is due to an increased inflow of warm Atlantic Water through Fram Strait and the Barents Sea (Zhang et al., 1998; Karcher et al., 2001). While the general flow pattern of this Atlantic layer does not seem to have changed in this period, the intensity of its cyclonic circulation at 200–800 m depth has increased synchronously with the warming. Still the flow intensity may vary interannually.

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Despite these changes, the generally stable location of the major flows allows the use of monthly mean climatological atmospheric data for a simulation of the spreading of radionuclides from Sellafield into the Nordic Seas and the Arctic Ocean (see also Chapter 10, this volume). These simulations compare very well with available observations, especially in the Nordic Seas and the mid-depth Arctic Ocean (e.g. Crane & Galasso, 1999). In contrast to the Atlantic Water layer, the surface mixed layer of the Arctic Ocean is subject to interannual variability of flow patterns, as can be seen from Fig. 10. This is partly due to the local surface stress conditions, partly due to the variable supply of water from adjacent ocean basins (Fig. 11). The apparent changes in surface mixed layer flow patterns on long time scales, manifest as variable supplies of shelf water to the Eurasian Basin and a relocation of the Transpolar Driftstream (Proshutinsky & Johnson, 1997; Karcher et al., 2001). The Transpolar Driftstream carries ice and water from Siberian shelves towards the Fram Strait and leaves the Arctic Ocean on its eastern side, along Greenland’s coast. Its relocation has consequences for example for the pathway and time scale for spreading of tracers from the Kara Sea or the Barents Sea via the central Arctic Ocean to Fram Strait (Fig. 10) (see also Gerdes et al., 2001a, 2001b).

5. Other transport modelling approaches In Section 3, only conservative transport modelling with advection-diffusion or particle tracking approaches, have been tackled. A consequent further development of the conservative approach is the simulation of non-conservative radionuclide transport which will be outlined briefly. The other approach that has to be mentioned deals with box or compartment modelling. 5.1. HD modelling versus box modelling A common approach in radiological assessment studies is the application of box or compartment models. In two- or three-dimensional box models, the area under investigation is covered usually with a horizontal grid much coarser than used in typical HD model studies. The size and the shape of the boxes do not have to be regular as with finite difference approximations in an HD model. Instead, the boxes are chosen according to geographic or hydrographic characteristics (flow regimes, water mass properties) or political or economic constraints (national territories, fishery territories). Similar to the Eulerian transport models, most box model approaches are based on the advection-diffusion equation, which is integrated in time, step-wise for all boxes (Scott, 1993). Since box models do not calculate hydrodynamics prognostically as HD models do, the flow fields need to be prescribed. They may be deduced from observations, budget estimations or from HD model simulations. An example for such a box-model flow field is shown in Fig. 12 (Karcher & Harms, 2000). Examples for successful box modelling in Arctic radiological assessment studies can be found in Nielsen et al. (1995), Nielsen (1997), Iosjpe et al. (1997) and Baxter et al. (1998). Although similar in their basic configuration, the skill and the philosophy of HD models and box-models are generally very different. HD models are able to calculate three-dimensional

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flow fields based on the realistic topography of the area and realistic forcing functions like wind or density. The dynamic behaviour of a fluid is incorporated through the momentum equations which basically reflect Newton’s law of motion. HD model results achieve a high spatial resolution ranging from kilometres down to metres. Time steps less than one hour are able to resolve a spectrum that ranges from tidal motions to decadal variability. In contrast to that, the goal of box model studies is to mimic the whole system rather than to simulate all relevant processes as HD model studies do. The numerical effort in HD models causes long computing times and limits the simulation period. Box models are much faster due to larger time steps and smaller computational effort. They are used to forecast decades, centuries or even thousands of years which is often a necessity in risk assessment. Usually, box models are based on a stationary flow field for the entire time period. Box models are simple in a sense that they exclude hydrodynamics. They mix instantaneously in each cell which implies that the ocean compartments should conform to the critical assumption that each compartment is well-mixed and homogeneous. However, an instantaneous and homogeneous mixing in each of the large boxes is the most critical assumption in marine box modelling. A remedy to this, is to include a term for time dependant dispersion in each box. This method has been successfully introduced in the NRPA-RISØ marine compartment model (Nielsen et al., 1995; Iosjpe et al., 1997). A common problem for calculating tracer dispersion in box-models and HD models is the parameterisation of mixing on unresolved spatial scales. There is much uncertainty about diffusion coefficients in both model types, which is especially unsatisfying since these values have a strong influence on the results. Box models can be very complex if they include geo-chemical or biological processes (Marietta & Simmons, 1988). Marine radiological box models allow for the incorporation of biological components, food chain processes and specific interactions or transfer of radioactivity like food uptake, inhalation etc. In this respect, radiological box models have larger capabilities than HD models. A comparison of box-model results with HD model results is difficult because the time span of forecast and the spatial resolution between both model types usually are very different (Scott et al., 1997). Typically, box-models provide results which are valid for boxes of several thousands of square kilometres. HD models are able to provide results with a much higher spatial and temporal resolution. The HD model is able to resolve e.g. tidal or daily cycles of tracer concentrations whereas a box model may deliver yearly averaged values, however covering thousands of years. All these differences have to be considered when comparing box model and HD model results. Both models have to be applied according to their different skills and the scope of the dispersion study. In assessment studies for example, box models present an indispensable tool. They can be a useful complement to HD models and vice versa. 5.2. Transport and dispersion of non-conservative radionuclides Radionuclides in the marine environment actually occur in two phases: dissolved radioactivity which is redistributed by the flow field in a conservative form more or less passively, and particle reactive radionuclides which tend to attach to particles or suspended material in the water column. Particle reactive and non-reactive radionuclides are classified according to a

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Fig. 12. Box structure and HD-model based flux estimations for the NRPA-RISØ marine box model for the North Atlantic/Arctic Ocean (Iosjpe et al., 1997; Karcher & Harms, 2000).

radionuclide specific distribution coefficient, Kd = Cs /Cw , which is defined as the ratio of the concentration of radionuclides in sediment Cs and in sea water Cw in an equilibrium state (Duursma & Carroll, 1996). Kd values are usually determined in laboratory experiments using sediments, collected from the sea bed. Recommendations for Kd values can be found e.g. in IAEA (1985). The conservative approach given in equation (9) is applicable only to the dispersion of dissolved radionuclides, i.e. with low distribution coefficients (Kd < 10 (m3 /kg), e.g. 137 Cs). As

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a first approximation it may be justified to treat these radionuclides as conservative. However, depending on the intention of the investigation, even for radionuclides with low Kd values a conservative approach might be critical. Although e.g. l37 Cs can be regarded as highly soluble, it remains to a small extent also particle reactive. Together with processes like deposition, erosion/remobilization, this might result in additional sources or sinks which influence the inventory on the longer time-scale. Radionuclides with high Kd values (e.g. 239 Pu) tend to attach to suspended particles or to the sediment intensively enough to change their dispersion behaviour drastically. There are several ways to include particle reaction and simulate the dispersion of nonconservative radionuclides. A simple approach, frequently used in box modelling, applies a partition coefficient P to determine the fraction of activity in the water Aw or attached to the sediment As (Carroll & Harms, 1999): Pw = Aw /At = 1/(1 + Kd Cs ), Ps = As /At = Kd Cs /(1 + Kd Cs ). At is the total activity consisting of Aw + As . Pw gives the fraction of activity in water that remains dissolved whereas the reciprocal or Ps gives the fraction that sorbs onto particles. The activity with particles in suspension, Ax is thus given by As = At Ps and the reduced activity in water is Aw = At Pw . The partition coefficient P ranges from zero to one and is a function of the distribution coefficient Kd in m3 /kg and the suspended load Cs in kg/m3 . For low Kd and suspended load values, Pw will not differ much from unity and Ps remains close to zero. In this case, the major fraction of activity remains in a dissolved phase. Conversely, high Kd and suspended load values will cause a considerable transfer of activity onto particles or the sediment. The application of a Kd based partition coefficient P is based on the assumption that the distribution of radionuclide activity on suspended particles and in the water is in an equilibrium state. Box models may fulfil this requirement due to their large time steps (days or more). However, this is usually not the case with HD models. Short time steps (minutes, hours) do not allow for an equilibrium assumption between the dissolved and the particulate phase. An exception from this may be in the case of stationary dispersion patterns where some sort of equilibrium can be assumed. Even if this approach is rather simple, it is may be a good first guess of radionuclide concentrations in dissolved or particulate phase, in particular if essential geo-chemical or sedimentological data for a more sophisticated approach are not available. A way to simulate non-equilibrium exchange processes, is to apply transfer coefficients which allow for simulations of the dissolved and the particulate phase simultaneously. However, this requires knowledge of the suspended load in the water column which results in a numerical solution of additional equations for the transport of suspended matter. During the past few years, progress has been made to develop approaches that include transport of non-conservative radionuclides in HD model based dispersion studies (Periánez et al., 1996a, b; Margvelashvily et al., 1997, 1998). Due to the larger computational effort, the applications are confined to local or regional scale models with moderate matrix sizes. Periánez (1999) for example applied a three-dimensional HD model together with a suspended matter transport model based on the advection-diffusion equation to the Irish Sea. Three transport equations describe the transport of non-conservative radionuclides in three phases: dissolved,

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particle bound and bottom sediment bound. The non-stationary transfer between all three phases is determined with empirical kinetic transfer coefficients. These successful approaches were also used in particle tracking models (Periánez & Elliott, 2001). In large scale model studies, the simulation of non-conservative radionuclides was often omitted. The main argument is that particle reactive radionuclides stay near the source and do not spread. However, recent observations showed that e.g. 239 Pu, originally released from the reprocessing facility in Sellafield, is transported over long distances in the North Atlantic. In fact, the sediment of the Irish Sea is the dominant source of 239 Pu in the Greenland Sea in the second half of the 1990’s (Herrmann et al., 1998). Ice transport in the Arctic Ocean may also play a significant role in redistributing particle reactive radionuclides (Dethleff et al., 2000). Therefore in future applications, the nonconservative approach should also be realised in large scale applications of HD models in order to investigate the long range transport of particle reactive radionuclides.

6. Summary The main intention of this chapter is to review different types of radionuclide transport models and to give an overview of state of the art applications. The review looks at HD models and transport models separately but with the focus equally weighted on both model types. Although the main topic of this book is transport of radionuclides, we want to stress that hydrodynamic circulation modelling has an important influence on radionuclide dispersion: there is no way to correct doubtful dispersion results, caused by inappropriate flow fields, with a sophisticated transport model. HD and transport models have been applied for several years now. Model physics and numerical methods have been validated in many applications and both model types present well established tools in marine environmental assessment studies. However, many detailed improvements have been made in the last years and even if the basic model physics have not changed very much, state of the art applications can be very different. In order to give guidelines for users, we would like to summarise the following points for consideration: • For transport modelling, the user may chose between Eulerian advection-diffusion approaches and Lagrangian particle tracking approaches. The main reason for preferring a particle tracking model is the absence of numerical diffusion. Eulerian methods appear to be more direct because they avoid the statistical particle approach and directly simulate concentration units. The numerical effort and hence the computational speed depends on various parameters and there is no general superiority of one of the cited methods. Both approaches have their strengths and weaknesses and they should be chosen depending on the objective of the model study. • The coupling mode of HD and transport models may affect dispersion results if a time or space interpolation of flow fields is involved. Transport models can be used in the online mode, coupled synchronously to an HD model or in the off-line mode as stand-alone models. Independent from off-line or on-line coupling, flow fields that enter the transport model should reproduce the dominant part of the existing flow variability in order to achieve realistic results.

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• We mentioned another widely known dispersion model type for marine radioactivity, the compartment or box model approach. However, our conclusion is that a direct comparison between box-models and HD models should be avoided because the philosophy of both model types is too different. HD and box models have to be applied according to their different skills and the scope and demands of the dispersion study. • Important recent advances for HD models concern the meteorological forcing in large scale and long term simulations. Time and space averaging of forcing fields, the use of atmospheric climatologies or realistic input data affects dispersion results considerably. The progress in circulation modelling for climate research during the last years, shows the important influence that meteorological forcing has for reproducing interannual or decadal variability of circulation and hydrography. This improvement has to be passed on to long range transport modelling of radionuclides in order to deepen the knowledge of tracer dispersion on long time scales. In this respect, radionuclides may help to indicate climate variability in ocean circulation. • Much progress was achieved in transport models concerning the inclusion of nonconservative characteristics of radionuclides. Processes leading to non-conservative dispersion characteristics mostly occur in shallow areas, where tidal friction and wind stirring provide enough turbulent mixing to erode sediment or to keep particles in suspension. This is the main reason why non-conservative dispersion was realised only in small scale applications. Being particle reactive, non-conservative radionuclides are supposed to stay close to the source area or at least in the near field. Long range dispersion of non-conservative radionuclides was therefore excluded in large scale simulations. However, observations show that e.g. 239 Pu from the Irish Sea disperses also in the North Atlantic and even in the Arctic Ocean. Future large scale applications should therefore include the non-conservative approach in order to investigate the effects of long range transport of particle reactive radionuclides in the world ocean. Finally, we would like to mention that numerical models benefit in general from increasing computer power. This allows better spatial resolutions which improves the results in nearly all aspects. However, higher resolution is not a universal remedy that cures all problems in numerical ocean models.

Acknowledgements Although we tried to write a review article on the application of hydrodynamic models to the transport of marine radioactivity, we do not claim to have presented all possible and existing applications in this field. We ask for some indulgence, if authors feel that their studies are missing or if a study was overseen. The results presented from the coupled ice-ocean model in Section 4.2 are based on work performed in close co-operation of M. Karcher with R. Gerdes and F. Kauker at the Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany. Figure 5 was kindly provided by the Ocean Circulation and Geochemistry Group of the Max-Planck Institute for Meteorology, Hamburg, Germany. Thanks to G. Umgiesser, Venice, who kindly provided Fig. 6. Thanks to all authors who provided us with their reprints.

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The authors want to express their gratitude towards the funding agencies who supported the presented work: Part of M. Karcher’s work has been funded by the EC-MAST II programme under grant MAS3-CT96-0070 (VEINS). Part of I. Harms’ and M. Karcher’s work has been funded by the International Atomic Energy Agency (IAEA), the German Ministry for Education, Research, Science and Technology (BMBF, 02-E-87054) and by the Norwegian Research Council in the framework of the ‘Transports and Effect’ programme.

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Schönfeld, W. (1995). Numerical simulation of the dispersion of artificial radionuclides in the English Channel and the North Sea. Journal of Marine Systems, 6, 529–544. Scott, E. M. (1993). Low level measurements of radioactivity in the environment. In M. Garcia-Leon & R. GarciaTenorio (Eds), Proc. of 3rd International Summer School, Spain 1993. World Scientific publishers. Scott, E. M., Gurbutt, P., Harms, I. H., Heling, R., Nielsen, S. P., Osvath, I., Preller, R., Sazykina, T., Sjoeblom, K. L. & Wada, A. (1997). Benchmarking of Numerical Models describing the Dispersion of Radionuclides in the Arctic Seas. Science of the Total Environment, 202, 123–134. Staneva, J. V., Buesseler, K. O., Stanev, E. V. & Livingston, H. D. (1999). Application of radiotracers to study Black Sea circulation: validation of numerical simulations against observed weapon testing and Chernobyl 137 Cs data. J. Geoph. Res., 104, 11099–11114. Tartinville, B., Deleersnijder, E. & Rancher, J. (1997). The water residence time in the Muroroa atoll lagoon: sensitivity analysis of a three-dimensional model. Coral Reefs, 16, 193–203. Webb, D. J., de Cuevas, B. A. & Richmond, C. S. (1998). Improved advection schemes for ocean models. Journal of Atmospheric and Oceanic Technology, 15, 1171–1187. Zhang, J., Rothrock, D. A. & Steele, M. (1998). Warming of the Arctic Ocean by strengthened Atlantic inflow: Model results. Geophys. Res. Lett., 25, 1745–1748.

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MODELLING RADIOACTIVITY IN THE ENVIRONMENT E. Marian Scott (Editor) © 2003 Elsevier Science Ltd. All rights reserved

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Chapter 4

Radioactivity in the terrestrial environment G. Pröhl GSF-Institute of Radiation Protection, Germany

1. Introduction The release of radionuclides to the environment from nuclear facilities during normal operation or during an unplanned situation leads potentially to a radiation exposure of the population. Due to the potential impact of ionising radiation on health, exposures due to the operation of nuclear facilities must be limited. Recommendations for dose limits are given by the ICRP and they are normally implemented by national regulatory bodies. It is common practice that compliance with dose limits has to be proven before the start of the operation. In this case, the exposure of the population has to be predicted by means of models. In other cases, radionuclides have already been released to the environment and the exposures to people have to be estimated in order to evaluate the radiological relevance of the doses and to decide whether and how such doses have to be minimised and how the minimisation can be optimised. In principle, two types of terrestrial models can be distinguished, namely equilibrium models and dynamic models. Equilibrium models are applied for the evaluation of the radiological consequences of continuous or quasi-continuous releases. They are applied e.g. to assess exposures during the normal operation of nuclear facilities or during weapons’ fallout. Under these circumstances, after a certain time, the activity concentrations in the environmental media are on average more or less constant. They are appropriate to estimate long-term averages for activities in soil, plants, animal products and activity intakes and doses to man. Especially after the reactor accident in Chernobyl, radioecology focussed on the transfer of radionuclides in the environment subsequent to single or short-term depositions. In this case, the activity concentrations in environmental media can vary drastically with time and equilibrium or quasi-equilibrium levels in environmental compartments are not achieved. Then, the system has to be modelled shortly after the input into the environment. The variations of the activity concentrations in the environmental media affected are very pronounced. For simulating the radionuclide behaviour in the environment, the activity concentrations in the environmental compartments have to be formulated as time-dependent functions. In many models, the system is simulated by a set of 1st order differential equations (Brown 87

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& Simmonds, 1995; Whicker & Kirchner, 1987). The transfer between the environmental compartments is quantified by first order rate constants. The advantage of this approach is that it represents a closed system. Problems may arise, since the rate constants are in general not accessible to direct measurements. Other models avoid the mathematical formulation as differential equations, since the data base is in many cases too poor to determine the rate constants with sufficient accuracy, e.g., the uptake of a radionuclide is often quantified as a simple ratio of the concentrations found in soil and plant. The derivation of a parameter that represents the transfer rate from soil to plant requires further assumptions. Then the complexity of the model would be more complex than the underlying data base. This means the integration of further parameters does not necessarily imply a gain in information. In this chapter, the pathways and processes that lead to a contamination of food are described and quantified. This chapter mainly considers short-term releases. However, continuous releases of radionuclides to the environment can be regarded as a series of single releases, that have to be superimposed for the estimation of the resulting activity concentrations in the desired endpoint. Numerous models radioecological models exist. However, in this chapter, to illustrate and quantify processes, attention was focused on the dynamic model ECOSYS (Müller & Pröhl, 1993). The focus on the ECOSYS model does not imply any judgement about other models and model approaches, it was selected because ECOSYS is integrated into the European decision support system RODOS (Ehrhardt et al., 1997), and in national nuclear emergency systems of Austria, Denmark (Hoe et al., 2000), Germany, Hong Kong (Poon et al., 1997) and Switzerland.

2. Pathways and processes Radionuclides may enter the terrestrial environment via atmospheric releases from nuclear facilities. Releases to aquatic systems may contaminate soils and plants through the use of contaminated freshwater. Radionuclides stored or disposed of as waste may also enter the terrestrial environment after release to soil or to water bodies that are used by man. An outline of the potential exposure pathways is given in Fig. 1. Radioactivity released to and dispersed in the atmosphere enters the terrestrial environment due to dry and wet deposition on soil and vegetation. Internal exposures to humans occur from the use of contaminated plants as food or as feed for domestic animals. Direct contamination of foliage also occurs, as well as – over the long term – uptake of radionuclides from soil and deposition of resuspended contaminated soil particles. 2.1. Deposition Radionuclides may enter the terrestrial environment subsequent to transport in the atmosphere as particles or gases. In many assessment models, the dry deposition of radionuclides from the atmosphere to soil and vegetation is quantified by the deposition velocity vg (m s−1 ), which

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Fig. 1. Outline of major transfer routes for modelling the migration of radionuclides in the terrestrial environment.

is defined as the ratio of the activity deposited on a specific surface (Bq m−2 ) Dr,i and the time-integrated activity concentration in air Ca,r of radionuclide r (Bq s m−3 ): vg =

Dr,i . Ca,r

(1)

The deposition velocity is an empirically determined quantity that depends on a variety of factors, such as the particle size, the characteristics of the interface surface/air the meteorological conditions and the chemical form of the deposit. For particles, a minimum deposition velocity is observed for diameters between 0.1 and 1 μm. The deposition of smaller particles is more effective due to diffusion (Jonas, 1984). For particle sizes > 1 μm, deposition increases with increasing particle size, because the effects of gravitational settling (sedimentation) and impaction become more important. The particle size spectrum is influenced by the source of the radioactivity in air. Routine releases are usually filtered, and the cut-off point of the filter device determines the maximum particle size. Unplanned releases may be unfiltered, and the whole particle size spectrum may be released. However, with increasing distance from the release point, the size spectrum approaches that of natural aerosols, i.e., is in the range of approximately 0.1 to 1 μm. Particles below 0.1 μm coagulate or “attach” to larger particles, whereas particles larger than about 1 μm are lost from the near-surface atmosphere due to sedimentation. Therefore, as a first approximation, at large distances from the source, the particle size from a release is similar to that of natural atmospheric background aerosols. Therefore the deposition velocity typically decreases with increasing distance from the release point, since the ratio of small to large particle concentrations increases due to the effective deposition of large particles.

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Table 1 Typical estimates for deposition velocities on grass (Pröhl, 1990) Chemical form

Deposition velocity (m s−1 )

Estimated uncertainty factor∗

Particles, 0.1–1 μm Elemental iodine Methyl iodide

0.001 0.01 0.0001

5–10 5–10 5–10

∗ The total range is given by the square of the uncertainty factor.

Gaseous radionuclides can be classified by their incorporation into reactive or non-reactive gases. Noble gases such as krypton or xenon do not interact with other substances or surfaces; therefore their deposition is negligible. Once released, noble gases are more or less globally distributed in the atmosphere, if their half-lives are long enough. Reactive gases (e.g., I2 , CO2 , and SO2 ) are characterised by intensive interactions with plants. These substances may enter plants through the stomata and subsequently be metabolised. For example, elemental iodine is bound by the starch produced during photosynthesis, CO2 is reduced to carbohydrates, and SO2 is also reduced. These reactions lead to pronounced gradients in air concentrations, which induce an effective flux of reactive gases from the near-surface atmosphere to the plant (Fitzjarrald & Lenschow, 1983). Due to the dependence on the metabolic activity of the plant, the deposition of reactive gases is subject to diurnal and seasonal variations. Iodine, which is released from nuclear facilities in three chemical forms, is a special case. Part of the iodine is bound to particles, part is in elemental form as I2 , and a third fraction is in organic form (predominantly methyl iodide, which is a relatively inert gaseous compound). For both particles and gases, the deposition velocity is influenced by the kind of plant canopy. In general, deposition is more effective for well developed canopies, since the area of interface between vegetation and atmosphere is increased (Jonas, 1984; Heinemann & Vogt, 1980; Roed, 1987). Although this effect is well known, it is considered in only a few assessment models. In models for routine releases, this problem can be avoided by the choice of appropriate long-term mean values. However, the deficiency is more serious for model applications to single releases, due to the pronounced seasonality of the standing biomass. Typical estimates of deposition velocities are summarised in Table 1; these estimates are associated with considerable uncertainty. Dry deposition is the result of the interaction of meteorology, chemical and physical characteristics of the contaminant, and the properties of the canopy. Taking into account these factors, the estimated uncertainty about the estimates in Table 1 is a factor of about 5–10, corresponding to a geometric standard deviation of 2–3. In more advanced approaches, the deposition processes are simulated in analogy to Ohm’s law as a sum of three serial resistances. This differentiates between the aerodynamic, the surface layer and the transfer resistance (Ra , Rs , Rt ). The relation of the resistances to the deposition velocity to a given surface is given by: vg ≈

1 1 1 + + . Ra Rs Rt

(2)

The aerodynamic resistance represents the flux of the substance considered from the atmosphere to the laminar layer of the surface. This transport depends on the meteorological pa-

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rameters such as the wind speed as well as on the structure of the interface atmosphere/surface. In this context, the soil structure as well as the development and the architecture of plant canopies are important factors, e.g., according to Hosker & Lindberg (1982), the aerodynamic resistance of tilled soil is a factor of 10 higher than for a forest canopy and about a factor of 4 higher than for a well developed maize canopy. The surface layer resistance represents the transport within the laminar layer of the surface. This process is determined by 3 processes, the importance of which depends in a specific case on the chemical-physical properties of the deposit: • For gases and particles smaller than 0.1 μm, the surface layer resistance is mainly controlled by Brownian movement. • For larger particles, impaction becomes more and more important. Due to the inertia of the particles, the particles do not follow the air on the interface, but they impact on the surface. • With increasing particle size, the deposition is more and more determined by sedimentation. The last step within the deposition process is represented by the transfer resistance. This resistance quantifies the strength of the interaction of the deposit and the interface. The interaction may be due to a chemical reaction such as the assimilation of carbon dioxide or the uptake of elemental iodine. For the deposition of gases, the transfer resistance strongly depends on the opening of the stomata, which is controlled by illumination, the relative humidity in air, and the water supply of the canopy. Due to chemical reactions of the deposit with the interface, a gradient of the deposit concentration in air is induced which lowers the transfer resistance. Furthermore, the actual conditions of the interface may favour the deposition, e.g. the deposition of water-soluble compounds is facilitated on wet surfaces. The complexity of these processes cannot be adequately represented in the application of deposition velocities. However, on the other hand, the resistance model requires a variety of parameters that undergo diurnal and seasonal variations. The measurement of these parameters is complicated and the parameters are difficult to handle. Therefore, the resistance model is mainly applied to investigate deposition processes in detail at a process level. In the Real-time Online DecisiOn Support System RODOS the resistance model has been modified (Gering & Müller, 2000). The deposition process is separated into the atmospheric resistance Ra – which is equivalent to the aerodynamic resistance – and the canopy resistance Rc which represents the surface layer and the transfer resistance: vgi =

1 . Ra + Rc

(3)

The atmospheric resistance is estimated from the wind speed u and the friction velocity (Jonas, 1984) which quantifies the turbulence in the near-surface atmosphere. The canopy resistance is assumed to decrease with the increasing leaf area index LAI according to: 1 LAI 1 = · . Rc LAImax Rc,min

(4)

Default values are defined for the deposition velocity for specific values of wind speed, canopy height and leaf area indices. These are modified by the atmospheric and the canopy

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Table 2 Deposition velocities for selected leaf area indices and wind speed as applied in RODOS (Gering & Müller, 2000) Surface

Leaf area

Wind speed

Deposition velocity (mm/s)

index

(m/s)

Particulates

Grass

1 1 1 1

0.2 1 4 15

0.19 0.22 0.22 0.23

0.9 2.5 3.7 4.2

Grass

5.4 5.4 5.4 5.4

0.2 1 4 15

0.6 1.0 1.16 1.2

1.1 4.6 12 19

Elemental iodine

resistance that is estimated from the meteorological conditions and the development of plants at the time of the year considered (Gering & Müller, 2000). Some results are given in Table 2. The deposition velocities reflect the findings of experimental observations (Jonas, 1984; Roed, 1987; Heinemann & Vogt, 1980). 2.2. Interception of wet-deposited radionuclides Radioactivity in air is washed out by rain and snow. Some fraction of radionuclides deposited with precipitation is retained by the vegetation, and the rest falls through the canopy to the ground. Although the activity retained is weathered off finally to soil and only temporarily retained by vegetation, the fraction that is initially intercepted is a very important quantity in all radioecological models. This is due to the fact that direct deposition may cause relatively high activity concentrations in feed and foods. In general, very simple approaches are used to estimate interception in assessment models (Pröhl & Hoffman, 1996). The interception fraction is often assumed to be constant for all elements, all plants, and all precipitation events. In some models, the interception of wetdeposited activity is modelled as a function of the biomass density according to the approach of Chamberlain (1970). However, investigations in recent years have demonstrated that interception of wet-deposited radionuclides is actually much more complex. Müller & Pröhl (1993) modelled the interception as increasing in proportion to the leaf area index and decreasing in proportion to the amount of rainfall according to:  ln 2  LAI · k · S · (1 − e− 3·S ·R ) , fw = min 1; R where: fw = interception fraction, LAI = leaf area index (total one-sided area of leaves per area of soil, m2 m−2 ), k = element-specific factor, S = water storage capacity of the plant surface (mm), R = amount of rainfall (mm).

(5)

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Table 3 Comparison of mass interception factors (interception fraction normalized for biomass) for different elements (from Hoffman et al., 1995) Radionuclide

Chemical form

Mass interception factor (m2 kg−1 )

109 Cd

cation cation cation cation uncharged anion anion

1.8 ± 0.19∗ 1.8 ± 0.24 1.3 ± 0.15 1.0 ± 0.17 0.94 ± 0.13 0.35 ± 0.059 0.27 ± 0.016

7 Be 51 Cr 85 Sr 141 Ce† 35 S 131 I

∗ Mean ± standard error of the mean. † Cerium-141 was used to label microparticles.

The exponential term accounts for the build-up of the water film on the plant leaves during the rainfall event. For low amounts of rainfall and fully developed plant canopies fw may be greater than 1.0; then 1.0 is taken. For cereals and grass, for the water storage capacity of the plant, in Müller & Pröhl (1993), a value of the 0.2 mm is applied; in Pröhl (1990), a range of 0.15–0.3 mm is given for the range of the water storage capacity. The element dependent retention factor k describes the ability of the element to be fixed on the leaves. A value of k < 1 means that the fraction of an element intercepted by the foliage is lower than the intercepted fraction of water in which this element is dissolved. A value of k > 1 means that the interception fraction of the element is higher than the interception of water. The retention factor k is mainly controlled by the chemical form. Since the plant surface is negatively charged, the absorption of anions is less effective than of cations. Within the group of cations, the retention by the leaves is more effective for polyvalent cations (Pröhl et al., 1995; Hoffman et al., 1995; Kinnersley et al., 1997). Presently, in ECOSYS, values for k of 0.5 for iodine, 2 for strontium and barium, and 1 for cesium and various other elements are suggested. Table 3 compares the mass interception factor (ratio of interception fraction to biomass) for various elements for leaves of different plant species (Hoffman et al., 1995). The differences among plants (festuca, lespedeza, spruce, maple and goldenrod) were of minor importance compared with the variations among the radionuclides; the interception of polyvalent cations is as much as a factor of 8 higher than for anions. An example of the results of equation (2) is given in Fig. 2, where the interception fraction for grass is plotted for a leaf area index of 1 against the amount of rainfall during which the activity is deposited. The model curve agrees well with experimental data obtained by Kinnersley et al. (1997). In dependence on rainfall, the interception fraction varies by nearly a factor of 10 for a given leaf area index. Including the variability of the leaf area index, the interception varies by a factor of the order of 50. Similar results were obtained by Hoffman et al. (1992). This underlines the necessity to characterise appropriately the deposition event in estimating its consequences. In Fig. 3, the relationship between leaf area index and interception as observed in experiments with cereals by Voigt et al. (1991) is shown; cesium was applied to cereals during a simulated rainfall of 1 mm.

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Fig. 2. Comparison of measured interception fractions for cesium against results of the radioecological model ECOSYS (Müller & Pröhl, 1993).

Fig. 3. Interception fraction observed in experiments with cereals as a function of leaf area index (from Pröhl & Hoffman, 1996).

Due to the complex interaction of rainfall amount, elemental properties, and stage of development of the plant canopy, interception is subject to pronounced variability. Figure 4 gives an idea of the range of the initial retention for mixed (wet and dry) deposits. The interception fraction (related to the total deposit) for Cs on grass decreases with an increasing fraction of wet deposition. Figure 4 represents model results, however, in a real situation, it is very likely that high fractions of wet deposition are associated with high amounts of rainfall, whereas low contributions of wet deposition probably coincide with low amounts of rainfall.

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Fig. 4. Interception fraction for Cs on grass (yield = 1.5 kg m−2 , LAI = 5.4) as function of the contribution of wet deposition and amount of rainfall.

Further research is especially needed to deepen the understanding of the influence of the chemical form, rainfall intensity, and canopy structure, including both the canopy architecture as well as the surface structure of the leaves. 2.3. Weathering Once deposited on vegetation, radionuclides are lost from plants due to removal by wind and rain, either through leaching or by cuticular abrasion. The increase of biomass during growth does not cause a loss of activity; however, it does lead to a decrease in activity concentration due to growth dilution. Since growth is subject to seasonal variations, the post-deposition decrease of the activity concentration of plants depends on the season. For example, for well managed pasture in Middle Europe, the mean daily increase in biomass varies from approximately 10 g d−1 m−2 (dry mass) in May to 2 g d−1 m−2 in October. Although growth cannot be described by a simple exponential function, at least in some parts of the growth curve these growth rates are equivalent to half-times on the order of 10 to 50 d. All these processes occur simultaneously, therefore, the exact contribution of each process to the removal of activity from plants is difficult to quantify. Therefore, the total loss is expressed by the weathering half-time Tw , which is the result of all the processes described above (including growth dilution). The weathering half-time is empirically determined by regression of the radionuclide concentration in vegetation with time. Weathering is simulated in assessment models by a single exponential function. However, observations indicate that the weathering loss is most rapid immediately after deposition (Ertel et al., 1989), and a multi-exponential loss function may be more appropriate. Although the effects of wind and rain cause the post-deposition activity loss, long-term retention is not, in general, correlated with weather conditions (Miller & Hoffman, 1983).

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The chemical form of the contaminant seems to be of minor importance for weathering loss. The analysis of observations after the Chernobyl accident (Kirchner, 1994) resulted in weathering half-times for iodine with a median of approximately 8 d (90% confidence interval, 4.5–15 d). For cesium, a median of 10 d (90% confidence interval, 5.4–20 d) was observed. The shorter half-times were observed primarily in regions with fast growing vegetation, whereas the longer half-lives were found in Scandinavia, where the growth rate was lower due to the later spring in this area (Kirchner, 1994). In general, longer weathering halftimes are observed for slowly growing or dormant vegetation (Miller & Hoffman, 1983). However, weathering is determined by several processes that have an effect over several weeks and that partly compensate each other. Therefore, the overall uncertainty of weathering is less than that for deposition and interception, which depend on specific particle and canopy conditions at deposition. Nevertheless, the approximation of the post-deposition behaviour by a single exponential function and the assumed independence of the weather conditions appear oversimplified. 2.4. Systemic transport of radionuclides Translocation has no or very little influence on the long-term-fate of radioactivity in the environment, since it describes only the distribution of radionuclides within the plant subsequent to foliar deposition and absorption by the leaves. However, for the purpose of estimation of radionuclides in foods and the assessment of dose to man, the systemic transport of radionuclides is a key issue. It is especially important for plants that are partly used as food or feed, such as cereals and potatoes. For plants such as leafy vegetables or maize silage that are used as a whole, translocation is relevant only in that it may reduce the activity that is lost by weathering processes. The activity concentration for the plant product i due to a single deposition event harvested at time t after the deposition is given by: Ci (t) = Dt · fI · Ti (t) · exp(−λr · t),

(6)

where: Ci (t) = contamination of edible product subsequent to foliar deposition (Bq kg−1 ), Dt = total (wet and dry) deposition (Bq m−2 ), fI = situation-specific interception fraction referring for total deposit, Ti (t) = translocation coefficient for plant type i (m2 kg−1 ). The translocation coefficient is quantified as the ratio of the activity concentration in the edible part (Bq kg−1 ) to the total activity retained by the plant canopy per unit ground area (Bq m−2 ) (IAEA, 1994), the unit of the translocation factor is then (m2 kg−1 ) as above. The most important factors influencing translocation are the physiological behaviour of the radionuclides in the plant and the time at which the deposition occurs during the growth period. Concerning the physiological behaviour, generally speaking, elements and hence radionuclides can be differentiated according to their mobility in the plants. There are two transport systems in the plant, the xylem and the phloem; mobile elements can be transported in either system. The xylem transports water and minerals from the soil to upper plant parts. The

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Fig. 5. Comparison of translocation coefficients for cesium (Voigt et al., 1991) and ruthenium in wheat (IAEA, 1994).

driving force for transport in the xylem is transpiration; therefore, the transport direction is from the bottom to the top of the plant. There is no discrimination between elements to be transported in the xylem once they have reached that system (e.g., by penetration from soil to roots). The main function of the phloem is to transport carbohydrates produced during photosynthesis from the leaves to the storage organs (e.g., grain, tubers). Transport in the phloem is possible in both upward and downward directions. Transport in the phloem is element-specific: Alkali elements, magnesium, sulphur, chlorine, and phosphate are transported in the phloem, whereas for physiological reasons, the alkaline earth metals, lead, plutonium, lanthanides, and actinides are not (Strasburger, 1978). The amount of translocated activity is highly dependent on the stage of development at which the deposition occurs. In Fig. 5, the translocation of cesium in wheat determined in experiments by Voigt et al. (1991) is compared with ruthenium, which is an immobile element. The results for cesium indicate that the amount of cesium translocated from the leaves to the grain varies by more than two orders of magnitude during the growth period, underlining the sensitivity of the physiological development for the quantification of this process. The maximum translocation can be observed, when the deposition occurs around the emergence of the ears. It is interesting to note that during this stage of development, the assimilation of carbon dioxide during the photosynthesis and the transport of carbohydrates from the leaves to the grain has its maximum (Geisler, 1983). However, the translocation for ruthenium varies by several orders of magnitude. The contamination of the grain by ruthenium subsequent to aerial deposition is only relevant, if the deposition occurs after the emergence of the ears. Then, in a strict sense, the grain is not really contaminated by translocation, but due to direct contamination of the grain. However, in experiments, both mechanisms occur simultaneously and can hardly be differentiated.

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The pronounced seasonality is especially important for estimation of the consequences of short-term deposition after accidental releases. For long-term releases, appropriate mean values can be used. Quantification of translocation as a function of the element, the plant, and the stage of development at the time of contamination is still a problem. Data are limited and most refer to cesium, some to strontium and very few to other elements. Most experiments were done with cereals, some with potatoes, and a few individual experiments with other species (Aarkrog, 1972, 1975, 1982; Voigt et al., 1991). In the recent years, various investigations were especially made on the translocation of cesium subsequent to foliar deposition on fruit (e.g. Anguissola Scotti & Silva, 1992; Baldini et al., 1987; Bengtsson, 1992; Carini & Lombi, 1997; Carini et al., 1999; Zehnder et al., 1993). 2.5. Uptake of radionuclides from soil The uptake of radionuclides from soil via the roots is difficult to quantify, since the soil is a highly complex system. The uptake of nutrients and contaminants is influenced by factors such as the soil type, pH, sorption capacity, clay content, organic matter, and management practice. The influences of these factors are qualitatively known, but hard to quantify for a specific site. For quantification of the root uptake of radionuclides, usually an empirically derived soil– plant transfer factor TFsp is applied, defined as the ratio of the activity concentration in plant Cp and the activity concentration in soil Cs : TFsp =

Cp . Cs

(7)

This parameter has several disadvantages: • The depth of the rooting zone is more or less arbitrarily defined to a depth of 20 cm. • The total activity in the soil is used to estimate plant concentrations without consideration of the chemical speciation in soil. • The experimentally determined transfer factors vary considerably. Nevertheless, it is used in most assessment models. In cases where the determination of the soil activity concentration is difficult due to pronounced activity gradients in the upper soil layer, an aggregated transfer factor is often applied that relates the activity in plants directly to the total activity per unit area (Howard et al., 1996). The uptake of radionuclides via the roots may become a relevant pathway only for radionuclides with half-lives of at least 1 year. During the past decades many experiments have been performed for the determination of transfer factors; a comprehensive compilation was made by the IAEA (1994). An impression of the range of transfer factors is given in Table 4. For the sake of simplification, the values are not differentiated according to soil type, plant species, or plant part. Although there are many exceptions, in general, transfer factors are higher for sandy soils than for soils rich in clay, and the transfer factors for generative plant parts (grains, tubers) are lower than those for vegetative plant parts (leaves, stems). Further details on the uptake of radionuclides from soil are given in Chapter 9, this volume. In general, the transfer of radionuclides from soil to plant is relatively low. In Table 5, the contamination due to root uptake is compared to the contamination due to direct deposition on

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Table 4 Typical ranges of transfer factors soil–plant (from IAEA, 1994) Element

Transfer factor (Bq kg−1 dry plant per Bq kg−1 dry soil)

Strontium Cesium

10−2 –1 10−3 –10−1 10−1 –10∗ 10−3 –1† 10−1 –10 10−3 –10−2 10−3 −10−1 10−3 −10−1 10−3 −10−1 10−5 −10−3 10−5 −10−3 10−5 −10−3

Iodine Technetium Lead Radium Uranium Neptunium Plutonium Americium Curium

∗ This range may be observed in natural and semi-natural ecosystems on very acid sandy soils with low sorption

capacity and a pronounced potassium deficit. † From Pel (1993).

Table 5 Comparison of the activity concentration of 137 Cs and 90 Sr in the following harvest of winter wheat and potatoes subsequent to a deposition of 1 kBq/m2 (it is assumed that all activity is either completely deposited on the leaves or on the soil) Deposition

Deposition

Contamination

Activity concentration (Bq/kg)

date

(kBq/m2 )

route

137 Cs

1st 1st 1st 1st 1st 1st

May June July Aug. Sept. Nov.

1∗ 1∗ 1∗ 1∗ 1∗ 1

Leaves Leaves Leaves Leaves Leaves Soil

90 Sr

Wheat

Potatoes

Wheat

Potatoes

9.4 150 200 150 –† 0.05

–† 10 30 38 17 0.03

0.6 0.6 1.8 130 –† 0.5

–† 0 0 0 0 0.1

∗ To ensure comparability, the interception factor is assumed to be 1. † Crop is not out on the field at this time of the year.

the foliage. The calculations were made with the radioecological model ECOSYS for 137Cs and 90 Sr for winter wheat and potatoes assuming depositions at different times during the vegetation period (1st May to 1st September) and for 1st November, where the contamination occurs due to root uptake only. For cesium, depending on the time of deposition, the resulting contamination of wheat subsequent to foliar deposition is up to more than a factor of 1000 higher than that due to root uptake. For potatoes, the difference is less, but still a factor of up to 100. For strontium which is an immobile element, the situation is different. Compared to

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cesium, the contamination subsequent to direct deposition is lower than for cesium, but the root uptake is higher. For potatoes, the contamination in the tubers after foliar uptake is zero due to the immobility of strontium. For cesium, the uptake from soil is low under intensive farm management conditions. However, on extensively used sites with acid sandy soils that are low in clay and potassium, the cesium availability in soil may be enhanced by orders of magnitude, and the cesium contamination of plants and animals may become a serious and persistent problem (Howard et al., 1996). The uptake from soil of actinides is lowest among the radionuclides given in Table 4. For these elements, contamination via root uptake is similar to or less than the contamination via deposition of resuspended contaminated soil particles on plants. Although important for all elements, the speciation of radionuclides in soil is especially complex for iodine, technetium and neptunium (Koch-Steindl & Pröhl, 2001). The availability of iodine is much less under aerobic conditions due to its increasing incorporation into and interaction with the organic matter. Technetium and neptunium are most available under aerobic conditions and are gradually reduced to less available chemical forms in badly aerated soils (Lieser, 1993). As is obvious from Table 4, the variability of transfer factors soil–plant is considerable. However, considering the complexity as well as the spatial and temporal variability of the soil–plant system, it will be very difficult to simulate the process of radionuclide uptake by roots and to reduce the uncertainty associated with modelling of root uptake. Therefore, the relationship between speciation in soil and nuclide uptake via the roots has not been sufficiently quantified yet. In radioecological assessment models, empirically determined transfer factors as defined by equation (2) are used. More advanced approaches try to estimate the flux of radionuclides from soil to plant. Then the root uptake is represented as a function of water flow to the root and the concentration gradient at the root surface. Those approaches use commonly the Michaelis–Menten formulation as (e.g., Epstein, 1966; Nye & Tinker, 1977; Barber, 1984; Butler & Wheater, 1990) as:  F=

 Fmax ca . k m + ca

(8)

The specific flux across the root surface (F , per unit area of root) is a function of the concentration at the root–soil boundary (ca ) governed by the two parameters Fmax , the maximum solute flux which can enter the root, and km , the Michaelis rate constant. More recently, it has been shown that the same relationship can be used to describe the root uptake of cesium by winter wheat (Shaw & Bell, 1989). If ca  km , then the above equation can be written as a linear relationship in the following form: F = αca ,

(9)

where α (cm s−1 ) is referred to as the root sorption coefficient or ‘root absorbing power’ (Nye, 1966). Although such approaches represent the root uptake at the interface soil–root, the parameters needed for quantification are very complicated to determine on a large scale with a

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sufficient resolution. If at all, they can be measured in experiments on plants cultivated in nutrient solutions or on isolated roots. Such experiments are very helpful to understand the processes, but they are difficult to apply for the quantification of activities in plants under field conditions. 2.6. Transfer to animals The transfer of radionuclides from feed to milk, meat, and other animal products is most often calculated using element-dependent transfer factors, defined as the ratio of the activity in milk or meat to the daily activity intake under equilibrium conditions. In a strict sense, such transfer factors are applicable only for a constant long-term rate of activity intake by adult animals. To account for time-dependent (dynamic) intakes, one or more biological halflives are introduced to simulate the build-up and loss of activity in milk, meat, or other animal products after start and stop of activity intake. In Müller & Pröhl (1993), the activity in animal food products is estimated according to:  J   Cm (T ) = TFm · amj · j =1

T



 Aa,m (t) · λb,mj · exp −(λb,mj + λr )(T − t) · dt ,

(10)

0

where: Cm (T ) = activity concentration in animal product m at time T (Bq kg−1 ), TFm = transfer factor for animal product m (d kg−1 ), J = number of biological transfer rates, amj = fraction of biological transfer rate j , Aa,m (t) = activity intake of animal m (Bq d−1 ), λb,mj = biological transfer rate j for animal product m (d−1 ). Sometimes more complex compartment models are used to describe the kinetics of radionuclides in animals by a set of differential equations to simulate additionally the recycling of radionuclides between the organs and to improve the understanding of the metabolism of radionuclides. However, such approaches require numerous transfer rates to quantify the activity exchange among the compartments. More details and some examples of such models are given in Chapter 9, this volume. In recent decades, much data on transfer factors have been accumulated. An impression of the data presently available is given by IAEA (1994). However, the quantity and quality of the available data are correlated with the past radiological significance of the animal productradionuclide combination considered. Thus, most data are available for the transfer of cesium, strontium, or iodine to milk. For many elements, the data base is very poor and is sometimes based on data for laboratory animals such as mice and rats, rather than on data for farm animals. In Table 6, transfer factors are summarised for iodine cesium and strontium that are used in the radioecological model ECOSYS. Furthermore typical dry matter intakes of domestic animals are given. For a prediction of the activity in animal food products subsequent to the deposition of radionuclides on farmland, a reliable prediction of the activity intake by the animals is a

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Table 6 Radioecological parameters for modelling the transfer of cesium, strontium and iodine to animal food products (from Müller & Pröhl, 1993) Animal

Dry matter intake

Transfer factor (d/kg, d/l)

product

(kg/d)

Cesium

Strontium

Iodine

Cow’s milk Sheep, goat milk Beef Pork Lamb Chicken Eggs

14 2 9 3 1.0 0.1 0.1

0.003 0.06 0.04∗ 0.4 0.5 4.5 0.3

0.002 0.014 0.0003 0.002 0.003 0.04 0.2

0.003 0.5 0.001 0.003 0.01 0.1 2.8

∗ For beef from lactating cows a value of 0.01 d/kg is applied.

key issue. The intake of radionuclides by animals is dependent on the animal species, the mass, the age, the growth rate of the animal and the digestibility of the feed. Furthermore, the use of feeds for animal nutrition depends on the season and the agricultural conditions and practices in a specific region. In order to illustrate the importance of the feeding practice and the interaction of feeding regime and season on the activity in animal products, the 137 Cs concentration in beef is shown in Fig. 6 following a single deposition on 1st June and 1st September. For both dates, the feeding of maize silage and grass is compared. A big difference between the 2 feeding regimes is observed for a deposition on 1st June. In beef fed on grass, the activity increases immediately, whereas in beef fed on maize the activity only increases in autumn to much lower levels, when the maize is fed that has been contaminated in June. In June, the development of maize is at a very early stage and the resulting contamination in autumn is very small. The difference of the activity in beef between the two feeding regimes is lower, when the deposition occurs in September. Then both crops, maize and grass are directly available for feeding. However, the activity in beef is more persistent in the case of maize feeding, since maize is harvested only once per year and then available during a whole year until the next harvest, whereas grass is continuously harvested throughout the whole vegetation period. The next spring grass is contaminated due to root uptake only, which leads to relatively low activity levels. This example underlines that the activity in animal foods is the result of a complex interaction of radionuclide, animal, season and agricultural practise and that reliable predictions can only be expected if the situation for which it is made is sufficiently well known. However, a remaining problem is the uncertainty concerning the dynamic behaviour of the radionuclides in the animals. Many transfer factors were derived from short-term experiments during which long-term components of retention and loss can hardly be detected. The application of such data for long-term studies might lead to underestimates of activity levels in meat and milk. Other open questions concern the influence of the chemical form, the role of stable elements, and the effects of antagonistic elements. The influence of soil ingested during graz-

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Fig. 6. Comparison of cesium activity in beef subsequent to an integrated air concentration of 1 Bq d/m3 on 1st June and 1st September for maize and grass/hay feeding (parameters: dry matter intake: 9 kg/d, transfer factor feed–beef: 0.04 d/kg, biological half-life: 50 d).

ing on the transfer of radionuclides to food also needs more consideration. Although these processes are known to be potentially relevant, they are presently not adequately considered in assessment models. 2.7. Resuspension Resuspension refers to the removal of deposited material from the ground to the atmosphere as a result of wind, traffic, soil cultivation, and other activities. Potentially, resuspension is a persistent source for radioactivity in air subsequent to deposition on the ground. Furthermore, resuspension may lead to redistribution of activity and to recontamination of cleaned surfaces. It is influenced by a variety of factors, such as the time since deposition, meteorological conditions, surface characteristics, and human activities. Due to the complexity of the process, relatively simple, empirically derived approaches are applied to quantify resuspension in assessment models. In a widely used approach, resuspension is quantified by use of a resuspension factor K (m−1 ), defined as the ratio of the activity in air (Bq m−3 ) to the surface deposit (Bq m−2 ). The resuspension factor can be easily derived from measurements; however, it must be decided at what height above ground the air concentrations are to be measured and to what soil depth the activity is assumed to be available for resuspension.

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Resuspension is influenced by various factors such as the soil type, texture and moisture, the vegetation cover, and the wind speed. The long-term resuspension factor K can be described by a time-dependent function (Garland et al., 1992): K = A · e−B·t + C,

(11)

where A + C gives the initial value of the resuspension factor (m−1 ), B is the rate of decrease of the resuspension factor (d−1 ), and C is the longterm resuspension factor (m−1 ). Under European conditions, values for the initial resuspension factor A + C measured after the Chernobyl accident range from 4 to 50 × 10−9 m−1 (Garland et al., 1992). The observed negative correlation between the initial resuspension factor and the concentration of the deposit indicates that the long-range transport of resuspended material is a relevant process. Within a heterogeneously contaminated region, such long-range transport leads to an apparently higher resuspension factor in areas with low deposits and vice versa. The loss rate B of the resuspension factor found after the Chernobyl accident corresponds to half-times between 200 and 800 d, with a median of approximately 1 year (Garland et al., 1992). Doses due to inhalation or to ingestion of resuspended material subsequent to re-deposition on crops are in general very small. Only for actinides is the inhalation of resuspended soil particles relevant as a long-term pathway. Resuspension is higher under arid conditions; for example, in the PATHWAY model (Whicker & Kirchner, 1987) developed for the southwestern U.S., a resuspension factor of 10−5 m−1 is applied. Resuspension leads to long-range transport of soil material, meaning that contaminated soil is removed. However, the overall annual average mass transport that can be derived from the resuspension factor is very small. Nevertheless, under certain circumstances, the mass transport due to wind erosion may be considerable; wind resuspension might become a relevant loss process for contaminated land for wind resuspension factors in the range of approximately 10−7 m−1 (BIOMOVS II, 1996b). Furthermore, during dry storms, larger amounts of soil may be removed in unvegetated, dry conditions. In particular, sandy soils are very vulnerable to wind erosion. In one area in Northern Germany, during such a storm, the removed soil mass was equivalent to a layer with a thickness of at least 1 cm (Blume, 1990). The probability of such events varies from region to region and is generally low in most areas, but just one such event every decade or two could cause a considerable loss of activity from soil. In the framework of safety studies of radioactive waste disposal, wind erosion might be a relevant long-term process for removal of activity from areas contaminated due to irrigation with radioactively contaminated water. It is interesting to note that sandy soils are characterised by both a high requirement for irrigation water and an enhanced vulnerability to wind erosion. Although the impact of wind erosion on agriculture has been studied intensively, the process is not yet appropriately considered in dose assessment models.

3. Open questions Although radioecological research has been ongoing for more than 4 decades, modelling of the transfer of radioactivity in the terrestrial environment is still associated with considerable

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Table 7 Priority for further research for different transfer processes for each field of application of environmental transfer models Transfer process

Deposition Interception Weathering Translocation Uptake from soil Migration in soil Resuspension Transfer to animals

Priority for further research (+ high, 0 medium, − low) Routine release

Accidental release

Dose reconstruction

Waste disposal

0 0 0 0 0 − − 0

+ + 0 + − − 0 +

+ + 0 + 0 0 − 0

0 0 0 + + + −

uncertainty. In Table 7, the different processes of radionuclide transfer are prioritised according to the necessity for data improvement for the 4 major applications of terrestrial transfer models. This ranking should be regarded as a general one that may be modified for specific radionuclides. Assessment of the radiological consequences of routine releases is not seen as a major problem. In general, the releases are low and lead to low doses, even when conservative models are applied for the dose calculations. However, for accident consequence assessment and dose reconstruction, the largest sources of uncertainty are due to the atmosphere-plant interface and the fate of contaminants subsequent to deposition. More research in deposition, interception, and translocation would most effectively lead to an improvement of model reliability. The large uncertainty in interception and translocation is also underlined by the results of probabilistic uncertainty analysis for accident consequences (USNRC & CEC, 1997). The reliability of predictions in the case of radioactive contamination of farmland could be considerably enhanced, if data on the stage of development of plants would be integrated. Such data are e.g. collected on a regular basis by national weather services and by satellites. In view of dose assessment issues, it is especially important to identify soil types and soil characteristics that cause high transfer of radionuclides from soil to plants. Furthermore, for performance assessments of waste disposals, the long-term behaviour of radionuclides in soil is a key topic. The transfer to animals is of special relevance for the implementation of appropriate countermeasures after accidental releases and in view of the behaviour of cesium in semi-natural ecosystems. For biosphere modelling within the framework of waste disposal, the long-term fate of radioactivity in the environment appears to be the most pressing problem. Therefore, due to the different radionuclides involved, the most important processes are root uptake, migration and resuspension. In particular, data on the flux of radionuclides from contaminated groundwater with a high water table to plants via the soil are extremely poor.

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References Aarkrog, A. (1972). Direct contamination of barley with Be-7, Na-22, Cd-115, Sb-125, Cs-134 and Ba-133. Risø-R256, 163–175. Risø, Denmark: Risø National Laboratory. Aarkrog, A. (1975). Radionuclide levels in mature grain related to radiostrontium content and time of direct contamination. Health Physics, 28, 557–562. Aarkrog, A. (1982). Translocation of radionuclides in cereal crops. Industrial Ecology Group Workshop, Ecological Aspects of Radionuclide Release, April 5–7. Aarkrog, A. & Lippert, J. (1969). The direct contamination of rye, barley, wheat and oats with Sr-85, Cs-134, Mn-54 and Ce-141. Rad. Bot., 9, 357–366. Aarkrog, A. & Lippert, J. (1971). Direct contamination of barley with Cr-51, Fe-59, Co-58, Zn-65, Hg-203, and Pb-210. Rad. Bot., 11, 463–472. Anguissola Scotti, I. & Silva, S. (1990). Foliar absorption and leaf-fruit transfer of 137 Cs in fruit trees. Journal of Environmental Radioactivity, 16, 97–108. Baldini, E., Bettoli, M. G. & Tubertini, O. (1987). Effects of the Chernobyl pollution on some fruit trees. Advances in Horticultural Science, 12, 77–79. Barber, S. A. (1984). Soil nutrient bioavailability: A mechanistic approach (398 p.). New York: John Wiley & Sons. Bengtsson, G. B. (1992). Mobility of superficially applied caesium-134 and strontium-85 in apple branches under precipitation-free conditions. Analyst, 117, 1193–1196. BIOMOVS II (1996a). Uncertainty and Validation: Effect of User Interpretation on Uncertainty Estimates. BIOMOVS II Technical Report No. 7. Stockholm: Swedish Radiation Protection Institute, BIOMOVS II (1996b). Atmospheric Resuspension of Radionuclides: Model Testing Using Chernobyl Data. BIOMOVS II Technical Report No. 11. Stockholm: Swedish Radiation Protection Institute. Blume, H. P. (1990). Handbuch des Bodenschutzes. Landsberg/Lech: Verlag Ecomed. Brown, J. & Simmonds, J. R. (1995). A dynamic model for the transfer of radionuclides through terrestrial foodchains; National Radiological Protection Board, Report NRPB-R273. Didcot, UK: Chilton. Butler, A. P. & Wheater, H. S. (1990). Model sensitivity studies of radionuclide uptake in cropped lysimeters. Nirex Safety Series Report NSS/R253 (p. 71). UK: Nirex Ltd. Carini, F. & Lombi, E. (1997). Foliar and soil uptake of 134 Cs and 85 Sr by grape vines. The Science of the Total Environment, 207, 157–164. Carini, F., Anguissola Scotti, I. & D’Alessandro, P. G. (1999). 134 Cs and 85 Sr in fruit plants following wet aerial deposition. Health Physics, 77, 520–529. Chamberlain, A. C. (1970). Interception and retention of radioactive aerosols by vegetation. Atmospheric Environment, 4, 57–58. Ehrhardt, J., Brown, S., French, S., Kelly, G. N., Mikkelsen, T. & Müller, H. (1997). RODOS: Decision-making support for off-site emergency management after nuclear accidents. Kerntechnik, 62, 122–128. Epstein, E. (1966). Dual pattern of ion adsorption by plant cells and by plants. Nature, 212, 1324–1327. Ertel, J., Voigt, G. & Paretzke, H. G. (1989). Weathering of 134/137 Cs following leaf contamination of grass cultures in an outdoor experiment. Rad. Environ. Biophys., 28, 319–326. Fitzjarrald, D. R. & Lenschow, D. H. (1983). Mean concentration and flux profiles for chemically reactive species in the atmospheric surface layer. Atmos. Environm., 17, 2505–2512. Garland, J. A., Pattenden, N. J. & Playford, K. (1992). Resuspension following Chernobyl. In International Atomic Energy Agency (Ed.), Modelling of resuspension, seasonality, and losses during food processing. IAEATECDOC-647. Vienna: IAEA. Geisler, G. (1983). Ertragsphysiologie von Kulturarten des gemäßigten Klimas. Hamburg and Berlin: Verlag Paul Parey. Gering, F. & Müller, H. (2000). Deposition calculation in RODOS PV 4.0. In RODOS: Decision support system for off-site emergency management in Europe, Report EUR 19144. Heinemann, K. & Vogt, K. J. (1980). Measurements of the deposition of iodine onto vegetation and the biological half-life of iodine on vegetation. Health Physics, 39, 463–474. Hoe, S., Müller, H. & Thykier-Nielsen, S. (2000). Integration of dispersion and radioecological modeling in ARGOS NT. In Proc. 10th IRPA Congress, Hiroshima, Japan, 14–19 May 2000.

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Hoffman, F. O, Thiessen, K. M., Frank, M. L. & Blaylock, B. G. (1992). Quantification of the interception and initial retention of radioactive contaminants deposited on pasture grass by simulated rain. Atmospheric Environment, 26A, 3313–3321. Hoffman, F. O., Thiessen, K. M. & Rael, R. M. (1995). Comparison of interception and initial retention of wetdeposited contaminants on leaves of different vegetation types. Atmospheric Environment, 29, 1771–1775. Hosker, R. P. & Lindberg, S. E. (1982). Review: Atmospheric deposition and plant assimilation of gases and particles. Atmospheric Environment, 16, 889–910. Howard, B. J., Johanson, K., Linsley, G. S., Hove, K., Pröhl & G., Horyna, J. (1996). Transfer of radionuclides by terrestrial food products from semi-natural ecosystems. In Modelling of radionuclide interception and loss processes in vegetation and of transfer in semi-natural ecosystems. IAEA-TECDOC-857 (pp. 49–79). Vienna: International Atomic Energy Agency. IAEA (1994). Handbook of Parameter Values for the Prediction of Radionuclide Transfer in Temperate Environments. Technical Report 364. Vienna: International Atomic Energy Agency. Jonas, R. (1984). Ablagerung und Bindung von Luftverunreinigungen an Vegetation und anderen atmösphärischen Grenzflächen. Bericht Jül-1949, 1949. Jülich: Kernforschungsanlage Jülich. Koch-Steindl, H. & Pröhl, G. (2001). Considerations on the behaviour of long-lived radionuclides in the soil. Radiat. Environm. Biophysics, 40, 93–104. Kinnersley, R. P., Goddard, A. J. H., Minski, M. J. & Shaw, G. (1997). Interception of caesium-contaminated rain by vegetation. Atmospheric Environment, 31, 1137–1145. Kirchner, G. (1994). Transport of iodine and caesium via the grass-cow-milk pathway after the Chernobyl accident. Health Physics, 66, 653–665. Lieser, K. H. (1993). Technetium in the nuclear fuel cycle, in medicine, and in the environment. Radiochimica Acta, 63, 5–8. Miller, C. W. & Hoffman, F. O. (1983). An examination of environmental half-time for radionuclides deposited on vegetation. Health Physics, 24, 731–744. Müller, H. & Pröhl, G. (1993). ECOSYS-87: A dynamic model for assessing radiological consequences of nuclear accidents. Health Physics, 64, 232–252. Nye, P. H. (1966). The effect of nutrient intensity and buffering power of a soil, and the absorbing power, size and root hairs of a root, on nutrient adsorption by diffusion. Plant Soil, 25, 81–105. Nye, P. H. & Tinker, P. B. (1977). Solute movement in the soil-root system (p. 342). Oxford: Blackwell Scientific Publications. Pel, E. (1993). The behaviour of iodine in the system soil–plant-air. Thesis, University of Karlsruhe, KFK-report 5118 (in German). Poon, C. B., Au, S. M., Pröhl, G. & Müller, H. (1997). Adaptation of ECOSYS-87 to Hong Kong environmental conditions. Health Physics, 72, 1–9. Pröhl, G. (1990). Modellierung der Radionuklidausbreitung in Nahrungs-ketten nach Deposition von Strontium-90, Cäsium-137 und Jod-131 auf landwirtschaftlich genutzte Flächen. München-Neuherberg, GSF-Bericht 29/90. Pröhl, G. & Hoffman, F. O. (1996). Radionuclide interception and loss processes in vegetation. In Modelling of radionuclide interception and loss processes in vegetation and of transfer in semi-natural ecosystems. IAEATECDOC-857 (pp. 9–47). Vienna: International Atomic Energy Agency. Pröhl, G., Müller, H. & Hoffman, F. O. (1995). Interception and post-deposition retention of radionuclides by vegetation and their importance for dose assessment. In IAEA (Ed.), Proc. of “Environmental Impact of Radioactive Releases”, IAEA-SM339-142. Vienna. Roed, J. (1987). Dry deposition in rural and urban areas in Denmark. Radiation Protection and Dosimetry, 21, 33–36. Shaw, G. & Bell, J. N. B. (1989). The kinetics of caesium absorption by roots of winter wheat and the possible consequences for the derivation of soil-to-plant transfer factors for radiocaesium. Journal of Environmental Radioactivity, 10, 213–231. Strasburger, E. (1978). Lehrbuch der Botanik. Stuttgart: Gustav Fischer Verlag. Thiessen, K. M., Thorne, M. C., Maul, P. R., Pröhl, G. & Wheater, H. S. (1999). Modelling radionuclides distribution and transport in the environment. Environmental Pollution, 100, 151–177. USNRC & CEC (U.S. Nuclear Regulatory Commission and Commission of European Communities) (1997). Probabilistic accident consequence uncertainty analysis – Food chain uncertainty assessment. NUREG/CR-6523, EUR 16771.

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Voigt, G., Pröhl, G. & Müller, H. (1991). Experiments on the seasonality of the cesium translocation in cereals, potatoes and vegetables. Radiation and Environmental Biophysics, 30, 295–303. Whicker, W. F. & Kirchner, T. B. (1987). Pathway: A dynamic food-chain model to predict radionuclide ingestion after fallout deposition. Health Physics, 6, 717–737. Zehnder, H. J., Kopp, P., Oertli, J. J. & Feller, U. (1993). Uptake and transport of radioactive cesium and strontium into strawberries after leaf contamination. Gartenbauwissenschaft, 58, 209–211.

MODELLING RADIOACTIVITY IN THE ENVIRONMENT E. Marian Scott (Editor) © 2003 Elsevier Science Ltd. All rights reserved

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Chapter 5

Modelling radionuclide transport in the geological environment: a case study from the field of radioactive waste disposal W. Russell Alexander a,* , Paul A. Smith b , Ian G. McKinley a a Nagra (National Co-operative for the Disposal of Radioactive Waste), Wettingen, Switzerland b SAM (Safety Assessment Management) Ltd, Edinburgh, Scotland, UK

1. Introduction This chapter provides a short introduction to modelling the distribution of radionuclides in the geological environment (or geosphere). The basic principles of radionuclide geosphere transport modelling are examined, with the emphasis on identifying the processes and structures of relevance to contaminant transport. Approaches for including these relevant features in transport models used to assess the long-term performance of radioactive waste repositories are then discussed, followed by an explanation of how such models should be tested to ensure that all relevant mechanisms have been included and are represented in an appropriate manner. There are several differences between repository performance assessments and the use of the models to assess groundwater resources or pure research on radionuclide chemistry in the geosphere and these will be examined and explained, mainly by use of examples from the deep geological repository programme in Switzerland1. The main difference is that, in a repository performance assessment (or PA), the radionuclide transport models are used as a tool to assess risk and safety and are applied in a relatively crude fashion. Obversely, building confidence in the results of the calculations employs series of tests which are much more thorough, better documented and requiring of much greater effort than anything generally considered in more traditional areas of radionuclide transport modelling. This dichotomy will be highlighted with appropriate examples from the literature and the subsequent effects on the progress of the modelling of radionuclide transport in the geological environment discussed. * E-mail: [email protected] 1 Radioactive waste repository designs vary depending on the precise waste type. This chapter focusses on what is

commonly termed ‘deep geological disposal’, meaning the waste is emplaced in a repository constructed at depths of several tens to over one thousand metres deep, so making use of the surrounding geological environment to both protect the waste itself and to retard any radionuclides released from the repository. 109

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1.1. Sources of radionuclides A wide range of radionuclides can be found in all components of the geological environment – the number of specific radioisotopes identified being primarily limited by the sensitivity of the analytical techniques used. These radionuclides include: • “Primordial” isotopes which are sufficiently long-lived to have persisted in measurable quantities since nucleosynthesis of the elements which make up the earth (>∼ 5 × 109 years ago). • Radioactive daughters (usually shorter lived) which are continually produced by the decay of primordial isotopes; of particular relevance are the “natural decay series” chains of daughters produced by the decay of 238 U, 235 U and 232 Th. • Radionuclides produced in situ by nuclear reactions; of particular importance are neutrons from the background of spontaneous fission of some actinide elements which can produce radionuclides by (n, α) and (n, p) reactions. • “Cosmogenic” radionuclides which are produced continuously in the upper atmosphere and enter the geosphere through the water cycle. • “Anthropogenic” radionuclides which are produced by the activities of man. Major sources to the geosphere include general atmospheric input from nuclear weapons testing and nuclear fuel cycle activities/accidents and direct underground input at specific locations due to disposal of radioactive wastes or, again, nuclear weapon testing. It should be emphasised that specific radioisotopes may have more than one source: for example, 3 H which can be produced in situ, is formed cosmogenically, is a component of anthropogenic atmospheric fall-out and is also a major component in many wastes considered for geological disposal. 1.2. Measuring and modelling radionuclide mobility From a purely academic point of view, the radioactive decay process has two key features which attract interest to the study of subterranean radionuclides: • The simple first order exponential decay law characterised by a single constant which can, in principle, be measured simply and accurately. This provides a “clock” which is the basis for an enormous range of geochronological techniques (see, for example, Faure, 1998). • The elemental transmutation and energy release associated with the decay process which can lead to partitioning of parent and daughter isotopes. This provides a valuable tool to assess in-situ elemental mobility/geochemistry over long periods of time (for examples, see Ivanovich & Harmon, 1992). Additionally, from a practical viewpoint, the radiation produced by the decay process provides a convenient method for quantifying specific isotopes at very low concentrations. Indeed, until the relatively recent development of mass-spectrometric techniques, the sensitivity of radioanalytical methods was an important consideration in the selection of radionuclides as tracers of environmental processes. The simplest application of radioactive decay considers “closed” systems in which no material exchange occurs. For example, a mineral which incorporates a radionuclide at a known

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concentration (or activity) or which excludes the daughter resulting from decay of this isotope can be readily dated by quantifying the extent of decay which has occurred. The principles involved and practical aspects of their application to a wide range of systems are discussed in detail elsewhere (e.g. Eicher, 1976; Ivanovich et al., 1992 and Press & Siever, 1998). Of more interest for the purposes of this chapter, however, are open systems in which a material flux exists. This chapter will focus exclusively on fluxes of radionuclides in groundwater although many of the basic principles could be applied to other fluids (magma or gas). For any open system, the parameters of interest could include one or more of the following: • Rate of radionuclide release from a specific source. • Rate of transport of the fluid or contained solute. • Rate of accumulation of radionuclide at a specific sink. Either the radionuclide examined could be of direct interest or it could be considered as an analogue for the behaviour of other isotopes of the same element or, indeed, chemically similar elements (see discussion on analogue and homologue elements in Petit, 1991 and Miller et al., 2000). Such information is clearly important for studies of geochemical cycling of elements, either for purely academic reasons or for commercial application in identification of the key factors influencing the formation and longevity of ore bodies (e.g. Plant et al., 1999). In other areas of the geological environment, the ability to trace and evaluate groundwater flow rates and provenance has long provided a valuable hydrogeological tool for groundwater exploitation (e.g. Land, 1966). Indeed, historically, this was the main reason for the development of many groundwater modelling tools earlier last century. These tools dramatically increased in power and definition with the introduction of stable and radioisotopic analytical methods, so allowing a better definition of groundwater ages and temporal evolution (see, for example, Osmond & Cowart, 1976 and Loosli et al., 1991, for an overview). The development and application of such models of radionuclide behaviour in the geological environment in these more ‘traditional’ areas of study are adequately covered in the literature (see, for example, Faure, 1998; Pearson et al., 1991, and Ivanovich & Harmon, 1992). Increasingly, however, another justification for such modelling studies is associated with the disposal of wastes (both radioactive and chemotoxic) or the remediation of areas already contaminated with such material and this, the modelling of contaminant transport in the geological environment, will form the main focus for this chapter. 1.3. Historical overview of radionuclide transport modelling in the geological environment (geosphere) Although the basic principles involved were established early last century, systematic studies of radioactivity in the geological environment (or geosphere) date from the latter half of the century, associated with considerable advances in analytical methodology. Initial emphasis was very much on the geochronology of closed systems and, in parallel, uranium-series geochemistry (linked, to a large extent, to the expanding military and civil use of nuclear materials). Quantitative studies of radionuclide mobility in the 1960s and 1970s were primarily associated with quantifying groundwater flow in aquifers or evaluating radionuclide transport

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from subterranean anthropogenic sources (nuclear weapon explosion cavities, waste disposal sites)2 . Mathematical models used were, of necessity, rather simplistic, limited by both the available databases and the power of computational tools. During the 1980s and 1990s, great efforts were made in the characterisation of rock formations previously simply regarded as aquitards and the true complexity of solute transport in a wide range of geological formations became increasingly evident. Key topics of interest during this period included: • The rôle of geological structure/heterogeneities/anisotropies (limitation of equivalent porous medium models and upscaling problems; see comments in Smith et al., 2001a, b and references therein). • The diversity of individual processes included in bulk terms such as “sorption” or “retardation” and the problems of measurement (see, for example, Sibley & Myttenaere, 1986). • Complexities added by the presence of colloids, organics and microbes in groundwater (see review of McCarthy & Zacchara, 1989). • Inherent errors and uncertainties associated with models and the need for verification and validation (see, for example, McCombie et al., 1991 and Pescatore, 1995). Much model development focussed on the development of “coupled” codes which coupled physical (groundwater transport) and chemical (contaminant retardation) processes in one code, so providing computationally-efficient tools to quantify increasingly complicated conceptual representations of radionuclide migration (e.g. Lichtner et al., 1996). Now, at the beginning of the new millennium, cheap computational power has changed the perspective on the quantification of radionuclide mobility completely. The limiting factors constraining more detailed quantification of radionuclide mobility in relevant rock formations are now set almost entirely by limitations in our ability to characterise the processes involved in situ and to validate model extrapolations made on the basis of laboratory-scale experiments to the temporal and spatial scales relevant to geological processes (see comments in Smith et al., 2001a, b and references therein).

2. Modelling radionuclide transport in the geosphere 2.1. Introduction For most applications of radionuclide transport modelling, the model and data requirements are very similar, namely that all relevant processes and, in the geosphere in particular, structures are represented in an appropriate manner. This is the case whether transport in the rock formation is predominantly controlled by diffusion (e.g. in a salt diapir or in a consolidated claystone) or by advection in porous media (e.g. in a sandstone aquifer or fluvial gravel deposit) or in fractured media (e.g. in a limestone formation or an uplifted granite massif). Here, those features which are basic to most models are listed and described and, where appropriate, comments are made on outstanding data requirements for transport models. 2 This was paralleled in marine radiochemistry with the use of isotopic signatures from atmospheric weapons testing and releases from nuclear accidents or reprocessing plants to trace ocean currents and to assess ocean residence times etc.

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Radionuclides are present in geological media in a variety of forms. These forms may be immobile, such as solutes in stagnant porewater in isolated pore spaces within the rock (and as fluid inclusions in minerals), and forms that are bound (sorbed) in some way to immobile mineral surfaces. They may also be mobile, such as solutes in connected networks of fractures and pores, volatile species in flowing gas3 and forms that are bound to groundwater colloids (which, for the purposes of this discussion, may be treated as mobile solid surfaces). Radionuclides may also be exchanged between forms: mobile radionuclides may, for example, enter and leave solution many times during migration from a deep source (e.g. an ore body or a waste repository) and the surface as they pass through various geochemical boundaries in the geosphere. In order to model how radionuclides are transported through geological media, an understanding must be developed of the structure of the pore spaces (including fractures) within the media, the transport processes that operate within these voids and the processes by which radionuclides are exchanged between immobile and mobile forms. The following sections give a general discussion of transport-relevant processes and geosphere structures, although, it should be noted, the actual structures that are present (and the relative importance of the different transport-relevant processes) vary considerably between different geological media. 2.2. Transport processes and geosphere structures 2.2.1. Processes Advection is the process by which radionuclides, which are associated, for example, with dissolved species or colloids, are conveyed by the bulk motion of flowing groundwater. The pressure gradients driving groundwater flow may arise, for example, from variations in the hydraulic head (due to variations in the height of the water table, for example), glacial rebound and variations in density associated with salinity and temperature contrasts. Groundwater flow rates vary considerably in most media, due to the heterogeneity in fracture and pore space structure and to friction on fracture or pore space surfaces. There will exist some paths for advection through the media or fractures that are faster than the average and some that are slower. The resulting spreading of transported solutes or colloids is known as mechanical dispersion. Longitudinal dispersion, the spread in the direction of bulk flow, is generally much stronger than transverse dispersion perpendicular to the flow. Diffusion is the process by which radionuclides migrate down gradients in chemical potential. Diffusion causes a spreading of transported solutes or colloids which, when combined with mechanical dispersion, is called hydrodynamic dispersion. The rate of diffusion is determined by the magnitude of the concentration gradient, and also by the diffusion coefficient of each given solute. The diffusion coefficient is itself a function of the properties of the rock, such as the tortuosity of pore spaces, the properties of the groundwater and, in particular, its temperature, and the properties of the diffusing species, such as their charge and size. In addition to the direct processes of advection, dispersion and diffusion, a range of coupled processes, including thermal, chemical and electrical osmosis, thermal diffusion, hyperfiltration and electrophoresis, can transport porewater and radionuclides in solution in response to gradients in temperature, pressure, solute concentration and electrical potential. Although 3 Gas-mediated radionuclide transport is not discussed further in this chapter (see overview of contaminant transport

in gas by Philp & Crisp, 1982).

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Table 1 The Onsager matrix of direct (diagonal) and coupled (off-diagonal) transport processes (from Horseman et al., 1996) Flux

Heat Fluid Solute Current

Potential gradient Temperature

Hydraulic

Chemical

Electrical

Thermal conduction (Fourrier’s Law) Thermal osmosis

Thermal filtration

Dufour effect

Peltier effect

Advection (Darcy’s Law) Hyperfiltration

Chemical osmosis

Electrical osmosis

Diffusion (Fick’s Law) Diffusion and membrane potential

Electrophoresis

Thermal diffusion or Soret effect Seebeck or Thompson effect

Rouss effect

Electical conduction (Ohm’s Law)

such coupling is generally negligable for most practical applications, some of these processes, including chemical osmosis and hyperfiltration, can be significant in argillaceous sediments, where the overlapping diffuse double layers of clay platelets result in the rock acting as a semi-permeable membrane. The complete Onsager matrix of direct and coupled processes is given in Table 1. Where there is significant groundwater flow, advection and mechanical dispersion are often far more effective transport processes than diffusion or the coupled processes mentioned above. This does not mean, however, that these processes can necessarily be neglected. In many media, significant groundwater flow is confined to discrete water-conducting features, such as fractures, sand lenses, thrust planes, alluvial or glacial channels and rock unconformities. Elsewhere in the rock body, groundwater may effectively be stagnant. In such cases, diffusion may, for example, transport radionuclides from water-conducting features into stagnant porewater “matrix” regions and vice versa. This process of matrix diffusion (see Fig. 1a) is referred to as a retardation process, since it results in slower transport than would be the case if only advection, mechanical dispersion and diffusion along water-conducting features operated. In some media, matrix pores may be accessible to solutes by diffusion, but larger molecules, ions and colloids may be excluded due to size and/or charge effects (Fig. 1b). The retarding effect of matrix diffusion may thus apply to radionuclides associated with solutes, but not to those associated with colloids (although colloids may be retarded or immobilised in other ways, such as by filtration). A number of chemical retardation mechanisms have been identified in natural systems, as shown diagrammatically in Fig. 1c and d: • • • •

adsorption, ion-exchange, precipitation, mineralisation.

Adsorption and ion-exchange are often collectively termed sorption, a term used generically to encompass chemical interactions with solids that retard transport. Sorption is generally modelled as a reversible process whereby sorbed radionuclides may be released to solution

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Fig. 1. The retardation mechanisms that may affect radionuclides in the geosphere (after McKinley & Hadermann, 1984).

if solution concentrations or composition change. Desorption kinetics may be slower than sorption kinetics and there may be instances where sorption could be considered irreversible4. Precipitation is not a sorption process but it can, on occasions, be difficult to discriminate between the two mechanisms in both field and laboratory studies. Generally, sorption would be expected to be the dominant process at low solution concentrations of radionuclides. As 4 Strictly speaking and according to thermodynamics, no chemical process, such as the sorption of radionuclides by mineral surfaces, can ever be truly irreversible. All chemical reactions are reversible, the significance is in the time taken and the conditions necessary for the reaction to be reversed. When irreversible sorption is discussed in terms of repository PA, it is generally meant that the kinetics of desorption are slow compared to the time period of interest to the assessment, assuming the physico-chemical conditions are constant.

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concentrations of radionuclides increase, precipitation of phases in which the radionuclide of interest is a stoichiometric component (i.e. an essential component of the mineral structure) may occur if saturation is reached in the groundwater. In circumstances where the total amount of dissolved solids in the groundwaters is high enough that precipitation occurs, a very complex chemical environment may develop in which radionuclides may co-precipitate as solid-solutions in a variety of mineral phases, or be scavenged by amorphous precipitates, such as iron oxyhydroxides, and also effectively be co-precipitated. The stability of newly precipitated minerals and amorphous phases depends on saturated concentrations being maintained. If concentrations fall below saturation, perhaps after a pulse of contaminated groundwater has passed, or as the radionuclide source area (e.g. the radioactive waste) is flushed with fresh groundwater, these minerals will begin to dissolve and release any radionuclides they contain back to solution. The situation can become even more complicated in microporous media, where particular solutes may be concentrated enough close to mineral surfaces to allow precipitation to occur – even when bulk solution is undersaturated. The kinetics of sorption and dissolution and precipitation processes are clearly important to long-term predictions of radionuclide behaviour. Irreversible sorption (slow desorption) and precipitation processes are obviously beneficial in terms of radionuclide retardation because they immobilise radionuclides very effectively. They are, however, difficult to demonstrate, and, consequently, many PA representations of radionuclide retardation make the assumption that all chemical retardation processes are instantaneously reversible5. Sorption processes generally occur sufficiently rapidly (relative to transport rates) in the geosphere to allow their kinetics to be ignored in transport models, although precipitation and mineralisation kinetics would clearly be relevant if these processes were to be included in an model. Where groundwater flows are relatively rapid (as may occur, for example, in a major fracture zone or in a shallow aquifer), then kinetic mechanisms become increasingly less significant in affecting the rates of transport of radionuclides. Radioactive decay and ingrowth are, of course, important processes to be taken into account in evaluating radionuclide transport. Radionuclides may decay, or be created by ingrowth, during transport through a geological medium. If the time taken for a radionuclide to be transported across the medium is significantly longer than the radionuclide half life (and no ingrowth from a parent radionuclide occurs, including that of recoil from radionuclides already present in the rock), then very little of that radionuclide will emerge. If, however, some transport paths exist for which the transport times are less than the half life, then these can result in some radionuclide migration to the surface, even though a large proportion of the radionuclide mass may decay during transport along other, slower paths. It was noted above that some radionuclides could be retarded and then released again to solution due to changes in the geochemical conditions of the geosphere. There are two other processes which contribute to changes in retardation potential of radionuclides in the geosphere, namely recoil and chemical transformation. Recoil is the process whereby a daughter radionuclide may be physically ejected from the site of the mother by the decay process 5 This aspect of PAs is often difficult for many people to understand: an important premise of a PA is to show that, despite assuming the worse case scenario for each process or mechanism (for example, ignoring geosphere retardation or irreversible sorption), a repository can be shown to satisfy the safety criteria set out by the regulatory authorities. Expressed in another way, the repository is deliberately ‘over-engineered’ to provide large margins of safety (i.e. belt and braces).

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(see Osmond & Cowart, 1976, for a detailed discussion). This may lead to direct transfer of the daughter from a mineral surface into solution or transfer of the daughter into another site within the mineral which may cause subsequent loss of the daughter to solution as the lattice is now damaged and so the daughter may be more readily leached from the mineral. In addition to the above chemical transformations, a change in speciation can occur (e.g. U4+ to U6+ ) either during the recoil process itself or due to a different oxidation potential between the displaced site (e.g. in the groundwater) and the original mineral lattice site6 . In chemical transformation the decay process can produce a daughter whose geochemical characteristics differ markedly from the mother, so changing the retardation properties of the daughter. A very good example is that of 226 Ra and its daughter and granddaughter. In a sulphate rich groundwater, for example, 226 Ra can be present as 226 RaSO4 and so may be co-precipitated in gypsum (as Ca226 RaSO4 ) or barite (Ba226 RaSO4 ). Upon decay, however, the daughter 222 Rn, in gaseous form, usually diffuses out of the mineral structure where it can be transported in the groundwater as it decays through a series of short lived daughter radionuclides to the relatively insoluble 210 Pb (which will sorb onto mineral surfaces, coprecipitate with iron and manganese oxyhydroxides or precipitate as lead sulphide). 2.2.2. Structures Radionuclide transport is sensitive to the geometrical, hydrogeological, mineralogical and geochemical properties of a geological medium. In many rock types, water-conducting features provide the dominant pathways for radionuclide transport. These features may be structural or sedimentary in nature and include, for example, joints, mineralised veins, complex shear zones, regional faults, sedimentary lenses and various forms of alluvial channels. As illustrated in Fig. 2, advection and dispersion are usually the dominant transport processes in deep geological media containing water-conducting features, with retardation by matrix diffusion, filtration and sorption7. The processes that dominate radionuclide transport often vary according to these properties, which are typically heterogeneous over a range of spatial scales. Even if it is the transport of radionuclides over a scale of tens or hundreds of metres that is of interest to the longterm performance of a deep geological repository, this does not necessarily mean that small scale structures and property variations can be neglected or “averaged over” (see comments in Smith et al., 2001a, b). Small-scale structures may, for example, provide relatively fast paths for advective transport, particularly if, say, they are lined with minerals that prevent or limit retardation by matrix diffusion and sorption. Alternatively, fracture infill material 6 Ku et al. (1992) noted that, in models of radionuclide transport in the geosphere, “The most appropriate formulation of recoil input remains an open question requiring further study.” They further noted that their formulation “. . .should provide an upper limit for input by recoil. . .”. Krishnaswami et al. (1982) and Scott et al. (1992) argue for much lower recoil-induced input on mathematical and chemical grounds, respectively. Indeed, Scott et al. (1992) further note that, in many cases, recoil loss per se will be minimal. See also the discussion in Short et al. (1988). 7 Although not included in Fig. 2, transport of radionuclides sorbed on natural groundwater colloids may also have to be taken into account in such deep geological media. In shallower geological media, transport in association with microbes and colloids must also be taken into account as should geochemical fronts. For example, redox fronts are well-known sites of radionuclide retardation (causing, for example, U roll-front ore bodies). Unfortunately, the current understanding of radionuclide entrapment at such fronts is limited to a few (economic) elements and, even then, the radionuclide trapping efficiency of such systems is unknown (see review in Hofmann, 1999).

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Fig. 2. Schematic illustration of transport processes in and around a water-conducting feature (after Fig. 4.2.2-1 in JNC, 2000).

Fig. 3. Overview of field-derived parameters used for radionuclide transport modelling and flowchart of data derived on different scales, using limestone beds of the Palfris Formation, Wellenberg, Central Swiss Alps, as an example (from Mazurek et al., 1997).

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and altered zones adjacent to fractures may have porosities and mineralogical properties that favour retardation (see examples in Smellie et al., 1986 and Moeri et al., 2002a, b). The characterisation of a geological medium for transport modelling purposes typically, therefore, has to examine features at the scale of decimetres or less (e.g. Moeri et al., 2002b, 2003), as well as those at a larger scale (e.g. Mazurek et al., 1997). Figure 3 shows an example, based on limestone beds in the Palfris Formation at Wellenberg in the Central Swiss Alps (a candidate location for a repository for low and intermediate-level radioactive waste), which illustrates the way in which information from hydraulic testing and geological characterisation, on scales ranging from 500 m to less than 0.1 m, provides input to radionuclide transport modelling. In other geological media, argillaceous sediments for example, the low hydraulic conductivity, lack of water-conducting features and immobility of colloids means that transport is dominated by diffusion of solutes, with retardation by sorption (e.g. Gautschi, 2001). Coupled processes may also play a role where, for example, strong thermal gradients are present (see case study by Soler, 1999 and the overview of Horseman et al., 1996, for example). 2.3. Modelling approaches in a repository PA 2.3.1. Introduction The approaches outlined above are appropriate to all types of radionuclide transport modelling in the geosphere. From this point, however, the modelling approaches employed in a repository PA will be described (for modelling approaches in more standard applications, see, for example, Osmond & Cowart, 1992, and Bruines & Genske, 2001) with particular reference to several Swiss studies on fractured rocks. It must be emphasised that all models and databases are simplifications of reality. The processes that are incorporated and their representation in the model, and the geometry (structures in the rock) within which these processes operate, must be simplified because of the complexity of natural systems and the impossibility of complete characterisation of the geosphere. Within a repository PA, it is necessary to have confidence either that the simplifications have a negligible impact on the results or, if bounding estimates are acceptable, that the simplifications are conservative, leading to an overestimation of radionuclide releases and consequently doses8 . It is also necessary to show that the parameters used in the PA are realistically (or, again, at least conservatively) assigned. In accordance with current understanding of geosphere transport processes (see, for example, NEA, 1999), the dominant processes governing solute transport in fractured rocks are generally assumed to be: • advection and dispersion within the water-conducting features, and • retardation due to matrix diffusion into the rock matrix and sorption onto mineral surfaces. In PA, these processes are assumed to operate in extensive heterogeneous networks of waterconducting features, although a detailed, small-scale understanding of the structure of the features is also required in order to model matrix diffusion (see, for example, Nagra, 1994). As 8 In the jargon of the radioactive waste disposal industry, this is called ‘conservative’ behaviour and may be com-

pared with the belt and braces approach noted above.

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Fig. 4. The use of supporting hypotheses and interpretative models to assess field and laboratory data in terms of input parameters of a transport model (from Smith et al., 2001b).

illustrated in Fig. 3, the processes (including the geometrical parameters and the parameters that are used to define the rates and spatial extent of processes) are derived from a broad base of information which includes the results of a range of characterisation techniques and general scientific understanding. The information is interpreted, in terms of transport-model parameters, by means of various supporting hypotheses and models (e.g. measured transmissivities are converted to advection parameters via groundwater flow models; sorption measurements are converted to transport model parameters via a Kd or sorption isotherm model: see, for example, Mazurek et al., 1992; McKinley & Alexander, 1992). It should be noted that there may be many specific differences in the structures that are relevant and the parameter values used depending on whether a model is applied in performance assessment or to the modelling of a field experiment. If, for example, the aim is to interpret

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experimental results or field observations, or to forecast the outcome of a future experiment, a model prediction is likely to correspond to a “best guess” of the evolution of radionuclide concentrations or fluxes. If, on the other hand, the aim is to demonstrate the safety of a radioactive waste repository, a model prediction is more likely to correspond to a pessimistic “upper bound” of the fluxes to, or concentrations in, the near-surface environment. Uncertainties arising from limited understanding of transport-relevant processes and from incomplete characterisation of geosphere properties and structure will inevitably limit the accuracy of “best guess” predictions, or necessitate the use of pessimistic model assumptions and data in making bounding predictions. The following sections discuss the modelling of transportrelevant processes and geosphere structures, including the influence of uncertainty on model formulation and parameter values. 2.3.2. Modelling transport-relevant processes Some of the processes relevant to radionuclide transport are well understood and can be modelled using fairly simple relationships based on fundamental physical and chemical principles (such as Darcy’s Law for advection, Fick’s Laws for diffusion and the Bateman Equations for decay and ingrowth) and these are incorporated, in some form, into most radionuclide transport models. Other processes are more complex to model and, in some cases, less well understood, examples being advection in flowing groundwater in highly heterogeneous geological media and the range of processes that are grouped together as “sorption”. Two distinct approaches have been adopted for incorporating such processes in transport models: (i) Incorporate detailed models for these processes in the transport models themselves. (ii) Incorporate the processes in a relatively simple form in the transport models (“model abstraction” or simplification). Develop separate detailed models for the processes and use these to derive input parameters for the transport models. The first approach is often favoured if the aim is to make best-guess predictions of radionuclide migration in, for example, aquifer studies. The second is frequently used for bounding estimates in a repository PA. The relative merits of the two approaches are discussed further in Section 3. Examples of the first approach are the computer codes that have been developed for both groundwater flow and radionuclide transport modelling. Typically, the groundwater flow part of the model is first set up and calibrated against field measurements. Radionuclide transport is then modelled, for example, by assessing radionuclide advection through the calculated groundwater flow field, while taking into account other transport-relevant processes such as retardation by matrix diffusion and sorption. Numerous such codes exist for porous media rocks and research is currently ongoing to integrate mechanistic sorption models in such codes for fractured media (see, for example, NEA, 1997, 1998). Examples of the second approach are the computer codes that require, as input, the results of groundwater flow modelling (e.g. in the form of specific flow rates along discrete transport paths) and parameters describing the relationship between the amount of a radionuclide sorbed on solid surfaces and the amount in solution (e.g. the sorption (or distribution)

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coefficient, Kd 9 , or sorption isotherms). Sorption coefficients, or sorption isotherms, are generally based on laboratory experimental observations. In theory, mechanistic sorption models (see review in Mangold & Tsang, 1991) can provide a tool for extrapolating such laboratoryderived sorption data into a representation of in situ radionuclide retardation in the geosphere. Unfortunately, the radionuclide sorption literature is, according to McKinley & Alexander (1993a) “. . . a morass of poorly defined concepts and inconsistent nomenclature” and so this approach must be treated with great care and a pinch of scepticism. Other processes are also relatively complex or poorly understood and have, up to now, also invariably been modelled in a highly simplified manner, justified by empirical observation, by arguments that suggest the process can be neglected or by arguments that suggest the approach, though unrealistic, at least leads PA modellers to err on the side of repository safety. Examples are the modelling of mechanical dispersion, coupled processes and colloidfacilitated radionuclide transport. In the case of mechanical dispersion, spreading is generally modelled empirically as a diffusion-like process, calibrated against field-scale tracer tests. Although this empirical approach may be adequate for fitting radionuclide breakthrough curves (i.e. the plot of the radiotracer concentration observed at the output borehole versus time since injection at the input borehole), it should be noted that it breaks down if used to model radionuclide fluxes at different points within the medium. Indeed, field observations suggest that the dispersion coefficient that must be used to fit breakthrough curves is a function of the mean travel distance (Gelhar et al., 1992). In spite of this concern, the modelling of dispersion as a diffusion-like process is common in repository PA, and justified, in part, by the fact that calculated radionuclide releases from the geosphere to the near-surface environment tend to be quite insensitive to the value assigned to the dispersion coefficient (e.g. Nagra, 1994). 2.3.3. Modelling geosphere structures: an example from fractured rocks In any fractured rock, there exists a continuum of discrete, water conducting features at all scales, from dekametres to millimetres. However, in PA modelling, there are two types of heterogeneity that, in general, need to be considered for transport-modelling purposes: • The larger-scale heterogeneity, arising from the fact that a typical transport path through a rock body often consists of a network of features with different properties. • The smaller-scale heterogeneity corresponding to the internal structural of the features, which may arise, for example, from a multi-phase history of deformation and mineralisation. In both cases, the structural complexity of most geological media, and the limited data to characterise this complexity, means that geosphere transport models almost invariably include a high degree of simplification. In modelling groundwater flow, smaller-scale heterogeneity can often be neglected. This heterogeneity determines the transmissivities of water-conducting features, but transmissivi9 Simply put, C = K · C , where C = the solute concentration in the aqueous phase (mol m−3 ), C = the r w w r d concentration ‘sorbed’ on the solid phase (mol kg−1 ) and Kd = the distribution coefficient (m3 kg−1 ). In a simple system, such a Kd may be thermodynamically based, but the values used in transport models are usually an empirical representation of kinetically fast, concentration independent, reversible uptake. Unfortunately, many transport modellers do not understand this distinction and consequently employ the Kd concept in totally inappropriate circumstances (see discussion in McKinley & Alexander, 1992, 1993a, b, 1996, 2003; Alexander & McKinley, 1994; Ku et al., 1998 and Payne et al., 2001 for examples and critical discussion).

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Fig. 5. Schematic representation of the relationship between a conceptual model of geological structures (in the crystalline basement of Northern Switzerland) and numerical models of groundwater flow at regional, local and block scales (from Fig. 3.5.5 in Nagra, 1994).

ties are usually measured directly (encompassing relatively large sections of a borehole and, therefore, usually including numerous water-conducting features), rather than derived from the direct characterisation of smaller-scale structure. It is also common in groundwater flow modelling to simplify larger-scale structures and to treat geological media either as continua or as discrete fracture networks. In the continuum approach, the actual medium, with its structural heterogeneity, is conceptually replaced by a more homogeneous “equivalent porous medium (EPM)”, characterised by macroscopic parameters, such as the hydraulic conductivity. In the discrete fracture network approach, flow in the volumes of rock between fractures is assumed to be negligibly small and the fractures themselves are treated as, often homogeneous, planar features characterised by macroscopic parameters, such as the transmissivity. In both cases, macroscopic empirical laws, such as Darcy’s Law, are used to describe the average groundwater flow through the medium as a whole or the flow through the discrete fractures in response to pressure gradients. Figure 5 shows an example of how a conceptual model of larger-scale heterogeneity in a geological medium can be simplified for the purposes of modelling groundwater flow. In this example from the crystalline basement of Northern Switzerland, a potential repository host rock, a set of three nested models is used. A regional-scale EPM model provides boundary

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conditions for a more detailed local-scale hybrid model, in which some of the larger linear structures (“major water-conducting faults (MWCF)”) are modelled explicitly. The local-scale model in turn provides boundary conditions for a block-scale discrete fracture network model, which is still more detailed. It is only on the relatively small block scale that the network of individual water-conducting features is modelled explicitly. In the context of transport modelling, it is important to emphasise that the groundwater flow modelling techniques outlined above yield average information about flow, either within the medium as a whole or within discrete features. A knowledge of the average flow within a water-conducting feature is necessary, but not sufficient, to model radionuclide transport. Channelling of the flow (which determines the actual rates of advection and gives rise to dispersion) and the effectiveness of matrix diffusion and sorption as retardation processes are strongly influenced by the internal structure of the features. Figure 6 gives an example of the internal structure of one type of water-conducting feature observed in the crystalline basement of Northern Switzerland, and the simplification of this structure that is made for transport modelling purposes. In this example, advection/dispersion occurs in channels, embedded within fractures that are themselves contained within the water-conducting features. Many paths for advective/dispersive radionuclide transport through a geological medium may exist, consisting, for example, of series of connected channels within a larger network of fractures, and these paths may vary widely in their transport-relevant properties, both along their lengths, and between paths (see, for example, Moeri et al., 2002b, 2003, for a clear example). Nevertheless, it is common in transport modelling to consider just a few parts (sometimes only one) as representative of the whole range. These representative paths are assigned uniform properties along their lengths, such as transmissivity, crosssection geometry, and properties relevant to retardation. Although an extreme simplification of reality, properties can be assigned such that the final results err on the side of safety (as defined above) and, furthermore, some numerical studies (e.g. Schneider et al., 1998, JNC, 2000) have indicated that, provided this assignment is carefully made, a small number of representative paths can be used to simulate closely radionuclide breakthrough for a more complex network. In Fig. 6, matrix diffusion and sorption would be expected to occur with fracture infill material between channels, and within altered and unaltered wallrock. Such supposition has recently been shown to be the case (Fig. 7) by an unequivocal data set from an in situ study by Moeri et al. (2002b, 2003). Such “matrix heterogeneity” is a feature of many fractured rocks. Until recently, most radionuclide transport models could not account for this heterogeneous nature of the matrix, and considered only one-dimensional matrix diffusion, in a direction normal to the flow through a fracture or channel, into a homogeneous matrix region. More recently, transport models (e.g. Barten et al., 1998; Vieno & Nordman, 1999) have become available that can account for matrix heterogeneity, thus making fuller use of the type of information on the internal structure of water-conducting features illustrated in Figs 6 and 7. 2.4. Governing equations and solution methods The governing equations and boundary conditions for radionuclide transport models vary, to some extent, depending on the processes of concern (whether or not is it possible to exclude coupled processes, the transport of radionuclides on groundwater colloids, etc.), and the de-

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Fig. 6. Tectonically disturbed zones in the crystalline basement of Northern Switzerland, as observed in drillcores and as simplified for the purposes of transport modelling (from Fig. 3.5.4 in Nagra, 1994).

gree to which detailed models of specific processes are incorporated directly into the transport model. Nevertheless, the starting point in the development of differential equations to describe the transport of radionuclides in geological media is to consider the flux of radionuclides into and out of a fixed fluid volume (or ‘element’) within the flow domain. A conservation of mass statement for such an elemental volume is: ⎡

⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ net rate of flux of flux of loss or gain ⎢ change of mass ⎥ ⎢ nuclide into ⎥ ⎢ nuclide out ⎥ ⎢ due to ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥±⎢ ⎥ ⎣ of nuclide within ⎦ = ⎣ the ⎦ ⎣ of the ⎦ ⎣ retardation ⎦ the element element element processes ⎡ ⎤ loss and gain ⎢ due to ⎥ ⎥. ±⎢ ⎣ radioactive ⎦ decay and ingrowth

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Fig. 7. Clear examples of radionuclide diffusion into and sorption onto fracture infill and wallrock in granodiorite from Nagra’s Grimsel Test Site, Switzerland (from Moeri et al., 2003). The sample has been impregnated in situ with fluorescent resin and then over-cored and sawn into slices for examination. (a) On the left: fault filled with fault gouge (yellowish colour) and fragments of wall rock (black) floating in the fault gouge. UV light, partial scale below in mm divisions. (b) On the right: alpha-autoradiograph of part of the sample on the left showing uptake of radionuclides in the fault gouge (white material) and on the fracture surface (or wallrock) and on the surface of the rock fragments (grey areas).

The flux into and out of the element typically arises from advection, mechanical dispersion and diffusion. Retardation processes include sorption and, in the case of fractured media, matrix diffusion. Where matrix diffusion is considered to provide retardation, two differential equations usually arise, one describing radionuclide transport in the matrix and the other describing radionuclide transport along a fracture or channel itself. A typical form for such equations, in the case of uniformly spaced, parallel fractures in a homogeneous, isotropic matrix, is: (i)

∂Cf ∂t

=

1 Rf(i)



(i)

−u

(i)

∂ 2 Cf ∂Cf (i−1) (i−1) (i−1) (i) (i) + aL u + Rf λ Cf − Rf λ(i) Cf ∂z ∂z2

 (i)  ∂Cp  1 + εp Dp b ∂y y=b

(1)

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for transport in the z-direction along fractures, where, on the right-hand side, the terms within square brackets represent, respectively, advection, hydrodynamic dispersion, radioactive ingrowth and decay, and diffusion into the porous matrix, and   2 (i)   ∂ 2 Cp(i) ∂ Cp ∂Cp(i) 1 (i−1) (i−1) (i−1) (i) (i) (i) = (i) Dp + λ C − R λ C + R p p p p ∂t ∂z2 ∂y 2 Rp

(2)

for transport in the matrix, where the terms on the right-hand side represent matrix diffusion, and ingrowth and decay. The parameters are defined as follows: Cf(i) and Cp(i) are the concentrations in solution, within the fracture and the matrix porewater, respectively, of the ith radionuclide in the decay chain 1 → 2 → 3 → · · · → i λ(i) is the decay constant for radionuclide i, (i) (i) Rf and Rp are the retardation factors for advection and dispersion within the fracture and for diffusion within the matrix, respectively, u is the water velocity within the fracture, is the longitudinal dispersion length within the fracture, aL εp is the matrix porosity, is the pore diffusion constant within the matrix, Dp b is the fracture half aperture, yp is the half distance between parallel fractures. In general, such equations are solved by numerical techniques, such as the finite element and finite difference methods. Many computer codes have been developed for this purpose. There are, however, a number of analytical solutions for special cases, such as steady-state conditions, which are useful for code verification and for performing simple scoping calculations (see, for example, Nagra, 1994; Vieno & Nordman, 1999).

3. Testing radionuclide transport models: an example from the Swiss radioactive waste management programme 3.1. Introduction Altough the aspect of testing is fundamental to all modelling activities (e.g. Whicker et al., 1999), special emphasis is put on clear and transparent model testing in repository PA. This is because repositories have to isolate radioactive waste (and some chemotoxic wastes) from the environment for hundreds to hundreds of thousands of years and, for many scientists and engineers, and especially the general public, such time spans are beyond comprehension and, consequently, they have grave doubts as to the safety of any such waste repositories. That repository performance over these long time scales is primarily assessed by the use of complex mathematical PA models only adds to the mistrust of many. How then can people be convinced that it is possible to assess the performance (and thus ensure the safety) of a repository over the long timescales of interest? One way is to address the robustness of the

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PA models, by clearly indicating the form and extent of model testing carried out within the repository PA. Not only can this show that the individual component parts (such as the models of radionuclide migration in the geosphere) of the complex structure which constitutes most PA models have been checked, but also that the ‘mathematical black boxes’ constitute an acceptable representation of the repository system. Part of the problem undoubtedly lies in the unusual nature of radioactive waste disposal: in most major engineering projects, such as bridge construction or aerospace engineering, the designs are tested against a range of laboratory experiments backed up by expert judgement based on experience with the same or similar systems. Here repository design deviates from standard engineering practice in that no high-level waste (and only a few low- and intermediate-level waste) repositories yet exist and, even when they do, testing their compliance to design limits will be, to put it mildly, somewhat difficult due to the time scales involved. In addition, the irrational fear of most things radioactive means that most people require some greater form of ’proof’ that a repository is safe than they are willing to accept for other engineered systems such as bridges or aircraft10. This being the case, significant additional effort must be expended within the radwaste industry to make it completely clear that the PA models can adequately predict the long-term behaviour (and consequent safety) of a repository. Here, a short overview is presented on the approach to testing geosphere radionuclide migration models in the radioactive waste industry using the Swiss national programme as an example. 3.2. Just what does ‘testing’ imply? Model testing in the academic world tends to be a relatively relaxed affair. After all, if your model is shown to be ‘wrong’ or inconsistent with new data, it is hardly the end of the world. Indeed, many would argue that this is simply what good science is all about: puting forward a thesis (in this case your model of radionuclide migration in the geosphere) and having it proven or challenged and replaced by a better, more appropriate, thesis is how science progresses. Unfortunately, in the field of radioactive waste disposal, the ability to make mistakes in the name of scientific progress is simply not acceptable. Not only are there the public fears and misconceptions (noted above) to be dealt with but, because of the seriousness of the job in hand, there are, inevitably, legal requirements to be met. Apart from general legal requirements pertaining to the use of predictive models in environmental protection (e.g. Hofmann, 1999), there are also specific national radioactivity dose requirements to be met (e.g. Frank, 1999). Consequently, a more rigid set of testing guidelines has developed and these are outlined below. 3.2.1. Testing: just what do we do? The direct testing of the results of a geosphere radionuclide transport model, as used in PA, is impossible, due to the scales of space and time involved. Rather, confidence is developed 10 Consider the case of a fully loaded Boeing 747-400: despite having some 6 million parts, flying at 10 000 m at 900 km h−1 , controlled by a crew, several computers and air traffic controllers, it has full acceptance from the public

– despite the massive consequences of catastrophic failure (to the passengers). For a radioactive waste repository: despite having very few main parts, being immobile, needing no control, it has no acceptance from the public – despite the fact that the consequences of even catastrophic failure will be small (see West et al., 2001, for further discussion of these points).

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through the consistency of the model assumptions, and associated databases, with a large number of diverse observations and experiments. Furthermore, the model should have the capability to make predictions, or at least bounding estimates, that can be tested, even if the scales differ from those relevant to PA. In the case of a geosphere radionuclide transport model, testing needs to address: • the adequacy of the catalogue of relevant processes incorporated in the model; • the adequacy of data-collection techniques and the transferability of laboratory data to in situ conditions; • the understanding and modelling of individual processes; • the modelling of combined processes in the transport model. In situ field experiments, with supporting laboratory programmes, can be used to address all of these issues. It should, however, be noted that there may be many specific differences in the structures that are relevant and the parameter values used depending on whether a model is applied in PA or to the modelling of a field experiment. For example, in a field experiment, the focus may be on an individual feature rather than on a network of water-conducting features and the scales may differ – most field experiments are carried out over distances of metres to tens of metres whereas distances of 100s to 1000s of metres are more applicable to a repository PA. The timescales of an in situ experiment also differ from that of a repository PA (hours to months compared to hundreds to hundreds of thousands of years). In addition, because of the large number of simplifications which have to be made for the PA calculations, the PA codes themselves cannot be directly tested: rather a sub-model (also known as a research model) of the PA model is used. The research model differs from the PA model only insofar that features and processes relevant to field and laboratory tests are also considered (such as small-scale porosity differences or kinetic effects during sorption) rather than only those relevant to the scales of a repository. Otherwise, the modelling approach outlined above applies equally in both applications. Thus, having achieved confidence in a research model through predictive model testing, a PA model developer is in a strong position to decide where simplifications can be justified in terms of either conservatism or insignificant effects. As an example of research model testing, a series of experiments carried out in one of Nagra’s underground laboratories (the GTS) in the central Swiss Alps (see Fig. 8) will be examined. Nagra began work on transport modelling at the GTS in 1985 with the hydrogeological characterisation of a water conducting shear zone in granodiorite (marked RRP in Fig. 8) and has continued, along with JNC (Japan Nuclear Cycle Development Institute), with a large series of in situ radionuclide tracer migration experiments. The radiotracers were injected into an artificially controlled flow-field in a complex fracture zone (Fig. 9) and the characteristic breakthrough curves of the tracers (see Fig. 10) were modelled in a multi-step process (described below). The behaviour of the radiotracers used increased in complexity with time from simple, nonsorbing tracers (fluorescein dye, 82 Br, 123I, 4,3 He and 3 H) through various weakly sorbing tracers (22 Na, 24 Na, 85 Sr and 86 Rb) to a long-term experiment with strongly sorbing 137Cs (see McKinley et al., 1988; Alexander et al., 1992 and Smith et al., 2001b, for details). Most recently, chemically complex tracers (99 Tc, 113 Sn, 75 Se, 234 U, 235 U, 237 Np, 60 Co, 152 Eu and stable Mo) have been utilised followed by the physical excavation of a part of the shear zone

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(a)

(b) Fig. 8. (a) Map of Switzerland. Nagra’s crystalline underground laboratory, the Grimsel Test Site (GTS), lies in the central Swiss Alps. (b) Overview of the GTS tunnel system. The experimental site is marked RRP/HPF/CRR denoting a long series of experiments with radiotracers (for details see Ota et al., 2001).

to recover these strongly retarded radionuclides (Alexander et al., 1996, 2001; Moeri et al., 2002b, 2003 and Ota et al., 2001, 2002). These radiotracer tests continue at present in the GTS with two ongoing experiments, one examining the effects of colloids produced during

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Fig. 9. Examples of dipole flow-fields in the experimental shear zone in the Grimsel Test Site (the shear zone lies in the plane of the paper). Radiotracers are pumped into the injection borehole (e.g. hole 9) and water (and radiotracer) is pumped out of the outlet borehole (e.g. hole 6) at 10–20 times the injection rate, so ensuring a ‘closed’ flow system (i.e. the radiotracers remain within the artificially constrained flow system). This has the benefit of simplifying matters for the hydrogeological modellers and ensures full recovery of the radiotracers. Note the deviation from the theoretical straight line flow between boreholes (due to the influence of both the detailed structure of the shear zone and the natural flow gradient).

the degradation of the man-made barriers on the migration of a suite of radionuclides and a second looking at the interaction of radiotracers with secondary cement phases produced in the geosphere by reaction with hyperalkaline cement leachates (see Biggin et al., 2003, for details). 3.2.2. The methodology of model testing The following sequence of tests are applied in the GTS projects to research models: 3.2.2.1. Determination of parameter values. The determination of parameter values describing rates and spatial extents involves an assessment of all the information available from the characterisation of the rock, from field and independent laboratory experiments, such as batch sorption and laboratory diffusion experiments and from natural analogues. These data may need to be interpreted by means of supporting, interpretative models (e.g. a linear or non-linear model of sorption) in order to provide parameter values for a transport model. If all parameters of a transport model can be determined in this way, then the model can be used directly in predictive testing (see below). There may be some parameters, however, that cannot be taken directly from field and laboratory measurements, although it is often possible to place bounds on these parameters by drawing on wider scientific understanding (e.g. from relevant literature or experience in similar rock types or hydrogeological environment). It is

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Fig. 10. Examples of measured breakthrough curves from the MI experiment (cf. Fig. 11). The 1.7 m flow distance relates to the boreholes 9-6 flow field and the 5 m to the 4-6 flow field.

then necessary to resort to inverse modelling to “calibrate” the model, providing parameter values for subsequent application in predictive modelling. Inverse modelling involves the following steps: • all “free” parameters that are not fixed by means of independent field and laboratory measurements are adjusted simultaneously (constrained within certain bounds, if these are known) until the best fit to a tracer-transport breakthrough curve (see Fig. 10) is obtained, • the best fit model curves are compared with experimental curves and assessed against success criteria, • if the comparison is unsuccessful, then either the derivation of fixed parameter values or the model formulation itself must be re-assessed, • if the comparison is successful, then the fitted parameters are tested for consistency with independent data, i.e. from the characterisation of the rock (e.g. Fig. 7) and from other field and laboratory experiments,

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Fig. 11. Illustration of the effects of different retardation mechanisms on radionuclide tracers in the geosphere (cf. Fig. 10). As noted above, the precise form of the breakthrough curves allows some deductions to be made as to the relative influence of the various processes (matrix diffusion, channelled flow etc.) occuring in the shear zone. See also Smith et al. (2001a) for comments on the limitations of this methodology.

• if there is inconsistency with independent data, then either the derivation of fixed parameter values or the model formulation itself must be re-assessed, • if the fitted parameters are consistent with independent data, then they can be taken into account when assessing available information, • to reduce the number of free parameters in further inverse-modelling exercises (e.g. with more complex tracers, as in the GTS experiments), • to determine whether free parameters can be eliminated and predictive modelling (e.g. of new experimental set-ups, with different flow fields, as in Fig. 10) is possible. 3.2.2.2. Predictive (or blind) modelling. This type of model testing involves the prediction of experimental breakthrough curves in advance of the experiments for a range of tracer experiments carried out under different flow conditions (pumping rates, separation of injection and withdrawal points, etc.) and using different tracers. For each predicted (model) break-through curve: • comparison is made of the model and experimental curves, with assessment against success criteria (based on an evaluation of experimental errors), • if the comparison is unsuccessful, then either the derivation of fixed parameter values or the model formulation itself is re-assessed, following successful comparison, selection of a new experimental set-up (with different flow conditions and/or tracers with different transport-relevant properties) for further predictions is carried out and the procedure is repeated until it is judged that “sufficient confidence” in the model has been attained.

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“Sufficient confidence” is not precisely defined. The aim, however, is to model a large number of experimental set-ups, since a “good model” should describe a large class of observations. The ability of the transport model to predict the behaviour of a variety of tracers in a range of different flow fields gives confidence that the model can be applied to the transport of radionuclides in a performance assessment, provided adequate characterisation of the repository system and suitable independent experimental data are available. Such predictive modelling, in particular, provides a sensitive test of the model concept and the numerical procedures precisely because the modeller is given only limited information. The method of testing is crucial: few people, even those involved in the disposal of radioactive waste, fully appreciate the difference between blind testing of model predictions and testing if a model can simulate particular observations – as can be clearly seen in the literature. This is a crucial point, as noted by Pate et al. (1994) “This aspect of blind (i.e. predictive) testing is particularly important as, in many cases, the manner in which the simulation is carried out can be very objective and, if the ‘answer’ is known, can be biased either consciously or subconsciously.” In a repository PA, simulation of data brings little or no confidence that the models involved can later predict repository evolution: confidence can be much better built by carrying out a series of predictive modelling exercises followed by experimental runs and a final assessment of the accuracy of the predictions (and, where necessary, improvement of the models). In addition, to ensure that the tests are scrupulously fair (and to further increase confidence in the tested models), it is strongly recommended that the predictions are published before the experimental runs are carried out in the field (or laboratory). In the GTS work, unfortunately, the predictions were only published in internal memos but open publication of predictions (in the scientific literature) has been carried out in two tests of thermodynamic databases for PAs11 . 3.2.3. The value of field experiments in model testing Given the scales of time and space that must be considered in a PA, direct testing of the realism (or the degree to which the models err on the side of safety) of a model in the system of interest is impossible. Field experiments, such as those performed at the GTS, are consequently invaluable in that: • the fundamental transport processes that operate in the system are expected to be the same or similar to those relevant to any fractured repository host rock; • the structures present, though differing in detail, are also similar enough to those of potential fractured repository host rocks; • the scales of time and space over which the experiments operate, though often considerably shorter than those of a PA, are often larger than those achievable in the laboratory; • the degree of characterisation of the system that is possible is greater than that achievable at a repository site (due to the smaller spatial scales involved in many field experiments and 11 In a study of a natural cementitious system, the modelling teams first predicted the behaviour of the elements of

interest in the system and only then was the measured data revealed for comparison. See McKinley et al. (1987, 1988) and Bath et al. (1987a, b) for the details. This was the first comprehensive blind predictive PA modelling exercise and the procedures developed for this study have formed the basis for most later studies (see Miller et al., 2000 and Bruno et al., 2001, for additional examples). However, in only one other case has the predictive work been published openly before the full analysis was carried out (see Alexander, 1992 and Alexander et al., 1992).

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the need, at a repository site, to avoid perturbing the favourable properties of the site by sticking too many holes in it); • field experiments, being performed in situ, are less subject to some experimental artefacts than laboratory experiments. However, it must be emphasised that the successful modelling of radiotracer tests (whether those in the GTS or in other in situ tests such as at the Aspö underground rock laboratory in Sweden (e.g. Winberg et al., 2000) or the Whiteshell underground rock laboratory in Canada (e.g. Vilks & Bachinski, 1996) gives support to the model representation of structures and processes that exist in, or operate on, spatial and temporal scales that are similar to, or smaller than, the tests themselves. A major difficulty with the tests lies in the extrapolation of this conclusion to the larger scales that are relevant to PA: no information is provided on processes that, though irrelevant on the spatial and temporal scales of field tracer tests, may be important over scales relevant to PA12 . The tests can, however, add to the body of observations and experiments with which the model is consistent and thus build confidence in its application in PA. This can be further strengthened by carrying out a series of complementary and interrelated experiments as has been the case at the GTS (see Biggin et al., 2003, for details) and backing these up with additional work in the laboratory and other natural systems (see also the comments on this approach in Alexander et al., 1998). 3.3. Model simplification and testing in performance assessment The issue of model simplification (or ‘abstraction’ in PA modelling terms) has been widely discussed in the context of confidence in the long-term safety of repositories for the deep geological disposal of radioactive waste (see, for example, NEA, 1999, 2000, 2000). In spite of recent refinements, the complexity of many geosphere transport models is still less than the state-of-the-art in mathematical modelling and the understanding of fundamental transport processes would permit. It would, for example, be possible in principle to incorporate a great deal of geometrical and process detail into these models. The main reasons why this is not more widely done in practice are: • Lack of data: characterisation of the geological environment is a stepwise process, and understanding evolves as characterisation moves from surface-based studies to more detailed studies underground. PAs are carried out periodically throughout this process, and 12 It is, of course, possible that different model representations of the structures and processes relevant to the GTS radiotracer tests would be similarly successful fitting the results of (at least some) tests and, for this reason, alternative models were developed by three independent research groups. The aim was, by attempting to falsify some of these alternatives, and, in particular, by making independent predictions of the results of tests in advance of the tests being performed, to narrow down the range of conceptual model uncertainty and to identify the processes that are most important in describing solute transport. This aim was only partially fulfilled, since: (1) The differences between the models were rather minor and thus the range of conceptual-model uncertainty was small. All models were built upon the dual-porosity concept, and no attempt was reported, for example, to understand the test results without invoking matrix diffusion. (2) New flow path structural data produced several years after the model testing was completed included several features (multiple flow paths, diffusion into fracture fill material) which had been explicitely excluded from the modelling approach. This shows that the models tested were insensitive to the fine detail in the flow path (which builds confidence in the overall PA approach) but also that the models, to put it bluntly, got the right answer for the wrong reasons (which is less good for confidence building). See Smith et al. (2001a, b) and Moeri et al. (2002b, 2003).

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inevitably have to cope with uncertainties associated with incomplete characterisation and so have to utilise models which, in terms of their degree of realism, are often a step behind the cutting edge of geological characterisation. It should also be noted that there are practical limits to the degree to which characterisation can be carried out without disrupting the favourable characteristics of a rock by simply drilling too many holes in the repository host rock. • A desire for clarity in performance assessment: a single, complex model that attempts to include all features and processes runs the risk of being used as a “black box”, which produces results in a manner that few people understand and which is difficult or impossible to check. Many parameters of an “all-inclusive” model might be poorly supported by sitecharacterisation and other evidence and the fact that a result could nevertheless be obtained may give some people a false confidence in its meaning. • Lack of communication: whilst initiatives such as those of the NEA (1997, 1998, 1999, 2000), which have attempted to involve as wide a range of disciplines and promote as much discussion as possible, are to be applauded, it remains a sad fact of life that interdisciplinary communication remains problematic. Even within a single national waste programme, it is always possible that simply too much information exists and so dichotomies arise. For example, in the Swiss programme in the 1990s, while the field exploration groups and the PA modellers were working well to keep each other abreast of new developments in geological and modelling efforts (see Mazurek et al., 1992), there was simultaneaously a break down in communication between the above groups and those working in the GTS, leading to lost chances to employ the most up-to-date geological data in the ongoing PA (see comments in Alexander et al., 1997). However, for PA modellers, these points may not be such a great problem due to the inherent simplifications necessary in carrying out PA calculations. The fundamental problem is that, to be acceptable in a repository PA, all models (and databases) employed must be ‘validated’ (in the IAEA sense of demonstrating that they are appropriate for the use envisaged). The whole process of validation remains highly controversial and here is not the place to discuss it in detail (instead, see Pescatore, 1995, and references therein), rather accept that there are often regulatory requirements to address (addressed, for example, in Frank, 1999, for the Swiss radioactive waste programme and defined in Luiten, 1999, for the European Union’s integrated environmental assessment). What is important to note is that validation is inherently subjective due to the difficulties in quantifying all the uncertainties in a slowly evolving repository and the host rock over geological time scales. This means that it is important that the validation procedures are carried out in a transparent and logical manner (hence the development of the blind testing approach detailed in Section 3.2.1). Thus for any PA model, the testing scheme can be focussed on answering the following questions: • Is the application clearly defined and scientifically reasonable? • Are the theoretical arguments/models used defensible? • What extent of validation is required to show that the models err on the side of (repository) safety by over-estimating radionuclide transport in the geosphere? • Are the selected data consistent with laboratory experiments, in situ studies and data from natural systems?

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It is clear that some aspects are too detailed to include in the models (see comments above on the representation of sorption or small-scale structural information) and so, where simplification is necessary, it must be demonstrated that the models will over-estimate radionuclide migration. Thus this model simplification (although, arguably, not scientifically rigorous in an academic sense), when it can be justified on the grounds of erring on the side of safety, is widely accepted by repository developers and by regulatory authorities. It is also generally accepted that numerical models that can be demonstrated to be either capable of at least setting upper limits on processes (such as estimating the maximum likely distance a given radionuclide will migrate through the geosphere) or to be erring on the side of safety (by using the worse case scenarios in any calculation) can yield results that are suitable for decision making (i.e. is the likelihood of substantial radionuclide migration in the geosphere high?). It may also be possible to demonstrate, using independent calculations or qualitative arguments, that certain features or processes have a negligible impact on radionuclide transport. Such arguments might, for example, be used to evaluate the importance or otherwise of coupled phenomena in argillaceous media (see, for example, Soler, 1999) or, as noted above, the relevance of matrix diffusion in a particular fractured rock. It has been pointed out that, although a degree of model simplification is inevitable and may even be desirable, simplification that leads to a clear loss of realism should be avoided (NEA, 1999, 2000). Confidence requires that PA models incorporate the essential features of the system, based on site understanding and conceptual models that are consistent with wide-ranging observations and evidence, even though all such information cannot always be incorporated directly into the models and databases (see comments above). Models of groundwater flow, for example, can be tested for consistency with the distributions of various natural tracers (salinity, heat, etc.), even though these distributions are not direct input to any given PA model for a repository site (NEA, 2000). 4. Overview of the state-of-the-art and directions for the future 4.1. Interpretation of radionuclide distributions Despite extensive efforts over the last three decades, the “Holy Grail” of a model which allows observed distributions of radionuclides in geological systems to be interpreted quantitatively in terms of the processes influencing their transport and immobilisation seems far from reach in most cases. Acknowledgement of the difficulties involved and rejection of overly-simplistic models must, however, be recognised as a significant advance. Only rarely now are natural radionuclide concentrations in groundwaters interpreted directly in terms of a “groundwater age”. Instead, information from a wide range of radioisotopes is combined with that from hydrochemistry, mineralogy, stable isotopes, structural geology etc. to derive one or more scenarios of groundwater evolution (see, for example, Edmunds & Smedley, 2000 and Castro et al., 2000). The radioisotope data themselves can be extensively refined to quantify potential perturbations from sampling, diversity of sources and rock/water interaction history (see Pearson et al., 1991, for examples). This requires not only more sophisticated models, but much more extensive analysis of very carefully obtained sets of samples.

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This trend can also be seen in the interpretation of tracer tests – carried out in laboratories, drillholes or underground test facilities such as the GTS. Simple advective flow/Kd models have been replaced by interpretations which emphasise detailed characterisation of both the flow system on a micro scale and the diverse range of processes which can influence retardation (various types of sorption, matrix diffusion, colloids, organics, reaction fronts, etc.). Very simplistic “tracer tests” have evolved into integrated, multi-disciplinary migration studies which include laboratory, in-situ, modelling and analogue components and may run on a timescale of decades at costs of tens of millions US$ (see information in Alexander et al., 1997). One specific area which seems to have come to a dead end involves attempts to derive key radionuclide transport parameters by interpretation of ratios of several natural series radionuclide isotopes (so-called “in-situ Kd s”). Despite grandiose claims for this methodology and extensive, often acrimonius, debate in the literature, there is no evidence that any such approach provides sensible results. Indeed, “reality checks” on such models often clearly demonstrate that they are incorrect – for example, when they predict orders of magnitude differences in sorption for different isotopes of the same element (McKinley and Alexander, 1996, 2003 and Ku et al., 1998). 4.2. Prediction of radionuclide migration As discussed in Section 2, for specific applications, the requirements in terms of accuracy/precision may be so limited that prediction of future radionuclide migration which err on the side of safety may be easier to meet than those for the interpretative studies considered in Section 4.1 above. In particular, many concepts for deep disposal of radioactive waste are so over-designed that uncertainties of several orders of magnitude in performance of the geosphere barrier (i.e. how well the geosphere retards radionuclides) are quite acceptable. For example, as shown in Fig. 12, different representations of retardation of a key radionuclide from high-level radioactive waste (135Cs) change its calculated release by more than 5 orders of magnitude. Nevertheless, even the worst model representation meets regulatory requirements and hence there is no driving need to develop a more realistic model which would undoubtedly give better performance. Although performance assessment of such deep geological repositories has been a driving force for development of the methodology for radionuclide transport prediction, such overconservatism is not acceptable for many other applications. The need for greater realism requires that uncertainties and complexities need to be addressed directly, rather than compensated for by simplification which errs on the side of safety. As yet, progress in this area has been very limited – constrained both by poor availability of the databases needed to apply more sophisticated models and the lack of systems which allow resulting models/databases to be adequately validated (see, for example, comments in Pate et al., 1994 and Bruno et al., 2001). At present, it must be acknowledged that the models used to support near-surface waste disposal or contaminated site remediation are generally very simplistic in their treatment of solute release while transport processes are often based on purely empirical models.

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Fig. 12. An example of the variation in the calculated retardation of 135 Cs in the geosphere. Here, six different representations were used to illustrate the sensitivity of the modelling concept to fracture geometry. Results calculated with non-linear sorption incorporated explicitly (◦) are compared with those calculated with an effective linear isotherm (+). For an explanation of the significance of the shading used in this plot, see Appendix 1 in Nagra (1994).

4.3. Conclusions and future developments Understanding of the processes of mobilisation and migration of radionuclides in deep geological environments has advanced considerably over the last couple of decades – even if most progress has been in the recognition that transport processes are much more complex than previously thought. Advances in computer hardware/software have effectively removed

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the inability to simulate defined systems as a constraint – the main stumbling block is characterisation of the system of interest in sufficient detail (especially if it is the site of a deep geological repository). For specific applications, fundamental concepts are already well established and practical methodologies are available – e.g. groundwater dating, prediction of the migration of simple solutes in saturated, porous media. In other cases, it may be acceptable to cover for uncertainties by always erring on the side of safety if the requirement is only to put bounds on maximum possible extent of mobilisation. There is a trend towards the incorporation of increasingly wide-ranging information when developing conceptual models for geosphere transport, including sometimes qualitative “soft information” such as the use of temperature and salinity distributions and other environmental tracers in the development, testing and refinement of conceptual and mathematical models for groundwater flow. Although often ignored in the past as being non-quantitative, such information can be used to test concepts and to reduce the range of alternative conceptual models and the range of parameter uncertainty that must be taken into account. A range of climatic and geological phenomena may lead to time-dependence in the transport-relevant properties of the geosphere. For example, after a glacial period, access to the rock matrix can increase dramatically as the rock expands (de-stresses) following the removal of the weight of the ice mass. In areas of significant uplift, the groundwater chemistry may change enormously over several millenia (e.g. in a coastal area, groundwaters may change from saline to fresh as the rock is uplifted above the zone of marine influence). In addition, in the case of PAs for deep geological repositories, the material from which the repository is constructed may induce changes (e.g. cementitious repositories could eventually leak hyperalkaline porewaters into the host rock so inducing the formation of secondary cement phases in the water-conducting zones). Codes are currently available or under development which allow for such time-dependence in transport-relevant properties and the main challenge for the immediate future will be testing the realism and relevance of these codes. In the case of geosphere transport modelling in fractured media, there is increasing appreciation of the importance of heterogeneity in the properties of the matrix, which may be accessible by diffusion from fractures, over a range of scales. Codes have been, or are being, developed to allow information on such heterogeneity to be incorporated in, at least, a basic manner. In representing sorption within geosphere transport models, it is, in principal, possible to replace the widely used empirical Kd approach with a more mechanistic approach that allows the effects of heterogeneity and temporal variability in, for example, groundwater composition to be taken into account directly. In the case of transport models for performance assessment, however, the trend seems to be rather to move towards the use of semi-empirical functions to represent the dependence of Kd s on (space- and time-dependent) geochemical conditions. Care must, however, be taken not to use such empirical relationships to describe radionuclide partitioning beyond the range of conditions to which they have been shown to be applicable. There is considerable interest in the possibility of incorporating “immobilisation processes” in geosphere transport models. Such processes include “irreversible sorption” (where timescale of desorption from a mineral surface is very slow), surface precipitation or coprecipitation, surface substitution and adsorption followed by solid-state diffusion. Although many of these processes are well understood for simple, idealised systems, the modelling

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of the processes in real geological system still presents considerable difficulties. Certainly, this would require some advances in conceptual understanding (e.g. for partitioning of natural series isotopes, treatment of colloids, description of reaction fronts, etc.), but the main rate-determining step will probably be provision of better databases from sophisticated experimental projects. Here, future work in underground rock laboratories could be used to provide the “missing data” of relevance to the new generation of radionuclide transport models. Now is the time to move on from the simple, short-term radionuclide migration experiments which have characterised the last two decades and to concentrate rather on long-term experiments in repository relevant sytems (e.g. complex fractures with low water fluxes) which are integrated with extensive laboratory support programmes and natural analogues. It is gratifying to see that this nettle is at long last being grasped and the plans to begin a further phase of work in the GTS underground laboratory which could last for up to 50 years and provide a venue for long-term radionuclide transport experiments seems to be a step in the right direction (further information on these plans is available on www.grimsel.com). Finally, practical applications also require rigorous testing of the model/databases used and this becomes increasingly difficult as the systems represented become more complex. However, in the context of increasing confidence in the applicability of transport models, regardless of their use, this remains the highest priority and must be implemented more widely in the immediate future.

References Alexander, W. R. (Ed.) (1992). A natural analogue study of cement buffered hyperalkaline groundwaters and their interaction with a sedimentary host rock, I. Source term description and geochemical database validation. Nagra Technical Report, NTB 91-10. Wettingen, Switzerland: Nagra. Alexander, W. R. & McKinley, I. G. (1994). Constraints on the use of “in situ distribution coefficients (Kd)” values in contaminant transport modelling. Eclogae geol. Helv., 87 (2), 321–324. Alexander, W. R., Dayal, R., Eagleson, K., Eikenberg, J., Hamilton, E., Linklater, C. M., McKinley, I. G. & Tweed, C. J. (1992). A natural analogue of high pH cement pore waters from the Maqarin area of northern Jordan II: results of predictive geochemical calculations. J. Geochem. Explor., 46, 133–146. Alexander, W. R., Frieg, B., Ota, K. and Bossart, P. (1996). The RRP project: investigating radionuclide retardation in the host rock. Nagra Bulletin No. 27 (June, 1996) (pp. 43–55). Wettingen, Switzerland: Nagra. Alexander, W. R., McKinley, I. G., Frick, U. & Ota, K. (1997). The Grimsel field tracer migration experiment – what have we achieved after a decade of intensive work? OECD Proceedings: Workshop on Field Tracer Experiments (Role in the Prediction of Radionuclide Migration), Cologne, Germany, 28–30 August, 1996. Paris, France: NEA/OECD. Alexander, W. R., Gautschi, A. & Zuidema, P. (1998). Thorough testing of performance assessment models: the necessary integration of in situ experiments, natural analogues and laboratory work. Extended abstract in Sci. Basis Nucl. Waste Manag., XXI, 1013–1014. Alexander, W. R., Ota, K. & Frieg, B. (2001) (Eds). The Nagra–JNC in situ study of safety relevant radionuclide retardation in fractured crystalline rock II: the RRP project methodology development, field and laboratory tests. Nagra Technical Report Series, NTB 00-06. Wettingen, Switzerland: Nagra. Barten, W., Robinson, P. C. & Schneider, J. W. (1998). PICNIC-II – A code to simulate contaminant transport in fracture networks with heterogeneous rock matrices. In K. P. Holz et al. (Eds), Advances in Hydro-science and engineering, Vol. III, Proc. 3rd Inter. Conf. on Hydrosci. Eng. (p. 152). Berlin, Germany: Cottbus. Bath, A. H., Christofi, N., Neal, C., Philp, J. C., Cave, M. R., McKinley, I. G. & Berner, U. (1987a). Trace element and microbiological studies of alkaline groundwaters in Oman Arabian Gulf: a natural analogue for cement pore

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NEA (1998). Water-conducting Features in Radionuclide Migration, Proceedings of the Third GEOTRAP Workshop (held in Barcelona, Spain). Paris, France: OECD/NEA. NEA (1999). Confidence in the Long-term Safety of Deep Repositories – its Development and Communication. Paris, France: OECD/NEA. NEA (2000). Confidence in Models of Radionuclide Transport for Site-Specific Performance Assessment. Proceedings of the Fourth GEOTRAP Workshop (held in Carlsbad, New Mexico, USA, 14–17 June 1999). Paris, France: OECD/NEA. Osmond, J. K. & Cowart, J. B. (1976). The theory and uses of natural uranium isotopic variations in hydrology. Atom. Ener. Rev., 14, 621–679. Osmond, J. K. & Cowart, J. B. (1992). Ground water. In Ivanovich, M. and Harmon, R. S. (Eds), Uranium series disequilibrium: application to earth, marine and environmental sciences (2nd edn). Oxford, UK: Clarendon Press. Ota, K., Alexander, W. R., Smith, P. A., Moeri, A., Frieg, B., Frick, U., Umeki, H., Amano, K., Cowper, M. M. & Berry, J. A. (2001). Building confidence in radionuclide transport models for fractured rock: The Nagra/JNC radionuclide retardation programme. Sci. Basis Nucl. Waste Manag., XXIV, 1033–1041. Ota, K., Moeri, A., Alexander, W. R., Frieg, B. & Schild, M. (2002). Influence of the mode of matrix porosity determination on matrix diffusion calculations. J. Contam. Hydrol. (in press). Pate, S. M., Alexander, W. R. & McKinley, I. G. (1994). Use of natural analogue test cases to evaluate a new performance assessment TDB. In von Maravic, H. & Smellie, J. (Eds), Natural Analogue Working Group, fifth meeting, Toledo, October 1992. CEC Nuclear Science and Technology Report, EUR 15176. Luxembourg: CEC. Payne, T. E., Edis, R., Fenton, B. R., & Waite, T. D. (2001). Comparison of laboratory uranium sorption data with “in situ distribution coefficients” at the Koongarra uranium deposit, Northern Australia. J. Environ. Rad., 57, 35–55. Pearson, J., Balderer, W., Loosli, H. H., Lehmann, B. E., Matter, A., Peters, T. J., Schmassmann, H. & Gautschi, A. (1991). Applied isotope hydrogeology – A case study in Northern Switzerland. Amsterdam, The Netherlands: Elsevier. Pescatore, C. (1995). Validation: the eluding definition. Rad. Waste Manag. Environ. Restor., 20, 13–22. Petit, J. C. (1991). Migration of radionuclides in the geosphere: what can we learn from natural analogues. Radiochim. Acta, 51, 181–188. Philp, R. P. & Crisp, P. T. (1982). Surface geochemical methods used for oil and gas prospecting: a review. J. Geochem. Explor., 17, 1–34. Plant, J., Simpson, P. R., Smith, B. & Windley, B. F. (1999). Uranium ore deposits: products of the radioactive earth. In Burns, P. C. & Finch, R. (Eds), Uranium: mineralogy, geochemistry and the environment, Reviews in Mineralogy, Vol. 38. Washington, USA: Min. Soc. Amer. Press, F. & Siever, R. (1998). Understanding earth (2nd edn). New York, USA: Freeman. Reardon, E. J. & Fritz, P. (1978). Computer modelling of groundwater 13 C and 14 C isotope compositions. J. Hydrol., 36, 201–224. Schneider, J. W., Zuidema, P., Smith, P. A., Gribi, P., Hugi, M. & Niemeyer, M. (1998). Novel and practicable approach to modelling radionuclide transport through heterogeneous geological media. Sci. Basis Nucl. Waste Manag., XXI, 821–828. Scott, R. D., MacKenzie, A. B. & Alexander, W. R. (1992). The interpretation of 238U-234U-230Th-226Ra disequilibria produced by rock–water interactions. J. Geochem. Explor., 45, 323–343. Short, S. A., Lowson, R. T. & Ellis, J. (1988). 234 U/238 U and 230 Th/234 U activity ratios in the colloidal phases of aquifers in lateritic weathered zones. Geochim. Cosmochim. Acta, 52, 2555–2563. Sibley, T. H. & Myttenaere, C. (Eds) (1986). Application of distribution coefficients to radiological assessment models. Amsterdam, The Netherlands: Elsevier. Smellie, J. A. T., MacKenzie, A. B. & Scott, R. D. (1986). An analogue validation study of natural radionuclide migration in crystalline rocks using uranium series disequilibrium studies. Chem. Geol., 55, 233–254. Soler, J. M. (1999). Coupled transport phenomena in the Opalinus Clay: implications for radionuclide transport Nagra Technical Report NTB 99-09. Wettingen, Switzerland: Nagra. Smith, P. A., Alexander, W. R., Heer, W., Fierz, T., Baeyens, B., Bradbury, M. H., Mazurek, M. & McKinley, I. G. (2001a). The Nagra–JNC in situ study of safety relevant radionuclide retardation in fractured crystalline rock I: The radionuclide migration experiment-overview of investigations 1990–1996. Nagra Technical Report Series NTB 00-09. Wettingen, Switzerland: Nagra.

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Chapter 6

Radioactivity in lakes and rivers Lars Håkanson a , Luigi Monte b a Inst. of Earth Sciences, Uppsala Univ., Villav. 16, 752 36 Uppsala, Sweden b ENEA, Dipartemento Ambiente, Centro Ricerche Casaccia, Roma, Italy

1. Introduction and aim During the last 10 years, there has been a “revolution” in ecosystem modelling. The generality and predictive power of our models have increased in a way that was inconceivable 10 years ago (see Håkanson, 2000; IAEA, 2000). This chapter describes a new generation of practically useful models that predict as well as one can measure - if one measures well. And yet, they are driven by readily available driving variables and have a general structure that applies to most types of pollutants in aquatic systems. The major reason for this development is, in fact, the Chernobyl accident. Large quantities of radiocesium were released in April/May 1986 as a pulse. To follow the pulse of radiocesium through ecosystem pathways has meant that important fluxes and mechanisms, i.e., ecosystem structures have been revealed. It is important to stress that many of these new structures and equations are valid not just for radiocesium, but for most types of contaminants, e.g., for metals, nutrients and organics. This means that the models, methods (of building and testing models) and equations described in this chapter for lakes and rivers should be of great interest also to other ecosystem modellers. “. . .if man is adequately protected then other living things are also likely to be sufficiently protected”. This statement from the International Commission on Radiological Protection (ICRP, 1977) is based on extensive reviews on the effects of ionising radiation, including radiocesium. The relative tolerance of different groups of aquatic organisms to the radiation dose required to kill 50% of exposed individual in a given period of time vary greatly among animals and plants. Mammals are most sensitive and hence the target group in this context. The effects on aquatic organisms from ionising radiation concern both individual and population levels and the variables of interest are, e.g., mortality rate, fertility rate and mutation rate (see Whicker & Schultz, 1982; Jimenez & Gallego, 1998). The recommended dose rate limits (from IAEA, 1988) are: • below 1 mGy/h, no population effects, • 1–10 mGy/h, decrease in reproduction, • above 10 mGy/h, increase in mortality. 147

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Environmental factors can also influence responses to radiation, e.g., in mammals, radioresistance increases under anoxic conditions. Salinity tends to increase radiosensitivity due to the metabolic stresses it places on the organism. Radiation response also depends on the stage of the cycle the cell is in when the exposure occurs. In general, the radiosensitivity of cells is directly proportional to their rate of division and inversely proportional to their degree of differentiation (IAEA, 1988). Lacking well-tested target effect variables related to the reproduction or survival of key functional organisms in real aquatic ecosystems, the concentration of radiocesium in fish muscle consumed by man is generally used as a target variable. The guideline value for commercial marketing of fish recommended by many National Food Administrations is 1500 Bq/kg ww (ww = wet weight). Aquatic ecosystems are very complex webs of physical, chemical and biological interactions (see Håkanson & Peters, 1995). It is generally both costly and laborious to describe their characteristics, and to predict them is even harder. To develop scientific programs in radioecology concerning conservation, management and remediation is an even greater challenge. Every aquatic ecosystem is unique, and yet it is impossible to study each system in the detail necessary for case-by-case assessment of ecological threats, and proposals for remedial measures. In this situation, quantitative models are essential to predict, guide assessment and direct intervention. This chapter will focus on three aspects related to radioactivity in lakes and rivers: (1) New approaches for modelling. (2) Modelling in lakes, with a focus of radiocesium, and experience related to the Chernobyl accident. (3) Modelling in rivers, with a focus on radiostrontium.

2. Modelling aquatic pathways Practically useful models in radioecology must satisfy some key features that make them simple and reliable tools for environmental management: • they must provide good predictive power; • they must be characterised by a relevant and simple structure, i.e., involve the smallest possible number of driving variables; • the values of the necessary driving variables should be easy to access and/or to measure; • the models must be validated for a variety of circumstances showing a wide range of environmental characteristics. In broad terms, the variables used in environmental models may be divided in two categories: (1) variables for which site-specific data must be available such as lake volume, mean depth, water discharge, amount of suspended particulate matter in water, etc.; (2) variables for which generic (= general) values are used due to the lack of site-specific data, e.g., the sedimentation rate and/or rates for internal loading (like diffusion and advection rates).

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The variables belonging to the first category are often called “site-specific variables”, or “environmental variables” or “lake/river-specific variables”. They can generally be measured relatively easily and their experimental uncertainty should not significantly affect the overall uncertainty of the model predictions of the target variable(s). The second category, the “model variables” or “model constants”, are often difficult to access for each specific system, such as the transfer rates from the sediment to the water, the deposition velocity of X from water to sediments, the migration rate from catchment to lake, etc. The model variables may contribute significantly to the model uncertainty unless validated and reliable sub-models for their generic values have been established from critical tests. Unfortunately, critically validated widely applicable models showing a high predictive power are difficult to develop. As a consequence, practically useful models are generally (see Monte et al., 1997) based on “collective parameters”. It has been shown that (Monte et al., 1996) in many circumstances the values of such important collective parameters integrate many compensatory effects of the different phenomena occurring in the very complex ecosystem where “everything depends on everything else” (Håkanson & Peters, 1995). Examples of such collective parameters in the freshwater environment are the “migration velocities” of the radionuclide from water to sediments, the “effective removal” rates and the “soil permeability coefficient” from the catchment to a water body (Monte, 1995a). Models based on such “collective parameters” show a unique and important feature: their predictions are characterised by a relatively low uncertainty despite the large range of the environmental characteristics and the lack of site-specific values of the model variables (see Håkanson, 1995). The use of models based on collective parameters is the result of a change in the perspective concerning the modelling of natural ecosystems (see Håkanson & Peters, 1995). The main lesson is that in predictive modelling, it is seldom necessary, or wise, to account for “everything”. The difficult task is to omit processes which may add more to the model uncertainty than to the predictive success for the given target variable. The traditional modelling philosophy is based on what has been called a “bottom-up” structuring of the set of the occurring processes (see Fig. 1). It is assumed that some fundamental processes, belonging at the top-vertex of the logical pyramid, may be modelled in terms of logical-mathematical primary principles from which all other natural processes may be derived. One can also call this structure “Euclidean-like”. Another well-known example of this approach is Newtonian mechanics. At least in principle, the equations controlling any mechanical system (it does not matter how complex) may be derived from three simple and fundamental principles of classical mechanics and may be expressed in terms of a few primary quantities (masses, forces, initial positions, etc.). This classical modelling approach is based on the principle that the knowledge of nature may be derived from the principles of such primary models. Obviously, according to this approach, the environmental models that do not cope with the “primary mechanistic principles” occupy a low level of the knowledge pyramid. The higher causal models should instead be based on more fundamental knowledge from disciplines like chemistry, mechanics, thermodynamics, etc. that belong to another hierarchical level in the pyramid. Environmental models based on collective parameters are structured differently, more similar to a web than to a pyramid. Each process is indeed related to a variety of other phenomena and there is no reason to use few of them as fundamental starting points for understanding and predicting all the others. As a consequence, if it is possible to find processes that may

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Fig. 1. Schematical illustration of hierarchical modes of thinking in predictive modelling. The target y-variables in this example are radiocesium and radiostrontium in fish eaten by man and in the water (modified from Håkanson, 1997).

be modelled in terms of mathematical formulae based on collective parameters, there are no special reasons for relegating this sort of model to the less “noble” role than the ones directly related to the fundamental principles of the knowledge of the nature. Moreover, a variety of past experiences (see Peters, 1991; Håkanson & Peters, 1995) demonstrate that complex models based on general principles are often more uncertain, and yield a lower predictive success than simpler models based on collective parameters. In a strict sense, there is no such thing as a general (= generic) ecosystem model, which works equally well for all ecosystems because all models need to be tested against reliable, independent empirical data and the data used in such validations must of necessity belong to a given restricted domain. If this domain is equal to the entire population of ecosystems of the given type, then and only then, is the model generic in the strict sense. The complexities of natural ecosystems always exceed the complexity and size of any model. Simplifications are always needed, and this entails problems. There are dynamic mass-balance models tested over such wide ranges (like the radiocesium model discussed later in this chapter) that it is tempting to label them generic, but there will always be an ecosystem with properties outside the given domain for which the model would yield poor predictions. This is why modelling can be pictured as a two-sided coin: One needs the equations as well as the range where the equations apply. Before the following models are presented, it should be noted that this chapter will not focus on critical model tests, such as sensitivity and uncertainty analyses, since many papers

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and books discuss such matters (see Hinton, 1993; Hamby, 1995; Håkanson, 1999, 2000; IAEA, 2000). The intention here is not to write a literature review but to cover fundamental structural components of models for radionuclides in aquatic environments. Naturally, when using models for the ecosystem level (for entire lakes, river streches, etc.), it is not possible to describe phenomena at the individual, organ or cell levels.

3. Lake modelling Nutrients, metals, radionuclides and many toxic substances are distributed in aquatic ecosystems by the same fundamental processes. This means that the same principles of modelling can be used for most chemicals, pollutants and/or natural substances in most aquatic systems. This modelling structure can be increased almost indefinitely by incorporating more processes for fluxes (mass/time), amounts (mass) and concentrations (mass/volume) to and from the defined compartments. The model discussed in this section focuses on some basic principles of dynamic mass-balance models. Models of this kind play an important role in radioecology and water management, e.g., for simulations of the type of remedial measures discussed by IAEA (2000), such as potash treatment, lake liming and fertilisation. Many of the processes regulating the flow and biological uptake of substances in lakes are controlled by hydrological and morphometrical factors, which cannot be influenced by remedies changing the water chemistry, e.g., liming and potash treatment. Remedies that could speed-up the recovery (e.g., reduction of the elements in fish) must aim at, e.g., reducing the load to the lake, or the uptake in biota by blocking the biouptake, increase bioproduction and cause a “biological dilution” of the elements in biota, and/or include specific substances, like potassium (K) that can be taken up in fish in about the same way as radiocesium and cause a “chemical” dilution. 3.1. The model for radiocesium How is radiocesium transported in soil, water and biological material? Which lakes are particularly sensitive to radiocesium contamination? Which measures can speed up the recovery? Such questions can be answered by the following model. It should be stressed that it is at least theoretically possible to model concentrations of substances in abiotic compartments, like sediments and water, in a rather straightforward manner. It is often much more difficult to apply causal models for biological variables and predict biouptake and concentrations in plankton and fish, and it is very difficult indeed to model, especially to have good predictive models, for the really interesting issues, namely for the ecological effects on the ecosystem level. The new model discussed in this section is based on elements from many other lake models for radionuclides and heavy metals. Several new structural components and sub-models have been developed in the ECOPRAQ-project and the MOIRA-project (see Comans et al., 1998; Monte et al., 1997). A great challenge is to construct a model based on causal principles and understanding of key processes, and then to simplify those parts that need to be simplified so that the model can be practically useful. This means that there is a logical development from understanding to simplification.

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Fig. 2. (A) Compartment model illustrating the fluxes (arrows; mass per unit time) in a traditional dynamic model for a given radionuclide (X) in a lake with compartments for top predator, small fish, zooplankton, phytoplankton, bacteria, macrophytes, benthos, water and sediments. (B) Illustration of a new type of model based on mass-balance concepts for all abiotic fluxes, a bioconcentration factor which is used to calculate concentrations of X in biota (so that one does not need to determine the biomasses, which is very difficult), and dimensionless moderators used to quantify how different water chemical conditions (K-conc., pH, colour, P-conc., etc.) affect biouptake.

If dynamic models are to be used in practice, e.g., to quantify fluxes of contaminants in lakes, the rates that govern the transport between the various compartments have to be known, simulated or guessed. A graphical illustration of a typical compartmental model is given in Fig. 2A. The figure illustrates top predator (like pike or large perch), small fish (e.g., small perch and roach), zooplankton, phytoplankton, macrophytes, bacteria and benthos, and abiotic compartments (like water and active and passive sediments) and the processes regulating the fluxes between these compartments. The figure also gives the fluxes to the lake (direct lake load and river input related to catchment load) and from the lake (outflow and sediment

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transport from the active to the passive sediment layer). This is a general modelling set-up applicable for any substance X in a lake. The concentration of X in any biological compartment is defined as the ratio between the amount of X in the compartment (in g or Bq) divided by the biomass of the species (e.g., kg pike in the lake). One needs biouptake rates and excretion rates for all biological compartments, variables to describe the compartments (like age, sex, organ or weight of the fish) and the biomasses of the species in the lake. All these model variables are difficult and expensive to access, they vary seasonally and are often just guessed. This means that models of this kind generally provide poor predictions. Better predictions can often be obtained for models structured in such a way that one uses a dynamic model for all important abiotic processes, which are relatively easy to handle compared to biological processes, so that the concentration of X in water and/or sediments can be predicted accurately. The bioconcentration factor (BCF) approach is then used to calculate the concentration of X in biota directly without having to estimate biomass. Often there is knowledge about the factors influencing the biouptake, like a lower biouptake for radiocesium in lakes with high concentrations of potassium. This knowledge can be expressed by dimensionless moderators operating on the BCF-value (see Fig. 2B). The crucial point in practical lake ecosystem modelling is the structuring of the model. The model structure should include all important processes regulating the transport and biouptake of X, no more, no less. Figure 3 gives the basic structure of the dynamic model for substances in lakes, which will be elaborated in detail in the following section. There are some fundamental structural components for all models of the type given in Fig. 3. When there is a partitioning of a flow from one compartment to two or more compartments, this is handled by a partition (= partitioning = distribution) coefficient. This could be a default value, a value derived from a simple equation or from an extensive sub-model. There are five such distribution coefficients (DC) in the following model: (1) The DC to distribute the fallout on the catchment area to inflow and outflow areas. (2) The DC handling the partition from inflow areas to outflow areas and lake water. (3) The DC differentiating between dissolved and particulate fractions. This DC is often referred to as lake Kd . (4) The DC regulating the sedimentation to either areas of erosion and transport (ET-areas) and/or to areas of accumulation (A-areas). (5) The DC describing the flux from ET-areas back to lake water and/or to A-areas. In traditional ecosystem modelling, it is often a goal to include as many relationships (fluxes and compartments) as possible and describe the complex relationships in a logical, causal manner. If, however, one would like to make a prediction about one or a few target y-variables, it is evident that it is impractical, costly and laborious to use big ecosystem models. In such cases, one would use a smaller model since in practical water management there are always problems to access the driving variables. The following model for radiocesium has been constructed as a tool (1) to model variations in time on a monthly basis, (2) to facilitate interpretations of remedial measures for high levels of radionuclides in fish, and (3) to highlight the most important processes regulating spread and biouptake of radionuclides in lake ecosystems more generally. The idea is not to include every possible process, but the most fundamental processes. There are 22 rates and 18 model variables in the model in Fig. 2A. To make a good prediction, one would need reliable lake-specific empirical data on all these 40 model vari-

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Fig. 3. Illustration of the basic structure of the mixed model for toxic substances in lakes. Modified from Håkanson (1999).

ables. Rates are sometimes called rate constants, but that might give the wrong impression that rates do not vary in time and space. Most rates are variables, just like weight and age of animals. This means that it is a great challenge to develop predictive models for the most fundamental rates and model variables, like lake Kd , the rate of sedimentation, and biouptake and retention rates.

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3.2. Equations 3.2.1. The catchment sub-model – inflow to the lake The catchment sub-model (see Fig. 4) handles the inflow to the lake via the tributaries. The direct deposition (fallout times area) is the net sum of dry and wet fallout onto the catchment. The tributary load is a function of the amount of X in the catchment times an outflow (transfer or retention) rate. The catchment area is divided into two parts: (1) Outflow areas (≈ wetlands) dominated by a relatively fast turnover of substances and horizontal (land overflow) transport processes. (2) Inflow areas (≈ dry land) dominated by vertical transport processes, first through the soil horizons, then ground water transport, and, finally, tributary transport to the lake.

Fig. 4. An outline of the catchment area sub-model. Modified from Håkanson (1999).

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3.2.1.1. Outflow areas. The approach to calculate the secondary input to the lake from outflow areas (Bq per month) is given by: MOA (t) = MOA (t − dt) + (FOA + FIAOA − FOAW − FPDOA) · dt,

(1)

where: MOA (t) = mass (= amount) of X in outflow areas at time t (Bq Cs-137); FOA = deposition on outflow areas (Bq per time unit); the time unit, dt, in this model for radiocesium is set to one month to get seasonal variations; FIAOA = flow from inflow areas to outflow areas (Bq per month); FOAW = flow from outflow areas to the lake (Bq per month); FPDOA = physical decay of radiocesium from outflow areas (Bq per month). The deposition on outflow areas is given by: FOA = FO · ADA · OA,

(2)

where: FO = fallout (= deposition) in (Bq/m2 ); ADA = area of drainage area (m2 ); OA = percentage outflow areas in the catchment (a dim. less distribution coefficient; value from 0 to 1; a typical value for catchments of boreal landscapes is 0.2, i.e., 20% of ADA is bogs, mires, etc. (see Håkanson & Peters, 1995). The approach to calculate the retention rate regulating the flow from inflow areas to outflow areas, FIAOA , is based on a default assumption that the retention rate for radiocesium in inflow areas is larger than in outflow areas. How much longer is given by the soil permeability factor (SPF). Unrealistically large fluxes will be obtained if this rate is set too high, and too low fluxes to the lake if the rate is set very low. FIAOA = DOAL · MIA · RIA ,

(3)

where: DOAL = the distribution coefficient used to partition the fluxes from inflow areas into outflow areas (OA) and/or directly into lake water (L). The default value of DOAL is 0.5, i.e., 50% of the transport goes directly to the lake water. MIA = mass (= amount) of radiocesium in inflow areas (Bq). RIA = the runoff rate for inflow areas, which is given by: RIA = RRd · (Prec/650)2/(12 · SPF).

(4)

Thus, to calculate the flux from inflow areas to outflow areas, one needs (1) a default runoff rate, RRd , (2) a precipitation factor (Prec) and (3) a soil permeability factor (SPF). The default runoff rate RRd is set to 0.04 for the first year after the Chernobyl fallout. This value emanates from studies presented by Håkanson et al. (1988). This gives the factor RRd /12 or 0.04/12. The runoff of radiocesium also depends on precipitation; the more

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it rains, the greater the transport from land to water of radiocesium. This is handled by the dimensionless moderator (Prec/650)2; 650 mm/yr is the reference value for the annual precipitation for European catchments/lakes and Prec is the actual precipitation in mm/yr. The exponent 2 means that there is a non-linear relationship between precipitation and runoff. If the actual precipitation markedly departs from the normal value, the soils will be saturated and the runoff increase. The soil permeability factor (SPF) is a dimensionless model constant accounting for the fact that soils of different grain size characteristics will have different retention rates for radiocesium: the more permeable, the larger the retention rate. The following default values are used for SPF: 1 10 50 60 80

for very permeable morainic and organic soils; for mixed morainic soils and sandy soils; for silty morainic soils; for silty soils; for clay soils.

If there are no data available on characteristic catchment area soil type, a SPF-value of 25 may be used as a general default value. For simplicity, this model has no distribution coefficient for soil types (i.e., no soil Kd ). The soil permeability factor (SPF) may be regarded as a collective parameter expressing many complicated relationships regulating the retention of radiocesium in soils. It should also be noted that there is no time-dependent function for the runoff rate from inflow areas as there is for outflow areas (see equation (6)). The rationale for this is that the flux of radiocesium from the inflow areas can be regarded as a slow-moving front. A certain fraction of the amount of radiocesium from the inflow areas is transported either to outflow areas or directly to the lake per unit time. The transport from outflow areas, on the other hand, is much quicker. Under default conditions, it is 25 times quicker than the transport from the inflow areas (i.e., for SPF = 25). Since the transport from outflow areas is relatively fast, the most mobile fractions of radiocesium are successively depleted from the outflow areas. This is handled by the time-dependent function (1/(TIME + 1)0.5 . Eight months after the fallout, the time-dependent rate is 1/3 [= 1/(8 + 1)0.5 ] of the initial rate; 99 months after the fallout, the rate is 1/10 [= 1/(99 + 1)0.5 ] of the initial value. The flux from outflow areas to lake water, FOAW , (Bq/month), is given by: FOAW = MOA · ROA ,

(5)

where ROA = the retention rate of Cs-137 in outflow areas, which, in turn is calculated from (see Håkanson et al., 1996). ROA = YQ · RRd /(12 ·

√ (TIME + 1)),

(6)

TIME = time after fallout initiated at the fallout month (1 is January). The actual value of the rate is modified by the seasonal moderator for water discharge (the moderator gives an increased transport of radiocesium from land to lake during spring and fall peaks in water flow). The rate is time-dependent. It decreases with time from the month of the fallout (given

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√ by 1/ TIME). A delay function initiates the runoff to the month of the fallout plus one month (TIME + 1). The equation will cause the initial outflow rate, RRd , (0.04/12) to decrease with time, and the seasonal moderator for Q will create a seasonal variability in the outflow rate. YQ = seasonal moderator for the water discharge or runoff (dim. less); YQ is defined as the ratio between mean monthly surface runoff rate (MMSR) and mean annual surface runoff rate (MASR), i.e., YQ = MMSR/MASR. It is also possible to define YQ from the ratio between the mean monthly precipitation and the mean annual precipitation or from the mean monthly water discharge and mean annual water discharge, as predicted from models, see Abrahamsson & Håkanson (1998). This model for water discharge (Q) has been used in the following calibrations and validations because it was not possible to access data on MMSR for all those lakes. The physical decay from outflow areas (FPDOA ) is defined as: FPDOA = MOA · PD.

(7)

The physical halflife of the radionuclide is the time required for the activity to decrease by 50%. For example, since the physical halflife (t0.5 ) of radiocesium is 30.2 years and the halflife constant (k) is given by: −k = 0.693/30.2.

(8)

In this model, we write −k = PD = 0.693/(30.2 · 12), since all calculations are done on a monthly basis. Thus, when one calculates a halflife or its corresponding retention rate, it is often assumed that the given state can be treated as a single compartment that loses a certain fraction of itself per unit time (dt is set to 1 month in this model). If T is the “full” time, then 1.386 (= 0.693/0.5) from 0.693/(0.5 · T ) is often referred to as the halflife constant. 3.2.1.2. Inflow areas. The following differential equation gives all the fluxes to and from the compartment “inflow areas”: MIA (t) = MIA (t − dt) + (FIA − FIAL − FPDIA − FIAOA) · dt,

(9)

where: MIA (t) = mass (= amount) of X in inflow areas at time t (Bq Cs-137); = deposition on inflow areas (Bq per month); FIA FIAL = flow from inflow areas to lake water (Bq per month); FPDIA = physical decay of radiocesium from inflow areas (Bq per month); FIAOA = flow from inflow areas to outflow areas (Bq per month), as given by equation (3). The deposition (FIA ) on inflow areas is given by: FIA = FO · ADA · (1 − OA).

(10)

The flow from inflow areas to lake water, FIAW , is: FIAW = MIA · (1 − DOAL ) · RIA .

(11)

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Fig. 5. An outline of the abiotic part of the lake model. Modified from Håkanson (2000).

The physical decay from inflow areas is given by: FPDIA = MIA · PD.

(12)

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3.2.1.3. Direct fallout. The direct fallout to the lake (FW ; Fig. 5) is simply fallout, FO (e.g., Bq/m2 ) times lake area (Area in m2 ), i.e.: FW = FO · Area.

(13)

These equations handle all primary load functions. Next, we will focus on internal lake processes. 3.2.2. Internal processes 3.2.2.1. Lake water. In this model, the entire lake volume is treated as one compartment. So, there is no distinction between surface water and deep water, no stratification and/or mixing. The fluxes of radiocesium to and from the compartment “water” are given by the following equation: MW (t) = MW (t − dt) + (FW + FAW + FOAW + FIAW + FETW − FWOUT − FWA − FWET ) · dt, (14) where: MW = mass of X in water (Bq); FAW = diffusion from A-areas (Bq/month); FETW = advective transport (= resuspension) from ET-areas to lake water (Bq/month); FWOUT = outflow of radiocesium from lake water (Bq/month); FWA = sedimentation from lake water to A-areas (Bq/month); FWET = sedimentation from lake water to ET-areas (Bq/month). The diffusive transport of radiocesium from A-areas to lake water, FAW , is given by: FAW = MA · Rdiff ,

(15)

where Rdiff = the diffusion rate (1/month). From Comans & Hockley (1992) (see also Hilton, 1997; Konoplev et al., 1997), the full equation for the partitioning of Cs-137 in sediments is: Kd = Kc1 · CFES /(CK + Kc2 · CNH4 ),

(16)

where: CFES = the “frayed edge site” concentration; CK = the K-concentration in the interstitial sediment water; Kc1 = the selectivity coefficient between radiocesium and potassium; Kc2 = the selectivity coefficient between potassium and ammonium; CNH4 = the concentration of ammonium in the sediments. Since it is generally very difficult to access reliable empirical data of the model variables in equation (16) for lake sediments, one can simplify the approach by assuming that one can

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use the K-concentration for the lake water also for the sediments and that the ammonium concentration can be estimated linearly from the rate of sedimentation and the hypolimnetic temperature. In that case, this sub-model does not require extra driving variables, just the driving variables already available in the model for other purposes. It is assumed (see Comans et al., 1998) that a default value of the diffusion rate for radiocesium in sediments is about 0.002 per year. The normal default value for the hypolimnetic temperature is set to 4 (◦ C). The diffusion rate is also set as a function of bioturbation, which influences the age of the A-sediments. If the sediments go anoxic, bioturbation is halted but the diffusion of radiocesium from sediments increases (Alberts et al., 1979). From these examples, Rdiff , may be estimated accordingly: If the sediments are oxic (and the Bioturbation factor, BF < 1, see later), the diffusion rate is given by: Rdiff = (0.002/12) · (SedA/100) · (HypoT/4),

(17)

where SedA = gross sedimentation on A-areas (in μg dw/cm2·day). The normal value for SedA is set to 100 μg dw/cm2 ·day. The critical SedA-value is set to 2000 μg dw/cm2·day. If SedA is higher than that, the sediments are likely to go anoxic and the bioturbation is halted. If the sediments are anoxic, the Rdiff -value is, for simplicity, set to be 20 times larger than under oxic conditions, as given by equation (17). SedA may be estimated from: SedA = SPM · vSPM · 150 · 104 /365,

(18)

where: SPM = suspended pariculate matter concentration in mg/l; vSPM = the fall velocity of suspended particulate matter; a default v-value of 500 m/yr may be used (see Håkanson, 1999). The other constants in equation (18) are dimensional adjustments to get SedA in μg dw/cm2·day. 150 is the duration of the growing season (days). The more stable the stratification is, the higher the hypolimnetic temperature (HypoT), the greater the microbiological activity, the greater the mineralization of organic matter, the greater the oxygen consumption, and the higher the potential diffusion of radiocesium from active sediments (Alberts et al., 1979). Accounting for hypolimnetic temperature in this manner means that many complex processes are lumped into one collective variable, HypoT. The default values used for HypoT and epilimnetic temperature (EpiT) in this approach are: Month: 1 2 3 4 5 6 7 8 9 10 11 12 EpiT: 0 0 3.0 4.0 7.0 11.0 16.0 20.0 15.0 7.0 4.0 1.0 HypoT: 4.0 4.0 4.0 4.0 4.5 6.0 8.0 10.0 8.0 7.0 4.0 4.0 These are meant to be typical mean monthly water temperatures in ◦ C. They should be regarded as standard, default values for this model. Using these values, one can note that homothermal conditions prevail during April and November. During the rest of the year, there is stratification. It is evident that this temperature regime is only valid for dimictic lakes. Ideally, these values should emanate from empirical measurements, or from models (see Håkanson & Peters, 1995; Ottosson & Abrahamsson, 1998). The model for lake water temperatures pre-

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sented by Ottosson & Abrahamsson (1998) has been used in the following calibrations and validations because it was not possible to access data on EpiT and HypoT for all those lakes. The advective transport from ET-areas to lake water, FETW , is given by: FETW = (MET · (1 − Vd /3))/TET,

(19)

where: MET = the amount of radiocesium in ET-sediments (Bq); Vd = the form factor (= 3·Dm /Dmax , where Dm = mean depth and Dmax = max. depth; m); Vd is often called the volume development (see Håkanson, 1981). It is used as a distribution coefficient to distribute the resuspended amount of radiocesium from ET-areas (i.e., wind/wave induced advective resuspension) either to the water compartment or to the compartment called accumulation areas (A-areas). If the lake is U-shaped, Vd is about 3 (i.e., Dmax ≈ Dm ) and all resuspended radiocesium will flow to the A-areas. If, on the other hand, the lake is shallow and Vd is small, most resuspended radiocesium will flow to the water compartment. TET = age of radiocesium on ET-areas. By definition (see Håkanson & Jansson, 1983), the materials which settle on ET-areas will not stay permanently where they were deposited but be resuspended by wind/wave activity and/or slope processes. If the age of the material is set very long, like 10 years, these areas will function as accumulation areas; if, on the other hand, the age is set to 1 week or less, they will act as erosion areas. Often (see Håkanson & Peters, 1995), it is assumed that the mean age of these deposits is about 1 year for lakes. This value is also used as a default value in this model. This means that the halflife, TET , is given by TET = (0.5 · 1 · 12)/0.693. A very important part of this model, or any similar model, is the structure for the partition  coefficient, often called Kd . Traditionally, this is the ratio between the particulate (Cpart in  Bq/kg dw) and the dissolved (Cdiss in Bq/1) phases, i.e., Kd = Cpart /Cdiss . Then Kd is given in l/kg. The total amount (in Bq/l) is equal to (Cdiss + Cpart ), where Cpart is given in Bq/l. The amount of suspended matter in the lake water, SPM, is given in mg/l. This means that the dissolved fraction (Cdiss ) can be written as: Cdiss = 1/(1 + Kd · SPM · 10−6 ).

(20)

This is a general definition of the dissolved phase based on lake Kd . So, the amount in the dissolved fraction in the lake is equal to Cdiss (Bq/m3 ) times the total lake volume (m3 ). Values of suspended particulate matter (SPM in mg/l) in the water phase should be empirically measured or predicted from a model (see Lindström et al., 1999). The default value for lake Kd for radiocesium is often set to about 800,000 l/kg or 0.8 l/mg suspended particulate matter (from IAEA, 2000). Another approach for radiocesium (see Comans & Hockley, 1992; Smith et al., 1997; Comans et al., 1998) is to express Kd as a function of “frayed edge site” concentration of the clay mineral illite (which has a unique potential to bind cesium), CFES , the K-concentration of the water, CK , and the selectivity coefficient between radiocesium and potassium (Kc ), where Kd = Kc · CFES /CK .

(21)

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Kc and CFES are often assumed to be constants since it is generally very difficult to access reliable empirical data of these variables for different lakes. This means that the approach may be written as: Kd = const/CK . Based on these arguments, the following approach is meant as a typical “collective” approach, i.e., as a simplification of many complex relationships, but it is also meant to be based on a firm mechanistic foundation. We have: Kd = 800 000/CK,

(22)

where CK is given in mg K/l. 800 000 is the Kd -constant. 3.2.2.2. Lake outflow. The lake outflow is a very important processes: “How open is the exit gate for X from the lake”? This is handled mathematically by a flux called, FWOUT , flux out of lake (Bq/month). FWOUT = (RPD + RW ) · MW ,

(23)

where: RPD = the physical decay (see equation (10)); RW = the lake water retention rate (1/month); MW = the amount of radiocesium in the lake water (Bq). RW is sometimes set to 1/Tw in mass-balance calculations. For large, deep lakes with small drainage areas, i.e., lakes with a long theoretical water retention time (TW ), it is evident that thermal and chemical stratifications, hydrological flow patterns and currents influence the retention time of water and contaminants. Then, should one use the entire lake volume, the volume of the epilimnetic water, or the volume of a defined fraction of the epilimnetic water in the calculation of the retention time? From lake eutrophication studies (Vollenweider, 1968), √ it is well known that better predictions of lake P-concentrations are obtained if one uses Tw rather than KT · TW in mass-balance calculations. In this model, we will define a function exp (from Håkanson & Peters, 1995) for the exponent in the expression 1/TW . This exponent (exp) should be about 1 in lakes with a quick water retention (if TW < 1 month). • For lakes with very fast water turnover (if TW < 1 month): RW = 1.386/TW.

(24)

• For very small lakes (if Area < 0.2 · 106 m2 ): ((50/(TW +50−1)+0.5)/1.5)

RW = 1.386/TW

,

(25)

,

(26)

where 50 is the retention rate constant. • For all other lakes: ((10/(TW +10−1)+0.5)/1.5)

RW = 1.386/TW

where 10 is the retention rate constant. This approach yields a faster and more adequate retention for lakes with long water retention as compared to 1/TW . If TW is 12 months, RW

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(equation (26)) is = 0.28; if TW = 60 months, RW = 0.24. The constant 1.386 is, as before, the ratio − ln(0.5)/0.5 = 0.693/0.5, and emanates from the definition of the halflife. 3.2.2.3. Sedimentation. We will first treat sedimentation on A-areas (FWA , as defined in equation (14)): FWA = MW · Rsed · (1 − ET) · (1 − Dres ) + MW · Rsed · (1 − ET) · Dres · 2,

(27)

where: MW = the amount of radiocesium in the lake water (Bq); Rsed = the sedimentation rate (1/month); Dres = the distribution coefficient describing the fraction of resuspended matter in the lake water; this fraction settles twice as fast as the primary matter (see equation (32)); ET = the fraction of ET-areas. The following processes influencing internal loading are accounted for in the sub-model for the ET-areas (see Håkanson, 1977; Håkanson & Jansson, 1983): (1) an energy factor related to the effective fetch and the wave base, (2) a form factor related to the percentage of the lake bed above the wave base, and (3) a lake slope factor related to the fact that slope-induced transportation (turbidity currents) may appear on bottoms inclining more than 4–5%. This information is used to estimate the bottom areas where processes of erosion (E), transport (T) and accumulation (A) prevail. E-bottoms are areas where the cohesive tine materials that follow Stokes’s law are not deposited. Such areas are dominated by coarse deposits. T-bottoms are, by definition, areas where the fine suspended materials are deposited discontinuously. In such areas, one generally finds mixed deposits. Soft A-bottoms appear beneath the wave base, where the fine suspended materials may be continuously deposited. By definition, there should be no net deposition of tine sediments on E- and T-areas over longer periods of time (1 year or more). One approach (from Håkanson & Jansson, 1983) is used to calculate ET for lakes larger than 1 km2 . This is the equation based on the dynamic ratio (DR), where: ET = 0.25 · DR · 41(0.061/DR) .

(28)

Another approach (see Håkanson, 2000) is used for lakes smaller than 1 km2 . That approach is based on the wave base and the form of the lake. The value used for the ET-areas is used as a distribution coefficient. It regulates the sedimentation of particulate radiocesium either to A-areas or to ET-areas. The sedimentation rate, Rsed , is given by: Rsed = ((1 − Cdiss ) · vCs )/Dm ,

(29)

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165

where: Cdiss = the dissolved fraction (see equation (21)); this means that (1 − Cdiss) is the particulate fraction, which is the only fraction that can settle out in the lake; vCs = the fall velocity for the particulate fraction (m/month); Dm = the mean depth (m). Burban et al. (1989, 1990) have demonstrated that changes in turbulence and concentration of suspended particulate matter are key regulatory factors for the fall velocity for suspended matter. In this model, the concentration of suspended matter (SPM) will be allowed to influence the settling velocity for particulate radiocesium. This is achieved by means of a dimensionless moderator. The amplitude value is calibrated in such a manner that a change in the concentration of SPM by a factor of 10, e.g., from 2 (which is a typical value for oligotrophic lakes) to 20 mg/l (which is typical for eutrophic lakes; see Håkanson & Peters, 1995), will cause a change in the settling velocity by a factor of 2. The borderline value for the moderator is set to 50 mg/l, since it is unlikely that lakes will have a higher suspended load. The default settling velocity for particulate radiocesium is set to 12 m/yr. Then, vCs , is given by: vCs = (12/12) · YSPM ,

(30)

where the dimensionless moderator expressing the influence of SPM on the settling velocity, YSPM , is given by: YSPM = (1 + 0.75 · (SPM/50 − 1)).

(31)

The calibrated amplitude value is 0.75. A simple and typical form of a dimensionless moderator is, e.g., the ratio between a mean monthly value, MM, and a mean annual value, AM. In traditional mass-balance models, one would multiply an amount (kg) by a rate (1/month) to get a flux (i.e., amount·rate or amount·rate·1). In this model, one multiplies kg·(1/month)·Y (= amount·rate·mod), where Y is a dimensionless moderator quantifying how an environmental variable (like SPM) influences the given flux (e.g., sedimentation of particulate radiocesium). Instead of building a large mechanistic sub-model for how environmental factors influence given rates, this technique uses a simple, general algorithm for the moderator. Empirical data can be used for the calibration of the moderator. The dimensionless moderator defined by equation (31) uses a borderline value, i.e., a realistic maximum value of SPM = 50, to define when the moderator, YSPM , attains the value of 1. For all SPM-values smaller than the borderline value, YSPM is smaller than unity. One can also build normal value moderators in such a way that the YSPM is 1 for the “normal” value and higher or lower than 1 for SPM-values higher and lower than the defined normal value (e.g., SPM = 5). The amplitude value regulates the change in YSPM when the actual SPM-value differs from the borderline value and/or the normal value. Sedimentation of radiocesium on ET-areas, FWET (Bq/month), is, accordingly, given by: FWET = MW · Rsed · ET · (1 − Dres ) + MW · Rsed · ET · Dres · 2.

(32)

3.2.2.4. Resuspension. This is the advective transport of radiocesium from ET-areas, either back to lake water or to A-areas. The form factor, Vd , is used as a distribution coefficient to

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regulate how much of the resuspended material from ET-areas that will go the lake water or to A-areas. The settling velocity of particulate radiocesium also depends on the amount of resuspended matter. The resuspended particles have already been aggregated; they have also generally been influenced by benthic activities, which will create a “glueing effect” (see Håkanson & Jansson, 1983). It is most probable that such particles have a higher settling velocity than the primary materials. In this model, these resuspended particles settle twice as fast as the primary materials. This is calculated by means of the distribution coefficient (Dres ) defined by the ratio between the resuspension (= advection) from ET-areas to water and to A-areas relative to the sedimentation on ET- and A-areas: Dres = (FETW + FETA )/(FWET + FWA ).

(33)

We will first treat resuspension back to lake water, FETA (advective flux to lake water; Bq/month): FETW = (MET · (1 − Vd /3))/TET,

(34)

where: MET = the amount of radiocesium in ET-areas (Bq); Vd = the form factor; TET = the age of radiocesium on ET-areas. As mentioned (see Håkanson & Peters, 1995), it is often assumed that the mean age of these deposits is about 1 year for lakes. Note that the calculation time, dt, is given in months, not years. TET was given by equation (21). Advective transport (= resuspension) from ET-areas to A-areas, FETA (Bq/month), is given by: FETA = (MET · Vd /3)/TET.

(35)

This equation completes the set of equations regulating the fluxes of radiocesium to and from the compartment lake water. Next, we will treat the two sedimentological compartments, Cs137 associated with A- and ET-areas. 3.2.3. Sedimentological fluxes 3.2.3.1. Active A-sediments MA (t) = MA (t − dt) + (FWA + FETA − FPDW − FAPS ) · dt.

(36)

All these fluxes, except for radiocesium transport (= burial) from surficial, active sediments to deeper, geological (= passive) sediments, FAPS , have been treated. The age of the active A-sediments is needed to calculate the retention rate of the substance in this compartment, and, hence, the flux out of the compartment to passive (geological) sediments (the retention rate is set to 1/age). The age is generally calculated as the ratio between the depth of the active sediments (in m) and the deposition of materials on A-areas (in

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m/month). Assuming a default value for the depth of the active layer as 5 cm and a default sedimentation rate of 4 mm/yr for A-areas (see (Håkanson & Peters, 1995)), we have a default value for the age of the active sediments, TA , of 150 months, i.e., the retention rate, 1/TA , is given by 1/TA = 0.693/(0.5 · 150). The biological mixing from benthos, gas ebullition, etc., will create a transport of old, previously deposited sediments into the compartment of the active A-sediments. This will influence the age of the active sediments. It is well-known from lake sedimentological studies (see Håkanson & Jansson, 1983) that benthos can eat the bottom sediments many times (up to 7 times). When new materials are deposited on the sediments, these will be compacted and the water content will decrease with sediment depth (see Håkanson & Jansson, 1983). This means that the actual age of the active sediments is generally significantly older than indicated by the simple ratio between sedimentation (cm/yr) and thickness of active sediment layer (cm). The correction factor for bioturbation is related to gross sedimentation (see Håkanson & Jansson, 1983). If the deposition is very high, the bottom-living animals are less likely to be able to create a complete mixing. If gross sedimentation (SedA) is larger than 2000 μg dw/cm2 ·day, it is assumed that the oxygen consumption is so high that the oxygen concentration is smaller than 2 mg/l and bioturbation is halted because meio- and macrobenthos die. The bioturbation factor (BF) is given by: if SedA > 2000 then BF = 1 else BF = (4 − 0.02 · (SedA/100 − 1)),

(37)

where 4 is the bioturbation and compaction constant, 0.02 is the amplitude value and 100 the norm-value for SedA. If actual SedA increases from the norm-value to 1000, the BF-value changes from 4 to 3.82. This means that the mean age of the sediments is likely to be about 5% lower just by accounting for these changes in bioturbation and compaction. This means that FAPS is given by: FAPS = MA · (1/(BF · TA ) + PD),

(38)

where PD is the physical decay rate for radiocesium. 3.2.3.2. Areas of erosion and transport. MET (t) = MET (t − dt) + (FWET − FETW − FETA − FPDET ) · dt.

(39)

All these fluxes, except for radiocesium transport from ET-sediments by physical decay have been treated before, and it is given by FPDET = MET · PD. With this, all abiotic fluxes are explained, and we can turn to the biological part of the model. 3.2.4. Biotic processes Technically in this model, the biotic processes are treated separately from the abiotic fluxes and the two pathways are linked by means of a bioconcentration factor (Fig. 2B). This is a simplification with many advantages in predictive and environmental engineering contexts:

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(1) One can predict the target concentrations in biota directly without data on biomass, which are generally very, very hard to access. The basic aim here is to predict the concentration of X in fish. This model uses compartments of one mass unit: X is transported from one mass unit of lake water to one mass unit of fish. This means that the model automatically yields concentrations. (2) One can relatively easily test (calibrate and validate) models of this type, since concentrations in biota are easily measured. This model structure generally gives better predictive success than traditional models using connected or parallel biotic compartments (like phytoplankton, zooplankton, benthos, prey fish and predator fish models of the type illustrated in Fig. 2A). The uptake of X from water to biota is still modelled dynamically. The bioconcentration factor is modified by factors known to influence biouptake of Cs-137: • • • • • •

The potassium concentration in lake water. The amount in dissolved and particulate phases of radiocesium. The feed habits of the fish. The weight of the fish. The trophic characteristics of the lake. The water temperature.

These factors are accounted for in this model, which is illustrated in Fig. 6. Approaches like this are most relevant in cases when the biotic fluxes of X are much smaller than the abiotic fluxes. This is the case for most metals and radionuclides in most lakes. The differential equation for the concentration (i.e., CF in Bq Cs-137 per kg wet weight of fish) is given by: CF (t) = CF (t − dt) + (FuF − FeF ) · dt,

(40)

where: FuF = biouptake (Bq/month·kg); FeF = excretion (Bq/month·kg). The biouptake is given by:

FuF = ALmod · AUmod · Kmod · BMF · SMTH(Cdiss + Cpar , BUD, Cdiss + Cpar ), (41) where: ALmod = a dimensionless moderator for allochthonous production; AUmod = a dimensionless moderator for autochthonous production; Kmod = a dimensionless potassium moderator; BMF = a biomagnification factor;

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Fig. 6. An outline of the biotic part of the model for radiocesium. Modified from Håkanson (1999).

SMTH = a smoothing function based on the concentration of radiocesium in dissolved phase, Cdiss , in particulate phase, Cpar , and a biouptake delay factor (BUD). Those factors affecting the biouptake will now be examined. The moderator for allochthonous production, ALmod , as well as the moderator for autochthonous production, AUmod , are meant to account for the principle of “biological dilution” (see Håkanson & Peters, 1995). That is, a given load of a radionuclide or a metal to a lake will be distributed over a larger biomass in a lake with higher bioproduction than in a lake with a lower bioproduction. The bioproduction in, especially “brown-water” lakes (see Jonsson, 1997) is not, however, a matter of just the supply of total phosphorus, handled by AUmod in this approach. Nitrogen can often be the limiting nutrient in such lakes, and ALmod is meant to account for this in a simple manner. It would, perhaps, have been logical to account for this by including more model variables, like total-N concentration, lake colour, etc. However, it is a well established fact (see Håkanson & Peters, 1995) that “brown-water” lakes generally have relatively large areas of wet land (mires, bogs, etc.), which supply the lakes with coloured substances (humus). The moderator for allochthonous production, ALmod , is based on the percentage of outflow areas, since this information is included in this model for other reasons (in the catchment area sub-model). This means that allochthonous production is accounted for without adding more driving variables, an important demand for this model. By incorporating this moderator for allochthonous production, the model becomes more general. Empirical calibrations have, however, demonstrated that it is not important to account for this effect in lakes little influ-

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enced by humic materials. That is, if the outflow area, OA, is smaller than the calibrated limit of 0.1 (10% of the catchment area), the allochthonous influences on the bioconcentration factor for radiocesium are likely very small. If OA > 0.1, it becomes increasingly important to account for this principle. Then, this moderator accounts for allochthonous production by the ratio 0.1/OA. One can note that this moderator will cause a decrease in biouptake by a factor of 2 if OA is 0.2 (values of OA > 0.3 would be very rare; see Håkanson & Peters, 1995). So, for AUmod , we have: • If OA < 0.1, then ALmod = 1, • If OA  0.1, then ALmod = 0.1/OA .

(42)

Empirical calibrations (for further details, see Håkanson, 1999, 2000) for radiocesium have demonstrated that it is important also to account for autochthonous production, especially for low-productive lakes; if lake total-P concentration is lower than about 14 μg/l. We have: • If TP > 14 (μg/l), then AUmod = 8, • If TP  14, then AUmod = (22-TP).

(43)

This moderator accounts for autochthonous production in a linear way. If, e.g., the total-P concentration is very low, say 6 μg/l, which is typical for ultra-oligotrophic lakes, equation (43) sets the moderator to 16. If the TP-concentration is higher than 14, the moderator is 8 and the expected concentration of radiocesium in fish becomes lower. The potassium moderator, Kmod , expresses the influence of lake K-concentrations on the biouptake for fish. The critical limit between sufficiency and deficiency is generally (see Fernandez et al., 1997) set at CK = 0.1 mM = 3.91 mg K/l. In natural lakes, one can expect CK to vary from about 0.1 to 150 mg/l (see Desmet et al., 1997). The dimensionless moderator is based on physiological mechanistic models for the bioconcentration factor for both sufficiency (CK > 0.1 mM) using the Nernst equation and deficiency (CK < 0.1 mM) using the Michaelis–Menton model. The entire realistic range for CK has been tested in the derivation of the K-moderator.   Kmod = 300/ 4.5 + CK0.75 .

(44)

Note that in this model CK must be given in μeq/l (= (mg/l)/0.0391). The biomagnification factor (BMF) is set (partly from Rowan & Rasmussen, 1994a, 1995) to 25 if the fish weight (WF ) is < 10 g ww for all types of fish; if WF > 10 g ww, then BMF is: 250/WF 2 3 4

for planktivores for benthivores for omnivores and for piscivores.

The biomagnification factor accounts for differences in biouptake/biomagnification from water to fish related to feed habits when all else is constant. Fish of the same weight but with different feed habits display different biouptake. Piscivores are, e.g., likely to take up more

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radiocesium than benthivores. If everything else is constant except for the feeding behaviour expressed by these trophic categories, factors 2, 3 and 4 should reflect the most likely differences in concentrations. Corresponding values for herbivores and detrivores are not available to the author. The BHL-value for planktivores seems to depend very much on the weight of the fish. The biouptake delay factor (BUD) accounts for the fact that large/old animals high up in the nutrient chain do not respond as quickly to changes in concentrations in water as small/young planktivores (see e.g., Håkanson, 1991). This delay is expressed as a function of the weight of the animal (WF in g ww). In this model where the calculation time, dt, is set to one month, the delay factor is set to 1 for small fish (weighing less than 100 g ww). For larger fish, BUD is given by: BUD = WF /100.

(45)

BUD is used as an averaging time within the framework of a smoothing function (see (Håkanson, 1999) for further information). The smoothing function gives an exponential smoothing of a given input and it works like a calculation of a running mean value. It can be written as: SMTH(input, averaging time, initial value). In this model we have: SMTH(Cdiss + Cpar , BUD, Cdiss + Cpar ), where Cdiss and Cpar are the concentrations of radiocesium in dissolved and particulate phases. The greater the weight of the fish (WF ), the greater the value of BUD, and the more pronounced is the delay in the uptake, as given by the smoothing function. Table 1 gives a very useful compilation of information (from (Brittain, 1998)) regarding common European species of fish, typical weight ranges and feed habits. This information can be used to identify feed habits and typical weights for all species of fish discussed in this chapter, and many more. The concentration of radiocesium in dissolved phase, Cdiss is calculated from (see also equation (22)): Cdiss = Cwa /(1 + Kd · SPM · 10−6 ),

(46)

where Cwa = the total concentration of radiocesium in lake water (Bq/l). The total concentration of all particulate phases, CparT , is given by: CparT = HA · ((Cwa − Cdiss ) + 0.001 · (MA + MET )/Vol).

(47)

Hence, CparT is the total concentration of radiocesium in (1) particulate phases suspended in the water and (2) in active A-sediments and ET-sediments. HA means feed habit. The values used for HA in this model are meant to reflect the influence on the bioconcentration factor when all factors except exposure to all types of particulate phases of Cs-137 are held constant. This factor should reflect the importance of radiocesium fluxes from sediments and suspended particulate matter. This means that the factor should be

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Table 1 Compilation of major European fish species in rivers, lakes and reservoirs (modified from Brittain, 1998). CM = consumed by man. I, II and II = stages Species

Typical weight range (kg)

Target Trophic weight range (kg)

Habitat

Food habit

Significance

Sturgeon Brown trout. I Brown trout. II Brown trout. III Arctic char Whitefish Smelt Pike Roach Minnow Asp Nase Barbel Bream Carp Wels Eel Burbot Perch. I Perch. II Perch. III Pike-perch Ruffe

20–50 0.05–0.1 0.1–0.3 0.5–1.5 0.01–0.2 0.1–1 0.01–0.05 0.5–3 0.05–0.2 0.001–0.01 0.5–3 0.1–1 0.5–3 0.5–2 0.5–3 2–20 0.1–1 0.1–1 0.01–0.1 0.1–0.3 0.3–0.6 0.5–3 0.005–0.02

30 0.1 0.2 1 0.1 0.3 0.01 1 0.1 0.01 1 0.3 1 1 1 5 0.5 0.5 0.1 0.2 0.5 1 0.01

benthic. riverine benthic. littoral benthic. littoral benthic. littoral pelagic pelagic pelagic benthic. littoral benthic/pelagic. litt. benthic. littoral pelagic. riverine pelagic. riverine benthic. riverine benthic. littoral benthic. littoral benthic benthic. riverine benthic. profundal benthic. littoral benthic. littoral benthic. littoral pelagic benthic. littoral

piscivore planktivore benthivore piscivore planktivore planktivore planktivore piscivore omnivore omnivore piscivore planktivore benthivore benthi-/detrivore benthi-/detrivore piscivore omnivore benthivore planktivore benthi-/omni. piscivore piscivore benthivore

CM, caviar CM, early CM CM CM CM prey CM prey prey species CM CM CM, angling CM CM CM CM CM CM, early stage CM CM CM prey species

meso- to eutrophic oligo- to mestotr. oligo- to mestotr. oligo- to mestotr. oligo- to mestotr. oligo- to mestotr. oligo- to eutrophic oligo- to mestotr. oligo- to hypertr. oligo- to mesotr. meso- to eutrophic meso- to eutrophic meso- to eutrophic meso- to eutrophic meso- to eutrophic meso- to eutrophic oligo- to eutrophic oligotrophic oligo- to eutrophic oligo- to eutrophic oligo- to eutrophic oligo- to eutrophic oligo- to eutrophic

highest for benthivores and lowest for planktivores, and in-between for other groups of fish. In this model, the following default values are used for HA: Benthivore: Piscivore: Omnivore: Planktivore:

HA = 1/2. HA = 1/8. HA = 1/20. HA = 1/200.

The concentration (in Bq/l) of radiocesium, which can be taken up by biota living in ETand active A-sediments (Csed ), is calculated as: Csed = 0.001 · (MA + MET)/Vol.

(48)

This is the second term in equation (47). The first term is Cpar . The excretion, FeF (in Bq/kg ww·month), from the fish is given by: FeF = CF · (BHL + PD),

(49)

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Fig. 7. Panel of driving variables. This panel uses lake-specific data for Lake Siggefora, Sweden.

where: CF = the concentration in fish (Bq/kg ww); BHL = the biological halflife of radiocesium in fish (1/month); PD = the physical decay rate (1/month). This expression for the biological halflife (BHL) comes from Rowan & Rasmussen (1995). BHL is given by: • If WF < 5 g ww, then BHL = 30 · (exp(−6.583 − 0.111 · ln(WF · 5) + 0.093 · EpiT + 0.326 · SS)). • If WF  5, then BHL = 30 · (exp(−6.583 − 0.111 · ln(WF ) + 0.093 · EpiT + 0.326 · SS)).

(50) (51)

That is, the BHL-value depends on the weight of the fish (WF ), the epilimnetic water temperature (EpiT in ◦ C) and whether steady-state conditions are met or not. If the model predictions are based on steady-state assumptions, then SS = 0, if not, SS = 1. Note that normally one cannot assume steady-state conditions. With this, all equations in the model have been presented. Figure 7 gives the panel of obligatory driving (lake-specific) variables. These are the only variables that have been changed in the following validations using independent data. This means that the validations are meant to demonstrate the predictive power of the model. 3.3. Validation The validation lakes are given in Table 2. They cover a wide range in lake characteristics, although they are all European; size from 0.042 km2 to 1147 km2 , mean depth from 2.9 to 90 m, characteristic pH from 5.1 to 8.5, trophic level from oligotrophic (TP = 10 μg/l) to

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Table 2 Data on the lakes used for validating the radiocesium model. (1) Marcus Sundbom, pers. comm., (2) IAEA (2000), (3) Jim Smith, pers. comm., (4) Håkanson (1991) (A) Lake

County

Fallout (kBq/m2 )

Area (km2 )

IJsselmeer Iso Valkjärvi Ø. Heimdalsv. Esthwaite Siggefora Hamstasj. (2217) Lill-Selasj. (2214) Zürich

Netherl. Finland Norway England Sweden Sweden Sweden Switzerl.

2.2 70 130 2 30 50 70 5.9

1147 0.042 0.78 1 0.73 0.18 0.07 537.4

Dm (m)

Dmax (m)

pH

TP (μg/l)

4.3 3.1 4.7 6.4 4.2 3.8 2.9 90

10 8 20 15 11 7.8 9.2 252

8.5 5.1 6.8 8 7.2 6.6 5.1 7

60 11 10 25 10 22.9 10.9 10

(B) Lake

Tw (yr)

Trophic level

Fish species

Abiotic var.

Reference

IJsselmeer Iso Valkjärvi Ø. Heimdalsv. Esthwaite Siggefora Hamstasj. (2217) Lill-Selasj. (2214) Zürich

0.41 3.0 0.17 0.19 0.78 0.21 0.02 4.2

Hypertr. Oligotr. Oligotr. Eutr. Oligotr. Mesotr. Oligotr. Oligotr.

Roach, smelt, perch Pike, whitef., perch Minnow, trout – Pike, roach, perch Perch – –

Water Water – Water SPM Water Water Water

2 2 2 3 1 4 4 3

hypertrophic (TP = 60 μg/l), fallout from 2 to 130 kBq/m2 ; and this data base includes seven fish species. The crucial question is: What can be expected in terms of predictive success? To give some background to a realistic expectation, one can start with the situation as it was about ten years ago. At that time many models predicted concentrations of radionuclides and metals in fish with an uncertainty of about a factor of 10. The BIOMOVS-project (BIOsperic MOdel Validation Study; see, e.g., BIOMOVS, 1990) has been instrumental in disclosing poor ecosystem models, which is a key to improving model behaviour. The BIOMOVS-project demonstrated that many model predictions were about one order of magnitude above measured values, and that the uncertainty limits around this prediction and around those of the other models tested in this project were more than two orders of magnitude apart. It is an understatement to say that such predictions are uncertain. Many ecosystem modellers today probably argue that good models yield uncertainties of about a factor of 2. The VAMP-project (IAEA, 2000) has made a significant contribution to the evolution of predictive aquatic ecosystem modelling. The interesting question now concerns the results for this model for validations not just for Cs-concentrations in water but also for fish. Csconcentrations in fish are important because such concentrations represent biological aspects of radionuclide contamination of aquatic ecosystems, which Cs-concentrations in water do not.

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The validation of the radiocesium model presented in this section has been done in the following manner: (1) The validation lakes were selected to cover a wide domain (see Table 2). (2) Empirical data were first directly compared to modelled values according to the procedure illustrated in Fig. 8.

Fig. 8. Illustration of the relationship between empirical data and modelled values for total Cs-concentration in lake water in Lake Esthwaite Water, U.K. This figure gives the direct comparison on a time-scale. Month 1 is January, 1986; month 5 is the month of the Chernobyl fallout. This figure gives the corresponding regression using log-scale. It also gives the empirical uncertainties using a standard characteristic CV of 0.3 (or rather 2·SD ≈ 95% confidence interval for empirical values).

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Figure 8A gives the empirical data for Cs-concentration in lake water in Esthwaite Water and the corresponding modelled values. These data were then regressed against one another, as illustrated in Fig. 8B. The r 2 -value in this case was 0.70, the slope 0.67 and the number of samples (n) 13. This regression only covers one lake and, hence, has a limited range. How does the model predict for all the validation lakes? Also note that the modelled monthy data for the entire lake are not independent of one another and that there are great empirical uncertainties for Cs-concentrations in individual water samples (CV ≈ 0.3; see Håkanson, 1999). This is illustrated by the uncertainty marks in Fig. 8B. Data-series like the one given in Fig. 8B have been compiled for all the validation lakes. The results for the Cs-concentrations in lake water are given in Fig. 9A. The r 2 -value is 0.923. The figure gives the 95% confidence intervals for the mean and the individual Cs-concentrations and the regression line. The slope is 1.087. This regression is based on 61 data points covering a range from 0.001 to 4.5 Bq/l. This is, of course, an amazing result for a validation. It actually means that the model predicts as well as one can measure if one measures very well – under the given conditions and in the given domain of the model. Figure 9B gives the corresponding results for Cs-concentrations in lake fish. The dataset comprises 185 values for seven species of fish covering a very wide range, from 2 to over 30,000 Bq/kg ww. There are 3 typical outliers. If these three outliers are omitted, the r 2 -value is 0.98. This is almost like an analytical solution. If the 3 outliers are included, the r 2 -value is 0.95. The slope is almost perfect. The r 2 -value for the actual data is 0.86. This demonstrates, even proves, that the structuring of this model is very good. The model gives better predictions than one would often obtain from a few samples from standard lake monitoring programs. This is also a clear demonstration of the model structuring advocated in this chapter. If almost “perfect” predictions can be obtained for radiocesium for fish in lakes, this holds great promise for other types of contaminants for future models. It also means that models of this type in the future are likely to become widely used and excellent tools in remedial contexts. 3.4. Comments It would be tempting to conclude that this is the ultimate model for radiocesium in lakes, but such a conclusion would be premature. There are several empirical components in this model and there are always uncertainties associated with such empirical approaches, e.g., for the soil permeability factor, the biouptake delay factor and the biomagnification factor. It would be a step forward, and it would add generic strength to this model, if it were possible to include more mechanistically-based approaches for these empirical model components. The sub-model for the K-moderator is an example of such a mechanistically-based approach. This approach is derived from the Michaelis–Menten equation and the Nernst equation, and it provides the type of structure to this model that helps to explain the excellent predictive power.

4. River modelling: basic concepts and equations The migration of contaminants in rivers is controlled by three main processes:

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Fig. 9. (A) Validation of the radiocesium model for lake water. (B) Validation of the radiocesium model for lake fish.

(a) Diffusion due to chaotic movements of the dissolved pollutant within the water; (b) Transport caused by the water current; (c) Pollutant interaction with sediments and suspended matter.

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These processes are modelled by well-known equations. 4.1. Diffusion The pollutant flux, F (Bq m−2 s−1 ), due to the diffusion process is related to the concentration gradient as follows: F = −K ∗

∂C . ∂x

(52)

The constant K ∗ is called the diffusion coefficient (m2 s−1 ). C is the concentration (Bq m−3 ) and x is the co-ordinate (m). Equation (52) is known as Fick’s first law of diffusion. The sign “−” in equation (52) implies that the pollutant flux is always directed to the region of low contaminant concentration. In principle, the diffusion process is due to the thermal motion of the molecules. Nevertheless, dispersion of pollutant due to turbulent motion of water is modelled by equation (52) as well. The order of magnitude of coefficient K ∗ ranges, ordinarily, from 10−10 to 10−8 m2 s−1 (see, for instance, Wesley, 1974). Turbulent motion of water is responsible for apparently higher values of K ∗ (eddy diffusion). 4.2. Transport The pollutant flux due to the water transport is related to the concentration C and to the water velocity vw (m s−1 ) by the following equation: F = vw C.

(53)

Formula (53) was obtained by dividing the total amount of substance flowing, per unit time, through the surface S by the surface area (see Fig. 10). 4.3. Migration to bottom sediment The process of pollutant migration to bottom sediment is due to the sedimentation of suspended matter in water and to the radionuclide diffusion through the sediment. The process of interaction of dissolved radionuclides in water with suspended solids has been investigated by many researchers (for instance, Benes et al., 1992; Cremers & Henrion, 1984). The radionuclide absorption by suspended matter can be modelled according to the well-know “kd concept” (kd = partition coefficient “particulate form/dissolved form”) based on the hypothesis of a reversible quick equilibrium between the dissolved (Cw , Bq m−3 ) and the adsorbed phases (Cs , Bq kg−1 ) of radionuclide Cs = kd . Cw

(54)

Of course the above hypotheses are not generally and rigorously true for every contaminant substance. The equilibrium between the concentrations of the dissolved and the attached

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Fig. 10. Pictorial representation of toxic substance migration due to water transport.

phases is not instantaneously achieved. Moreover, the adsorption–desorption processes are not generally reversible. Many authors have discussed and analysed the dynamic behaviour of sorption–desorption processes of dissolved pollutants with sediments and suspended matter (Karickoff, 1986). Several models accounting for complex sorption–desorption kinetics and non-reversible interaction processes have been developed (Comans & Hockley, 1992). 4.3.1. Sedimentation The flux of radionuclide migrating to sediment due to the sedimentation is F = Sr Cs ,

(55)

where Sr is the net sedimentation rate (kg m−2 s−1 ) that accounts for the dynamic balance of the particle settling to bottom sediment and of the particle resuspension from sediment to water. From equation (54) it follows: F = Sr kd Cw . vs = Sr kd has the dimension of a velocity.

(56)

4.3.2. Diffusion through bottom sediment The migration of a radionuclide through sediments may be modelled by the diffusion equation (NCRP, 1984). Suppose that bottom sediment is sub-divided in a set of layers of thickness dx.

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Fig. 11. Pictorial representation of toxic substance migration through sediment.

From the balance of radionuclides in each layer we get (see Fig. 11): dxS

∂ ∂ ∂ (θ Cw (x, t) + kd ρCw (x, t)) = SK ∗ θ Cw (x + dx, t) − SK ∗ θ Cw (x, t) ∂t ∂x ∂x − λ dxS(θ Cw (x, t) + kd ρCw (x, t)), (57)

where K ∗ is the diffusion coefficient, Cw (x, t) is the concentration of the contaminant in sediment interstitial water at time t and at depth x from the sediment surface, ρ is the sediment density (kg m−3 ), θ is the sediment water content (m3 m−3 ) and λ is the radioactive decay constant. Equation (57) controls the time behaviour of the total amount of radionuclide in a sediment layer of thickness dx and surface area S. The first two terms on the right side of the above equation are the fluxes of radionuclide from and to the contiguous sediment layers. The last term accounts for radioactive decay.

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After simple algebraic calculations, supposing that ρ, θ and kd are independent of x, we get the well-known diffusion equation for a pollutant in porous media (dx tends to 0): ∂ ∂2 Cw (x, t) = K 2 Cw (x, t) − λCw (x, t). ∂t ∂x

(58)

K, the effective diffusion coefficient, is related to K ∗ by the following formula K=

K∗ . R

(59)

R (dimensionless) is the so called retardation factor: R =1+

Kd ρ . θ

(60)

If ρ, θ and kd depend on x, it is possible to obtain a more general equation by a similar assessment of the contaminant balance in the sediment layers. 4.4. The “omniscient model” The tools and the ingredients for structuring the model to predict the migration of radionuclide in rivers seem now all available. The model is at hand! We have “only” to get the values of the coefficients in the model equations. For instance, we need the values of ρ and θ as functions of the sediment layer depth. Moreover it is necessary to determine the values of kd as a function of the sediment characteristics, of the chemical characteristics of the river water, of the size of the suspended particle, etc. On the other hand, K ∗ in equation (52) is a function of the water turbulence, of the water depth (thermal stratification may influence K ∗ ), etc. We need also “some other” data relevant to the water fluxes, the water velocity as a function of x and a plethora of morphometric data of the water body. It is necessary to have estimates of the sedimentation rates. If we decide to account for complex sorption–desorption kinetics we need values of the parameters controlling these processes. Are all these data available? If the answer is not, we can try to develop suitable sub-models for predicting them. For instance, it is possible, in principle, to determine the sedimentation rates from the particle settling velocity calculated by Stokes formula. Such a formula requires evaluations of the radii and the specific weight of the particles and of the dynamic molecular viscosity. Obviously, some of the above parameters depend on the environmental conditions. Moreover, any formula is characterised by an intrinsic uncertainty as it is obtained for ideal cases that are very different from the real conditions of a water body. Therefore, we have to develop, repeatedly, further sub-models that require additional parameter values and whose results are affected by non negligible uncertainty. This traditional strategy for developing predictive models seems inappropriate when applied to complex environmental systems. Indeed, as anticipated in Section 2, it is based on the belief of a “pyramid” structure of the set of the natural phenomena. It is assumed that some fundamental processes, belonging at the top-vertex of the logical pyramid, may be modelled in terms of logical-mathematical primary principles from which all other natural processes may be derived. This modelling strategy is based on the assumption that, starting from the

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Fig. 12. The “pyramidal structure” of knowledge. Each “statement” has its own place and can be derived from few fundamental principles.

actual system to be modelled, it is possible to climb the “bottom-up pyramid” to reach a small set of fundamental equations that can be used to model the system itself. The following example illustrates such a procedure. Let us suppose that we have to calculate the velocity of water discharged through a small hole in the bottom of a large bucket. It is well known that this problem may be solved by a direct application of Bernoulli’s theorem (see Fig. 12). This theorem can be derived from the main principles of the mechanics through very familiar concepts and laws, such as mechanical work and continuity equation. Figure 12 shows the position of the Bernoulli’s theorem in the knowledge pyramid. It is interesting to notice that the calculated √ velocity is only a function of the acceleration of gravity (g) and of the bucket height (h): 2gh. The values of h and g are affected by very small uncertainties. Therefore the velocity of water through the hole can be accurately predicted. At a first sight, there appears little doubt that the development of reliable environmental models requires the totality of the processes occurring in the examined system. Such “omniscient” models have the merit of framing, in rational structures, the phenomena and their relationships. Unfortunately, this means that knowledge concerning a great deal of environmental parameters is needed, and these are often difficult to measure and evaluate. As a consequence, these models often cannot be used in practical circumstances. On the other hand, model complexity does not guarantee, necessarily, the accuracy of results (Håkanson, 1995; Håkanson et al., 1995). Model uncertainty can increase if more and more parameters are accounted for in the model (IAEA, 2000). Indeed, the overall model uncertainty increases as result of the contribution of non negligible uncertainties from a large number of parameters. Figure 13 shows an example of development of an environmental model by trying to climb the “knowledge pyramid”. The ratio “total radionuclide concentration in water/dissolved radionuclide concentration” may be related to kd and Wss (the weight of suspended matter per cubic metre of water) by a simple formula: f = 1 + kd wss . Wss and kd may be related to some environmental characteristics such as the size of suspended particles, the erosion, the sedimentation rate (this, indeed, controls the balance of the suspended matter). Once suitable sub-models

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Fig. 13. The effect of an attempt for developing an environmental model by climbing the “knowledge pyramid”. At each step the model becomes more and more complex and requires more and more parameters showing non negligible uncertainties.

have been obtained to predict f in terms of the above quantities, it is necessary to predict these quantities as functions of many other environmental characteristics such as the actual erosion mechanisms, the characteristics of the rocks in the catchment, the water regime, etc. This procedure may be repeated for an innumerable set of process “shells” around the target variable f . Therefore, as a result of this model structuring strategy we get ever more complicated models that are difficult to manage. We can conclude that it is practically impossible to find the exact position of “f ” in the knowledge pyramid. To assess if the effort of developing complex models is really justified in view of the uncertainty of the model results is a crucial point in structuring environmental models. Therefore, it is wiser to develop simplified models including some degree of “aggregation” by grouping system variables and parameters. For instance, f may be used as a primary, aggregated parameter in a model. Aggregation influences the uncertainty of the model output. An overaggregated model may ignore some details of importance for a reliable prediction of the behaviour of a system. Nevertheless, it is easier to manage aggregated models having simple structures especially for practical applications. In some cases, aggregation can take advantage of the mutual compensation of some processes occurring in the environment (“collective aggregation”; Monte, 1996a). Indeed, mutual compensation effects of different phenomena occurring in a system can lead to collective behaviours that are less variable, and consequently may be predicted with less uncertainty, than each single process. Systems that may be decomposed in sets of homogeneous sub-systems are the most obvious examples: indeed, the processes originating as “ensemble averages” over the class of these sub-systems show somewhat regular behaviour due to statistical effects (“statistical aggregation”). Competitive effects can reduce the variability of the migration parameters despite the large ranges of the environmental conditions. Examples of collective aggregated parameters will be discussed in Sections 4.5 and 4.6.

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4.5. Coupling the models: water transport and radionuclide migration from water column to sediment Models for predicting the behaviour of radionuclides in water bodies are based on the advection-diffusion equation (Onishi, 1994). According to the required degree of details of the results, the model solutions are supplied as averages over different time and spatial scales (Shukla, 1993; Zheleznyak et al., 1992, 1997). We focus our attention to an infinitesimal part of a watercourse and to the migration processes occurring within and through it. We consider the following components of the system: • • • • •

The water column; An upper sediment layer strongly interacting with water (“interface layer); An intermediate sediment layer below the “interface layer” (“bottom sediment“); A sink sediment layer below the “bottom sediment”; The catchment.

Figure 14 shows the system components and the fluxes of radionuclide which are due to the following migration processes: • • • • •

Sedimentation; Radionuclide migration from water to sediment; Radionuclide migration from sediment to water (resuspension); Radionuclide migration from catchment; Radionuclide transport.

The total amount, T (Bq), of radionuclide in water (dissolved form), in suspended matter and in the sediment interface layer is (a list of some of the symbols used in the present chapter is available in Table 3): T = Cw l(x)h(x)x + Cw kd wsm l(x)h(x)x + Cw kd Dil δl(x)x.

(61)

x is the length of the infinitesimal element of the watercourse. Equation (61) may be written in the following form T = Cw [1 + kd wsm + kd Dil δ/ h(x)]h(x)l(x)x.

(62)

heff (x) = h(x) + h (x),

(63)

Using

where: h (x) = kd wsm h(x) + kd Dil δ,

(64)

we get T = Cw heff (x)l(x)x.

(65)

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Fig. 14. Fluxes of radionuclide within an infinitesimal section of a water body.

The time variation obeys the partial differential equations (Fig. 14) when x tends to 0  R(x, t) 1 vCw (x, t) ∂Cw (x, t) ∂ =− CT (x, t)(x, t) + − ∂t l(x)heff (x) ∂x l(x)heff heff −

vs Cw (x, t) CT (x, t)P (x, t) Ksw Dep (x, t) − + − λCw (x, t), heff l(x)heff heff

(66)

∂Dep (x, t) = vs Cw (x, t) + vCw (x, t) − Ksw Dep (x, t) − Kds Dep (x, t) ∂t − λDep (x, t),

(67)

CT (x, t) = f Cw (x, t).

(68)

The radionuclide flux from water to sediment due to diffusion processes has been calculated as v (the radionuclide migration velocity) multiplied by the concentration of dissolved radionu-

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Table 3 List of symbols Symbol

Definition

Units

CT (x, t) Cw (x, t)

concentration of radionuclide (total = dissolved + particulate form) in water concentration of dissolved radionuclide in water or attached to soil particles (Section 4.6) concentration of radionuclide in suspended matter radionuclide deposit in the bottom sediment thickness of sediment “interface layer” ratio “total radionuclide concentration/dissolved radionuclide” depth of the water body effective depth of the water body = water body depth + incremental depth incremental depth width of the water body partition coefficient “particulate form/dissolved form” radionuclide migration rate from sediment to water radionuclide migration rate from bottom sediment to deep sediment water withdrawal rate per unit length of water course radionuclide flux from the catchment per water body unit length time Total amount of radionuclide in water and sediment interface layer radionuclide migration velocity from water to sediment (diffusion) radionuclide sedimentation velocity distance of the observation point from the origin weight of suspended matter per unit volume of water density of sediment interface layer radioactive decay constant water flow

Bq m−3 Bq m−3

Cs Dep (x, t) Dil f h(x) heff (x) h (ξ ) l(x) kd Ksw Kds P (x, t) R(x, t) t T v vs x wsm δ λ (x, t)

Bq kg−1 Bq m−2 m dimensionless m m m m m3 kg−1 s−1 s−1 m3 s−1 m−1 Bq s−1 m−1 s Bq m s−1 m s−1 m kg m−3 kg m−3 s−1 m3 s−1

clides in water. The fluxes of radionuclide from sediment to water and from bottom sediment to deep sediment were calculated, respectively, as the migration rates Ksw and Kds multiplied by the radionuclide deposit in bottom sediment. Equation (67) can be obtained by the discretisation of the diffusion equation controlling the radionuclide migration through the sediment and from the water column to the sediment. The interface layer was introduced to account for the boundary condition of such an equation. The concentration in the interstitial water of the upper layer of sediment is supposed to be approximately equal to the concentration in the water column above the sediment. Obviously, the discretisation reflects on the spatial and time resolution of the model. The total amount of a toxic substance per square metre in the sediment (total deposit) can be evaluated integrating the concentration of the pollutant (dissolved +attached form) per cubic metre of sediment over the sediment depth. We will assess the total deposit in the ideal case of a simple two-phase system composed of a water column and of bottom sediment. Suppose that the concentration of the toxic substance in water is approximately constant during the period of observation

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(that is, h is very large) the concentration of radionuclide in interstitial water at depth x at instant t is, approximately,   x Cw (x, t) = C0 erfc √ . 2 Kt

(69)

The above equation has been obtained by solving equation (58). The effect due to radioactive decay can be included in the previous equations by multiplying the right side by e−λt . C0 is the concentration of radionuclide in the water column. Equation (69) is an exact solution of equation (58). If kd ρ θ (this hypothesis is true in case of toxic substances that, like 137 Cs, are characterised by a high value of kd ), the total amount of pollutant in sediment is 2C0 T (t) = √ S K ∗ ρθ kd t, π

(70)

where S (m2 ) is the surface of the water body. The balance of the contaminant in the water body implies SC0 h + T (t) = P S.

(71)

P is the amount of pollutant deposited per square metre (for instance in g m−2 or, in case of radioactive substances, in Bq m−2 ). After simple algebraic calculations we get C0 =

P h+

T (t ) SC0

.

(72)

√ hal = T (t)/SC0 = √2π K ∗ ρθ kd t has the dimension of a length. It corresponds to the term kd Dil δ in formula (64) and accounts for the effects of radionuclide sorption on the sediment interface layer during a time interval t. Using kd = 15 m3 kg−1 (a typical value for 137 Cs in water bodies), θ = 0.5, K ∗ = 10−9 m2 s−1 , ρ = 1 kg dm−3 , t = 30 days, we get T (t)/SC0 ≈ 5 m. As kd increases by factors of 10, 100, 1000 the above parameter increases by factors 3, 10, 32, etc. Therefore a large uncertainty of kd does not reflect proportionally on the values of the incremental depth when wsm is negligible. hal is an example of aggregated parameters showing slight variability when the environmental conditions change (sediment characteristics that reflect on kd , θ and ρ) (Monte, 1996a). The above equations have been obtained on the basis of some specific hypotheses. The definition and the use of hal have heuristic foundation. When equation (66) is numerically solved by subdividing the watercourse in “boxes” (discretisation) the value of hal depends on the transit time of water in each box. If t is of the order of few days or few hours, hal is, in principle, negligible for many radionuclides depending on the value of kd . Nevertheless, generally, turbulent motions of water perturb bottom sediment and enhance the sorption of radionuclides on the sediment interface layer. As consequence, the actual values of hal are larger than the ones calculated by the previous formula. The use of hal was somewhat useful in predicting the initial average concentrations of 137 Cs in some European lakes contaminated following the Chernobyl accident (Monte, 1995a).

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Obviously, the subdivision of the sediment in three layers (interface, bottom and deep sediments) corresponds to a particular discretisation of the diffusion equation (58). The time resolving power of a model (TRP) is the interval of time necessary to assure the damping of some transient processes that are not intended to be modelled in a sufficiently detailed time scale. For instance, the model is based on the approximation of a homogeneous distribution of the contaminant in water, a condition that is not immediately approached but depends on the dimension of the model “boxes” used for the discretisation of equation (66). Moreover, the time constants of the sorption–desorption processes of pollutants on sediment may be of the order of months whereas, the model predictions are simulated on the hypothesis of an “instantaneous” equilibrium between the dissolved and the particulate radionuclide phases. The model is designed to evaluate estimates of the experimental data averaged on an interval of time equal to the TRP. TRP is, therefore, an important characteristic of a model in relation to the output reliability. The performance of a model output is more reliable when TRP is close to an optimal value that depends on the hypotheses and the structure of the model. If fine time and spatial resolutions of the model are required, a diffusion term must be added to the right hand side of equation (66). On the contrary, when results averaged on a suitable time and spatial scales are sufficient, the diffusion term may be neglected. In case of a point source contamination, this condition is accomplished when the distance of the point 2 of observation from the source is larger than 3 Lh where L and h are, respectively, the width and the depth of the watercourse (Sayre, 1973). At any rate, the generalisation of equation (66) is matter of mere mathematics. Considering the diffusion process (52), the three-dimensional equation may be obtained assessing the contaminant balance in a lattice composed of infinitesimal systems similar to the one depicted in Fig. 14. The model we have described is not aimed to reach such a fine level of resolution. It may be used to get values averaged over suitable spatial and time scales. 4.6. Migration through catchments The assessment of radionuclides migrating from catchments to rivers is of paramount importance to develop predictive models for watercourses. The present Section describes an example of a collective approach to model such a complex migration problem. In the past years a variety of models, based on the most important processes affecting the transport of pollutants in drainage areas, were developed to estimate the migration of toxic substances from catchments of fresh water bodies (Jolánkai, 1983; Shukla, 1993; BIOMOVS II, 1996; Konoplev et al., 1999). The flux of radionuclides migrating from a catchment basin may be modelled by the “Transfer function” (TF) methodology (Monte, 1998a). The TF is defined as the amount of radionuclide flowing per unit time from a drainage basin to a water body following a single pulse deposition of radioactive substances on the catchment (impulse response). TF is a function of time and is related to the radionuclide deposition per square metre. The TF can be approximated by a sum of some exponential components (Monte, 1995b): r (t) = εD

 i

 Ai αi exp −(λ + λi )t ,

(73)

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where r (t) (Bq s−1 ) is the TF, ε is the transfer coefficient from the catchment (m−1 ), D is the radionuclide pulse deposition per square metre (Bq m−2 ),  is the water flow from the catchment (m−3 s−1 ), λ is the radioactive decay constant (s−1 ), Ai and λ + λi are, respectively, the relative weight and the effective decay constants of the ith exponential component in the  TF. The sum of the relative weights of the exponential components must be 1: i Ai = 1. The mathematical form and the parameter values of TF were obtained by fitting experimental data of radionuclide concentrations in rivers contaminated following the Chernobyl accident. The real exponents αi give a reason for possible non-linearity of TF as a function of water flow. The analysis of data obtained from some European rivers (Monte, 1996a) suggests that the values of exponent α for the medium term component (1–6 years from the pulse deposition) range from 0.53 to 1.02 (geometric mean = 0.86) and from 1.12 to 1.41 (geometric mean = 1.3) for 137 Cs and 90 Sr respectively. A value αi = 1 is used when the model is aimed to predict average yearly concentrations of radionuclides in water. If the deposition D(t) is a function of time (Bq m−2 s−1 ), the radionuclide flux from the sub-catchment may be calculated by the convolution integral: r (t) = ε

 i



t



 D(τ )Ai exp −(λ + λi )(t − τ ) dτ .

(74)

0

From simple mathematical calculations it follows that r (t) = ε



Si ,

(75)

i

where dSi = −(λ + λi )Si + Ai D(t). dt

(76)

Therefore, it is possible to calculate the radionuclide contribution from a catchment following a time dependent radionuclide deposition by a simple system of first order differential equations. The methodology is described, in detail in Monte (1997). Si may be regarded as the inventory (Bq m−2 ) of radioactive substance in the radionuclide storage compartments. These last may schematically represent the various soil layers and the vegetation cover in the sub-catchment. Experimental estimates of parameters in TF are reported in the international literature (Hilton et al., 1993; Santschi et al., 1990; Sundblad et al., 1991; Monte, 1996b). Some applications of the above sub-model to Italian contaminated rivers following 90 Sr deposition due to the nuclear weapon tests in atmosphere are described in the literature (Monte, 1997). There are many examples of parameters whose ranges of variability are lower than the ones of the primary parameters characterising the properties of the specific aquatic system analysed. These “collective” parameters are characterised by reduced variability when the environmental conditions change. Indeed, some phenomena controlling a process in the environment seem to compensate each other reducing the range of variability of the process itself. The migration of radionuclides through a catchment is an example of complex process that can be modelled by statistical “collective” aggregation.

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Let L be an elementary component of a catchment. Suppose that the soil in L shows homogeneous characteristics and, moreover, that the contaminant migrating through the catchment due to the surface water transport (surface runoff) is, prevailingly, confined to the upper layer of soil of thickness ζ . Such a hypothesis is true when the contaminant is characterised by high values of the partition coefficient (kd ). We define:  I (t) s ρ Cw Cs ω

the area of L (m2 ); the radionuclide inventory, at time t, in the soil upper layer of thickness ζ (Bq m−2 ); the average water flux per square metre flowing through the catchment (m3 s−1 m−2 ); the soil density (kg m−3 ); the concentration of the contaminant in the interstitial water (Bq m−3 or g m−3 ); the concentration of the contaminant attached to the soil particles (Bq kg−1 or g kg−1 ); the volumetric content of water in L (m3 m−3 ).

The inventory may be calculated as function of the concentration in the interstitial water: I (t) = Cw ζ ω + Cs ζρ = Cw ζ (ω + ρkd ).

(77)

The contaminant flux, GF (Bq s−1 ), from the element L is GF = s Cw .

(78)

The inventory is controlled by the following first order differential equation: dI (t) I (t) = −s . dt ζ (ω + ρkd )

(79)

If kd ρ ω and I (0) is the initial inventory, which corresponds to the initial deposition, the contaminant flux from L is:   s s I (0) exp − t . (80) GF(t) = ζρkd ζρkd Putting ξ =

ζ s

GF(t) =

we get   t I (0) exp − . ξρkd ξρkd

(81)

Equation (81) is similar to the formula used by Joshi & Shukla (1991). A large catchment is composed of an “ensemble” of sub-catchments characterised by different values of ξ, ρ and kd . If A is the area of the catchment and f (ξρkd ) is the distribution function of ξρkd in the catchment, the total flux of the contaminant becomes  GF(t) = AI (0) 0

∞

  f (ξρkd ) t d(ξρkd ). exp − ξρkd ξρkd

(82)

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The previous formula shows that the GF is composed of an infinite number of infinitesimal exponential terms. Putting ξρkd = θ1 , equation (82) may be written as follows:  GF(t) = A

∞

0

  1 1 f exp(−tθ ) dθ. θ θ

(83)

The (modelled) effective removal rate, λER (t) of the toxic substance at time t is: λER (t) = −

1 dGF(t) . GF(t) dt

(84)

After easy calculations and an integration by parts we get, from formulae (83) and (84): ∞ 1 1 0 θg  (θ ) exp(−tθ ) dθ λER (t) = + ∞ , t t 0 g(θ ) exp(−tθ ) dθ

(85)

where g(θ ) = θ1 f ( θ1 ). The second term in the right hand side of the previous equation may scarcely affect the order of magnitude of λ, due to the presence of g  (θ ) in the integral on the numerator. This means that both positive and negative values may mutually compensate when the function is integrated over (0, ∞), whereas g(θ ) is a non-negative function. In the present paper the calculations of λER (t) were carried out by supposing that kd values show log-normal distributions (Sheppard & Thibault, 1990):   1 1 1 exp − 2 (ln kd − μ)2 . f (kd ) = √ 2σ σ 2π kd

(86)

From distribution (86) it follows, after simple calculations: λER (t) =

ϕ(t) , t

(87)

where ∞   (ln θ+μ∗ )2 ) ln θ exp(−θ t) dθ 1 0 exp(− ∗ 2σ 2 ϕ(t) = 1 − 2 μ + . ∗ 2 ∞ σ exp(− (ln θ+μ ) ) exp(−θ t) dθ 0

(88)

2σ 2

μ∗ is the mean value of ln ξρkd . Formula (88) shows that ϕ(t) is close to 1 when σ is very large (limσ →∞ ϕ(t) = 1). Of course it is possible to account for the radioactive decay (or for any other first order degradation process) by adding to the right hand side of equation (79) the term −λI (t) where λ is the radioactive decay constant.

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Fig. 15. ϕ(t) is, approximately, a linear function of lnt. ϕ(t) is scarcely sensitive to the variation of the environmental characteristics related to the interaction of radionuclide with soil in catchment (σ ). The function is, moreover, slightly variable (in the figure t ranges from 0.1 to 50 years).

Function (88) has been evaluated for 137 Cs and 90 Sr using the experimental log-normal distribution of kd in organic soils from Sheppard & Thibault (1990): Cs: μ = 5.6, σ = 3.6; Sr: μ = 5, σ = 1.8. μ and σ are, respectively, the mean value and the standard deviation of the distribution of the natural logarithm of kd measured in litre kg−1 . The simulation results have been obtained using ξρ ≈ 0.1 kg litre−1 year. Such a value corresponds, for instance, to ρ ≈ 1.3 kg litre−1 , ζ ≈ 0.04 m and  ≈ 0.5 m year−1 . For the above radionuclides (88) is, approximately, a linear function of the natural logarithm of time ϕ(t) = a ln t + b. Figure 15 shows ϕ(t) as a function of ln t for some values of σ . The following values of a and b were calculated: Cs: a = 0.069, b = 0.808; Sr: a = 0.22, b = 0.505. The value of the removal rate of Cs is broadly comparable to the removal rate of Sr although the kd distributions of these two elements are very different. The previous numerical evaluations show that ϕ(t) is almost independent of μ∗ and σ . As consequence λER (t) is, approximately, a function of t. The radionuclide removal rate λER (t) is, therefore, slightly variable when the catchment characteristics change. Function (73) is an approximation of GF. Values of λER (t) (formula (87)) averaged over suitable time intervals correspond to λi in formula (73).

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4.7. Some examples of model validation The comparison of the model predictions with experimental data sets (model validation) is a crucial test for supporting the credibility of a model (Davis et al., 1999; Peterson & Kirchner, 1998). Some examples of validation of the model here described are reported in Figs 16 and 17. Figure 16 shows the application of the model to river Marta (approximate length = 50 km), the outlet of lake Bolsena a volcanic lake in central Italy. Lake Bolsena (surface 113.6 km2 , average depth 81 m, maximum depth 151 m) has a very long mean water retention time (120.6 years). The lake was contaminated by 137 Cs from the Chernobyl accident. The figure reports the 137 Cs concentrations in the outlet water 6–8 years after the Chernobyl accident at various distances from the lake. Figure 17 shows the application of the model to river Tiber (Tevere), the largest river in central Italy. The length and the area of drainage basin of the river are, respectively, 396 km and 17 156 km2 . The river and its catchment were contaminated by the deposition of 90 Sr due to the nuclear explosions in atmosphere of past decades. The predictions have been obtained using yearly average values of the water flows through the water bodies. The model is aimed at simulating the yearly averages of the radionuclide concentrations in water. The values of the parameters used for the above simulations are reported in Table 4.

Fig. 16. Comparison of the model predictions with experimental data of 137 Cs concentration in river Marta.

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Fig. 17. Comparison of the model predictions with experimental data of 90 Sr concentration in river Tiber.

Table 4 Generic values of model parameters Parameter

Description

Units

90 Sr

137 Cs

h

Parameter accounting for the quick interaction of radionuclide in water with sediment upper layer Transfer coefficient from catchment. Weight of the first component of the transfer function from the catchment. Weight of the second component of the transfer function from the catchment. Decay constant of the first component of the transfer function from the catchment. Decay constant of the first component of the transfer function from the catchment. Decay constant of the first component of the transfer function from the catchment. Migration velocity to sediment due to diffusion Migration velocity to sediment due to sedimentation Migration rate from sediment Migration rate to deep sediment

m

0

≈6

m−1 dimensionless

0.2 0.935

0.2 0.96

dimensionless

0.045

0.04

s−1

2.3 × 10−7

2.3 × 10−7

s−1

4.2 × 10−9

1.5 × 10−8

s−1

4.2 × 10−10

–

m s−1 m s−1

3.5 × 10−7 –

1 × 10−6 1.2 × 10−7

s−1 s−1

3.0 × 10−8 –

1.5 × 10−8 1.2 × 10−8

ε A1 A2 λ1 λ2 λ3 v vs Kws Kds

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4.8. Sensitivity and uncertainty analyses of the model The relative variation of radionuclide concentration in water Cw due to a variation μi of a parameters μi can be calculated as follows Cw  μi ≈ Si , Cw μi

(89)

where Si =

μi ∂Cw Cw ∂μi

(90)

is the sensitivity coefficient of Cw to μi . Equation (89) is valid for small variations of parameters μi . The above formulae show that, if Si  1 and the variability of a parameter is small (as in the case of a collective parameter), the model may successfully predict the radionuclide concentration in different water bodies although a generic value of such a parameter is used. Indeed, the small relative difference (μ/μ) between the “generic” value of the parameter used in the model and the relevant unknown “site specific” value is poorly reflected on the radionuclide concentration in water (Si  1 in formula (89)). According to definition (90), an extensive sensitivity analysis of a model requires the evaluation of a large number of time functions. Moreover, functions Si depend on the water body characteristics and on the contamination dynamics. Therefore, it is more realistic to appraise the effects of the variation of a parameter by analysing the model results obtained for different values of the parameter itself. Among the various parameters of the model, the radionuclide sedimentation velocity plays an important role in watercourses. We analyse now the sensitivity of the model output to such a parameter for a pulse release of contaminant into river water. We will assume, for simplicity, that the radionuclide interaction with sediment is negligiK Dep (x,t ) in ble. Therefore the radionuclide resuspension from bottom sediment (the term sw heff equation (66)) is set equal to 0 and h = heff . Equation (66) becomes (the following equations are valid for both CT and Cw )  (x, t) ∂ ∂C(x, t) =− C(x, t) − g(x)C(x, t), ∂t l(x)h(x) ∂x

(91)

where g(x) is g(x) =

 (v + vs ) 1 ∂ + (x, t) + λ. f h(x) l(x)h(x) ∂x

(92)

The pulse moves downstream with velocity vc =

(x, t) . l(x)h(x)

Let suppose, for convenience, that such a velocity is constant.

(93)

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We get  ∂ ∂C(x, t) = −vc C(x, t) − g(x)C(x, t). ∂t ∂x

(94)

Therefore   t  C(x, t) = C0 δ(x − vc t)exp − g(vc ξ ) dξ

(95)

0

as the pulse reaches a distance x from the source at time t = x/vc .

(96)

For a continuous contaminant release into a river from a point source, equation (91) becomes (steady state conditions): g(x) ∂C(x) =− C(x). ∂x vc

(97)

Thence    1 x g(ξ ) dξ . C(x, t) = C0 exp − vc 0 As a consequence, in both cases, the sensitivity is    x 1 1 vs vs ∂C dξ . − = S= C ∂vs f vc 0 h(ξ )

(98)

(99)

The absolute value of S is very large when x (the distance of the point of observation from the contamination source) is large and the velocity of the water is small. Therefore, radionuclide concentration in water at points far from the contamination source is very sensitive to the sedimentation velocity. To obtain reliable model results, site specific values of the radionuclide sedimentation velocity are required when the above conditions occur (large distance from the source, small value for the water velocity). The sensitivity analysis of the model output to the generic parameters is of importance to assess the model uncertainty. Uncertainty and sensitivity analyses are somewhat related. Indeed, if μi are the uncertainty of the model parameters, the uncertainty Cw of the radionuclide concentration in water may be calculated from the sensitivity matrix Si according to formula (89). A different approach for the uncertainty analysis of a model was described in scientific literature (Monte et al., 1996; Monte, 1997, 1998b). We will briefly summarise the principles of such an approach. The uncertainty level may be obtained from the statistical analysis of the ratios “Measured value/Predicted value” for simulations relevant to various aquatic systems and contamination

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scenarios. This methodology is based on the direct comparison of the predicted results with the experimental data to evaluate the probability that a prediction is within a specific interval around an experimental observation. Such a probability is derived from statistical measures (geometric mean and standard deviation) of the frequency distribution of the calculated ratios “Measured value/Predicted value”. The described methodology for the assessment of the model uncertainty (EBUA, Empirically Based Uncertainty Analysis) is conceptually different from the traditional uncertainty analysis technique. This last evaluates the uncertainties of model output from the assumed statistical distributions of the model parameters. Traditional uncertainty analysis is, therefore, an “a priori” technique of computation. It, indeed, is based on the hypothesis that the statistical distributions of the parameter values are known. On the contrary, the empirically based uncertainty analysis is based on the comparison of the model output with independent experimental data sets. As previously stated, the probability of predicting an experimental value, within a certain confidence range, is evaluated by means of statistical techniques applied to the ratios “measured values/model predictions”. The technique does not require an “a priori” knowledge of the statistical distributions of the model parameters and, moreover, implicitly accounts for the effects of the uncertainty caused by the insufficiency of the model structure. EBUA estimates the “overall” uncertainty of the model. For the model here described the uncertainty of the output at the 68% confidence level is a factor of 2 when site specific values were used for the radionuclide sedimentation velocity. 5. Conclusions Lakes and rivers are extremely complex. Every aquatic system is unique, and yet it is impossible to study each in the detail necessary for case-by-case assessment of ecological threats, and proposals for remedial measures. In this situation, quantitative models are essential to predict, guide assessment and direct intervention. A decade ago ecosystem models were rightly regarded with suspicion. Then, many lake models predicted, e.g., concentrations of toxins in fish within a factor of 10. Today, lake models can predict such targets as well as one can measure, within a factor of 0.25 to 0.5. One reason for this is, in fact, the Chernobyl accident. Large quantities of 137Cs were then released as a pulse. To follow that signal through ecosystems has improved the possibilities to identify fundamental transport processes and, hence, to structure models. The predictive power of a model is not governed by the strength of the model’s strongest part, but by the weakness of its weakest part. • In the future, it would be very interesting for example to try to find general equations for the settling velocity for different categories of carrier particles, like humic matter, detritus and clays. • In the future, it would also be interesting to find better general equations for one of the most fundamental model components, the particulate fraction, for key substances in lakes and rivers. This value regulates the fraction of X in particulate form, which by definition is the only fraction that can settle out, and the dissolved form. This value is important not only for abiotic transport routes, it also regulates benthic and pelagic pathways. • It would also be very important for science and society to have access to decision support systems (DSS) incorporating validated models applicable over a wide domain of ecosys-

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tems, and including possibilities to make multi-attribute analyses (MAA) so that realistic remedial scenarios can be simulated and the outcome evaluated in a holistic and rational manner from environmental, social and economical perspectives.

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Håkanson, L., Brittain, J., Monte, L., Heling, R., Bergström, U., Suolanen, V. & Sjoeblom, K.-L. (1995). Modelling of Radiocaesium in lakes. Results from the VAMP project. Proceedings of an International Symposium on Environmental Impact of Radioactive Releases organised by the International Atomic Energy Agency. Held in Vienna, Austria, 8–12 May 1995 (pp. 337–353). Håkanson, L., Brittain, J., Monte, L., Heling, R., Bergström, U. & Suolanen, V. (1996). Modelling of radiocesium in lakes – the VAMP-model. J. Environ. Radioactivity, 33, 255–308. Hamby, D. M. (1995). A comparison of sensitivity analysis techniques. Health Physics, 68, 195–204. Hilton, J. (1997). Aquatic radioecology post Chernobyl – a review of the past and a look to the future. In Desmet, G. et al. (Eds), Freshwater and Estuarine Radioecology (pp. 47–73). Amsterdam: Elsevier. Hilton, J., Livens, F. R., Spezzano, P. & Leonard, D. R. P. (1993). Retention of radioactive caesium by different soils in the catchment of a small lake. The Science of the Total Environment, 129, 253–266. Hinton, T. G. (1993). Sensitivity analysis of ecosys-87: an emphasis on the ingestion pathway as a function of radionuclide and type of disposition. Health Physics, 66, 513–531. IAEA (1988). Assessing the impact of deep sea disposal of low level radioactive waste on living marine resources. Tech. Rep. No. 288. Vienna: International Atomic Energy Agency. IAEA (2000). Modelling of the transfer of radiocaesium from deposition to lake ecosystems. Report of the VAMP Aquatic Working Group (chairman: Lars Håkanson). IAEA-TECDOC-1143 (343 p.). Vienna: International Atomic Energy Agency. ICRP (1977). International Commission of Radiological Protection. Recommendations of the International Commission of Radiological Protection, Publication 26. Oxford: Pergamon Press. Jimenez, F. & Gallego, E. (1998). Effects of radiation on aquatic organisms. Preprint, from the MOIRA project (9 p.). Spain: Univ. de Politech. de Madrid. Jolánkai, G. (1983). Modelling of non-point source pollution. In S. E. Jørgensen (Ed.), Application of Ecological Modelling in Environmental Management, Part A (pp. 283–385). Amsterdam: Elsevier. Jonsson, A. (1997). Whole lake metabolism of allochthonous organic material and the limiting nutrient concept in Lake Örträsket, a large humic lake in northern Sweden. ISBN 91-7191-382-3. Sweden: Diss. Dept. of Physical Geography, Univ. Univ. Joshi, S. R. & Shukla, B. S. (1991). The role of water/soil distribution coefficient in the watershed transport of environmental radionuclides. Earth Planetary Sci. Lett., 105, 314–318. Karickoff, S. W. (1986). Pollutant sorption in environmental systems. In W. Brock Nelly & G. E. Blau (Eds), Environmental Exposure from Chemicals, Vol. I (pp. 49–64). Konoplev, A., Bulgakov, A., Hilton, J., Comans, R. & Popov, V. (1997). Long-term kinetics of radiocesium fixation by soils. In G. Desmet et al. (Ed.), Freshwater and Estuarine Radioecology (pp. 173–182). Amsterdam: Elsevier. Konoplev, A. V., Bulgakov, A. A., Hoffman, F. O., Kanyár, B., Lyashenko, G., Nair, S. K., Popov, A., Raskob, W., Thiessen, K. M., Watkins, B. & Zheleznyak, M. (1999). Validation of models of radionuclide wash-off from contaminated watersheds using Chernobyl data. J. Environ. Radioactivity, 42, 131–141. Lindström, M., Håkanson, L., Abrahamsson, O. & Johansson, H. (1999). An empirical model for prediction of lake water suspended matter. Ecol. Modelling, 121, 185–198. Monte, L. (1995a). A simple formula to predict approximate initial contamination of lake water following a pulse deposition of radionuclide. Health Physics, 68 (3). Monte, L. (1995b). Evaluation of radionuclide transfer functions from drainage basins of fresh water systems. J. Environ. Radioactivity, 26, 71–82. Monte, L. (1996a). Collective models in environmental sciences. The Science of the Total Environment, 192, 41–47. Monte, L. (1996b). Analysis of models assessing the radionuclide migration from catchments to water bodies. Health Physics, 70, 227–237. Monte, L. (1997). A collective model for predicting the long-term behaviour of radionuclides in rivers. The Science of the Total Environment, 201, 17–29. Monte, L. (1998a). Predicting the migration of dissolved toxic substances from catchments by a collective model. Ecological Modelling, 110, 269–279. Monte, L. (1998b). Predicting the long term behaviour of 90 Sr in lacustrine systems by a collective model. Ecological Modelling, 106, 141–159. Monte, L., Håkanson, L., Bergström, U., Brittain, J. & Heling, R. (1996). Uncertainty analysis and validation of environmental models: the empirically based uncertainty analysis. Ecological Modelling, 91, 139–152.

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Monte, L., Håkanson, L. & Brittain, J. (1997). Prototype models for the MOIRA computerised system. ISSN/11205555 (90 p.). Roma: ENEA. NCRP (1984). Radiological assessment: predicting the transport, bioaccumulation, and uptake by man of radionuclide released to the environment. Recommendations of the National Council on Radiation Protection and Measurements. Report No. 76 (299 p.). Bethesda, MD: NRPB. Onishi, Y. (1994). Contaminant Transport Modeling in Surface Water. In M. H. Chaudrhy & L. W. Mays (Eds), ‘Computer Modeling of Free-Surface and Pressurised Flows’, NATO ASI Series E, Applied Sciences, Vol. 274 (pp. 313–341). The Netherlands: Klumer Academia Publishers. Ottosson, F. & Abrahamsson, O. (1998). Presentation and analysis of a model simulating epilimnetic and hypolimnetic temperatures in lakes. Ecol. Modelling, 110, 223–253. Peterson, S. R. & Kirchner, T. B. (1998). Data quality and validation of radiological assessment models. Health Physics, 74 (2), 147–157. Peters, R. H. (1991). A Critique for Ecology (366 p.). Cambridge: Cambridge Univ. Press. Rowan, D. J. & Rasmussen, J. B. (1994a). Bioaccumulation of radiocesium by fish: the influence of physicochemical factors and trophic structure. Can. J. Fish. Aquat. Sci., 51, 2388–2410. Rowan, D. J. & Rasmussen, J. B. (1995). The elimination of radiocesium from fish. J. Appl. Ecology., 32, 739–744. Santschi, P. H., Bollhander, S., Zingg, S., Luck, A. & Farrenkothen, K. (1990). The self-cleaning capacity of surface waters after radioactive fallout. Evidence from European waters after Chernobyl, 1986–1988. Environ. Sci. Technol., 24, 519–527. Sayre, N. W. (1973). Natural mixing processes in rivers. In H. W. Shen (Ed.), Environmental Impact on Rivers. Colorado: Fort Collins. Sheppard, M. I. & Thibault, D. H. (1990). Default soil solid/liquid partition coefficients, kds, for four major soil types: a compendium. Health Physiscs, 59, 471–482. Shukla, B. S. (1993). Watershed, River and Lake Modelling through Environmental Radioactivity (225 p.). Canada: Environmental Research & Publications Inc. Hamilton, Ontario. Smith, J. T., Leonard, D. R. P., Hilton, J. & Appleby, P. G. (1997). Towards a generalized model for the primary and secondary contamination of lakes by Chernobyl-driven radiocesium. Health Physics, 72, 880–892. Sundblad, B., Bergström, U. & Sverker, E. (1991). Long term transfer of fallout nuclides from the terrestrial to the aquatic environment. In L. Moberg (Ed.), The Chernobyl fallout in Sweden. Results from a research program on environmental radiology (pp. 207–238). Stockholm: The Swedish Radiation Protection Institute. Vollenweider, R. A. (1968). The scientific basis of lake eutrophication, with particular reference to phosphorus and nitrogen as eutrophication factors. Tech. Rep. DAS/DSI/68.27 (159 p.). Paris: OECD. Wesley, J. P. (1974). Ecophysics – The Application of Physics to Ecology (340 p.). Springfield, Illinois, USA: Charles C Thomas Publisher. Whicker, F. W. & Schultz, V. (1982). Radioecology: Nuclear Energy and the Environment, Vol. 1 (228 p.). Boca Raton: CRC Press. Zheleznyak, M., Demchenko, R., Khursin, S., Kuzmenko, Yu., Tkalich, P. & Vitjuk, N. (1992). Mathematical modelling of radionuclide dispersion in the Prypiat-Dnieper aquatic system after the Chernobyl accident. Sci. Total Environ., 112, 89–114. Zheleznyak, M., Shepeleva, T., Sizonenko, V. & Mezhueva, I. (1997). Simulation of countermeasures to diminish radionuclide fluxes from the Chernobyl zone via aquatic pathways. In U. Bäverstam, G. Fraser & G. N. Kelly (Eds), Decision making support for off-site emergency management. Proceedings of the Fourth International Workshop, Aronsborg, Sweden, 1997. Radiation Protection Dosimetry, 73 (1–4) 181–186.

MODELLING RADIOACTIVITY IN THE ENVIRONMENT E. Marian Scott (Editor) © 2003 Elsevier Science Ltd. All rights reserved

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Chapter 7

Radioactivity in aquatic biota Tatiana G. Sazykina∗ Scientific & Production Association “Typhoon”, 82 Lenin Ave., Obninsk, Kaluga Region, 249020 Russia

1. Experimental ground for radioecological models 1.1. The differentiation between the chemical composition of aquatic biota and aquatic media Aquatic radioecology as a science of the radionuclide migration in aquatic ecosystems was started in 1920s with the discovery of the high accumulation of some natural radionuclides (226 Ra, 238 U) by aquatic organisms (Vernadsky, 1929a, b). During subsequent years, the environmental behavior of numerous natural and artificial radionuclides was studied; at present a great amount of data are available on the radionuclide accumulation by different species of aquatic organisms, both in laboratory and natural conditions. In parallel, the biological role of stable elements and their accumulation by living organisms has been studied for a long time in biogeochemistry and biochemistry. It is well known, that living organisms have a fundamental capacity to accumulate selectively from the environment some specific elements, which are necessary for constructing the organic tissues of living organisms, and are essential for metabolism. Carbon, nitrogen, hydrogen, oxygen, and sulphur are major elements for life; they are present in any part of living cells; phosphorus is the key element in the processes of energy transfer in organisms; potassium and sodium are necessary for the electrochemical processes; calcium is necessary for constructing bones/shells; some elements, such as Fe, Zn, Mg, Co function as core components of ferments, which catalyze the metabolic processes in organisms (Lehninger, 1982). Normally the elemental chemical composition of organisms is specific for each biological species and is kept, on average, constant in order to maintain optimal concentrations of elements in the internal body’s media. The degree of chemical element accumulation depends on the biological function of a particular element and its amount, which is necessary for metabolism. The elemental chemical composition of aquatic organisms differs considerably from the composition of aquatic media and bottom sediments. Table 1 demonstrates the difference between the ocean water composition, and composition of different groups of marine biota. The ∗ E-mail: [email protected]

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Table 1 Elemental chemical composition of ocean water and marine organisms (according to Goldberg; Bowen, 1979) Element Ocean water Phytoplankton Zooplankton (mg kg−1 d.w.) (mg kg−1 d.w.) (mg l−1 ) (soft tissues) H S Ca K C Sr P I Fe Zn U Co Cs Ra

10.8 × 104 8.85 × 102 4 × 102 380 28 8 0.07 0.06 0.01 0.01 0.003 0.0005 0.0005 1 × 10−10

4.6 × 104 (3–6) × 103 6.1 × 103 1.3 × 104 2.25 × 105 70–700 (4–18) × 103 270 220–1500 20–280 0.7 380 0.11 (0.4–2) × 10−6

Crustaceans Mollusks Fish (mg kg−1 d.w.) (mg kg−1 d.w.) (mg kg−1 d.w.) (soft tissues) (soft tissues)

5 × 104 – (2–4) × 104 1 × 104 4.16 × 105 340–720 7.5 × 103 – 100–1800 180–530 0.38–0.56 0.08–13 0.032 (0.08–0.13) × 10−6

6 × 104 6 × 103 (2–38) × 103 1.3 × 104 4 × 105 500 9 × 103 1 10–160 52–230 – 0.02–7 0.02–0.04 7 × 10−9

6 × 104 1.6 × 104 (1–11) × 103 1.2 × 104 4 × 105 12–80 6.6 × 103 4 90–780 40–8500 – 0.1–8 0.02–0.32 1.5 × 10−7

6.8 × 104 4.7 × 103 76–2 × 104 1.5 × 104 4.75 × 105 0.2–20 1.8 × 104 1 9–88 9–82 0.04–0.08 0.006–0.005 0.04–0.2 1.5 × 10−8

Table 2 Radionuclides of biological importance to aquatic organisms Chemical element

Radionuclide

Hydrogen Carbon Phosphorus Potassium Calcium Strontium Cesium Iron Cobalt Zinc Manganese Iodine Sulfur

3H

Ruthenium Cerium Plutonium Yttrium Uranium

106 Ru

14 C 32 P 40 K 45 Ca 90 Sr 137 Cs 59 Fe 60 Co 65 Zn 54 Mn 131 I 35 S

144 Ce 239 Pu 91 Y 238 U

Decay period, T1/2

Type of decay

Comments

12.34 years 5730 years 14.3 days 1.28 × 109 years 163 days 28.6 years 30.17 years 43.1 days 5.27 years 244.1 days 312.3 days 8.04 days 88 days

β−

β− β− β−, β− β− β−, β−, β−, β−, β−, β−

Chemical elements, which represent main components of body’s tissues, as well as elements important for the metabolism of organisms

368.2 days 284.3 days 24 000 years 58.5 days 4.47 × 109 years

β− β−, γ α β−, γ α

γ

γ γ γ γ γ

Other important radionuclides

concentrations of major biologically important elements in the organisms are considerably higher than those in the ocean water; on the other hand, the concentrations of some major components of ocean water, such as Na and Cl are considerably lower in the marine organisms than those in water.

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Living organisms interact with radionuclides as certain chemical substances and do not distinguish radionuclides from chemically similar stable compounds. Radioactive isotopes of a chemical element have the same chemical properties as stable isotopes of this element. Bioaccumulation or sorption of a radionuclide by living organisms is not associated with its radioactivity, but clearly demonstrates the difference between the concentration of analogous stable element in the environment and in the organism. So, as the organisms accumulate stable bioelements (C, P, H, Co, Zn and others) from the environment, they consume also their radioactive “twins”, such as 14 C, 32 P, 3 H, 60 Co, 65 Zn and others. Table 2 presents the list of major biologically important radionuclides and their characteristics. Besides, organisms consume not only the “key” bioelements most essential for metabolism, but also some other elements, with similar chemical properties. Thus Cs (and also 137 Cs, 134 Cs) is chemically similar to potassium, Sr (and 90 Sr) to calcium, and so on. Living organisms demonstrate partial discrimination in consuming elements, which are less suitable for metabolism. The substitute elements are consumed more actively, when the concentration of main elements in aquatic media is low, and is not sufficient for organisms. 1.2. Bioassimilation and passive adsorption of radionuclides by biota Traditionally, the radionuclide accumulation by aquatic organisms is estimated in relation to radionuclide concentration in water, using the radionuclide concentration factors CF =

y , X

(1)

where CF is the concentration factor; y is the specific radionuclide activity of the organisms’ tissues, Bq kg−1 d.w. or fresh weight; X is the radionuclide concentration in a filtered water taken from the habitat, Bq kg−1 . The “concentration factor” theory is based on a fundamental assumption about the equilibrium in the radionuclide distribution within the system “aquatic media–organism”. When such equilibrium takes place, the concentration factor for a given radionuclide and a given organism in its habitat is constant and does not change with time. The concentration factor approach was initially developed and employed for natural radionuclides, whose distribution between aquatic media and biota is indeed in equilibrium. Typical values of concentration factors, estimated for marine and freshwater organisms are summarized in Table 3. The radionuclide concentration factors are not universal characteristics of aquatic organisms, and can vary considerably depending on environmental factors. First of all, radionuclide concentration factors are strongly dependent on concentration of stable analogous elements which are available for uptake from the aquatic environment. Figure 1 demonstrates in graphical form the inverse proportion between 137 Cs concentration factors and concentration of calcium in aquatic media, obtained in experiments with several species of freshwater plants – Elodea, Cladophora and some others (Timofeeva & Koulikov, 1968). Similar results in experiments with other radionuclides and other organisms are described in publications (Hiyama & Shimizu, 1964; Whicker & Schultz, 1982). Actually, stable and radioactive isotopes with identical or similar chemical properties form a common “pool” of bioelement in the aquatic environment. Organisms uptake a radionuclide in parallel with its stable analog from an available pool of the element, the share of the

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Table 3 Concentration factors (Bq kg−1 fresh mass) (Bq l−1 )−1 for radionuclides in marine and freshwater biota (IAEA, 1975; Gromov & Spitsin, 1975; Timofeev-Resovsky et al., 1960) Element Marine organisms

C P Cs Sr Zn Fe Co Ru

Freshwater organisms

Algae

Invertebrates

Fish

Non-calciferous Calciferous species species

Soft tissues

Shells

Soft tissues

Bones

1 × 104 1 × 104 50 5–20 2 × 104 3 × 104 1 × 104 2 × 103

5 × 104 1 × 104 10–30 1–10 3 × 104 3 × 104 5 × 103 2 × 103

– 1 × 104 – 1 × 103 1 × 103 1 × 105 – 50

2 × 104 4 × 104 10–100 2 1 × 103 3 × 103 1 × 103 20

– 2 × 106 – 50 3 × 104 5 × 103 – –

1 × 104 5 × 104 10 200 100 5 × 105 1 × 104 2 × 103

Aquatic plants

Fish

1 × 104 1400 500 530 7240 6800 4600 1 × 106

1 × 104 2500 250 760 830 550 2000 50

Fig. 1. The dependence of the concentration factor for 90 Sr in aquatic plants and calcium concentration in water: (1) Elodea; (2) Cladophora; (3) Myriophyllum; (4) Ceratophyllum. From: (Timofeeva & Koulikov, 1968).

radionuclide in the uptake is equal to its relative share in the pool. In aquatic media with a low concentration of the stable analogous element, the quota of radionuclide uptake by the organism is relatively high; and it is relatively low in a medium with a high concentration of the stable “twin” element. That is why the radionuclides’ concentration factors for freshwater organisms are often higher than those for marine organisms. Bioassimilation of bioelements and their radioactive “twins” by organisms is an active multi-factorial process, which depend on many factors, influencing the normal course of biochemical reactions within organism. For instance, green plants assimilate some elements (Zn, Fe), using the energy of photosynthetic process, so light is an essential factor for assimilation of 65 Zn, and 59 Fe by plants. Figure 2 illustrates the important role of light exposure in

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Fig. 2. The effect of light intensity on the uptake of 65 Zn from seawater by algae Ulva lactuca. Initial parameters of seawater in the experiment: stable Zn – 8 mg l−1 ; 65 Zn – 296 kBq l−1 ; pH = 8.2; temperature 22.5◦ C. Light intensity (foot-candeles): (1) no light; (2) 80; (3) 275; (4) 600. From Gutknecht (1965), IAEA (1975).

the bioassimilation of 65 Zn by algae Ulva lactuca from sea water (Gutknecht, 1965; IAEA, 1975). Bioassimilation of a particular element by an organism occurs simultaneously with many other elements, which are necessary for constructing living tissues and maintaining metabolism. Since the availability of bioelements in the environment does not fit the optimal requirements of organisms, some elements are in excess and some in biological deficiency. An element, which is the most deficient for an organism is called the limiting bioelement; the growth of the organism is directly limited by such a bioelement (Liebig, 1847). The deficiency in the limiting bioelement may become an obstacle for assimilation of radionuclides, as well as for stable bioelements. Figure 3 illustrates the end of 134 Cs assimilation by an algae Rhodamenia palmata following the exclusion of the limiting compound CO2 (carbon dioxide) from the air, aerating the system (Scott, 1954). The physico-chemical form of an element in aquatic media is of key importance for its bioassimilation by living organisms. From a biochemical point of view, some chemical forms of a bioelement are more convenient for inclusion into metabolic processes than others; that is why either preference or discrimination may be observed in biological uptake of different chemical forms of the same element. The results of physiological experiments, where radionuclides were used as tracers of certain chemical elements, clearly demonstrate the difference in assimilation rate for ions of different valency, organic and non-organic compounds, etc. It should be noted, that even within one and the same organism different organs and tissues are specialized in selective assimilation particular chemical forms of elements. Figure 4 shows the uptake of cobalt by shells and soft tissues of mollusk Donax denticulatus; it is clearly seen

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Fig. 3. The effect of limiting factor (carbon dioxide) on 134 Cs uptake by algae Rhodamenia palmata in the laboratory conditions. The arrow indicates the moment when the CO2 supply to the algal population was excluded. From Scott (1954).

Fig. 4. Uptake of cobalt in form of ions (58 Co) and cobalamine (60 Co) by shells (1 and 1 respectively) and soft tissues (2 and 2 respectively) of the mollusk Donax denticulatus. From Lowman & Ting (1973).

from the graphs, that the soft tissues of the mollusk assimilated cobalt preferably in the form of vitamin B12 (cobalamin) traced by 57 Co, whereas for shells the more preferable was the ionic form of cobalt, traced by 58 Co (Lowman & Ting, 1973).

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A radionuclide, entering the aquatic media in a chemical form, which differs from the natural form of the stable analogous element in the media, is expected to be accumulated by organisms either with preference, or discrimination, depending on the extent of compliance with the optimal needs of the organisms. The concentration factors for a radionuclide in this case may differ considerably from those for stable analogues. 1.3. Bioassimilation and passive adsorption of radionuclides by biota Many radionuclides are isotopes of heavy elements, such as 144 Ce, 106 Ru, 239 Pu, 235,238U. Organisms do not need these chemical elements for metabolism, so such radionuclides are not bioassimilated actively. In aquatic media, heavy radionuclides usually exist in the form of colloids or suspensions, which are easily adsorbed on any submerged surfaces. The specific surface of small aquatic organisms, especially small planktonic species is very high. Since adsorption is directly proportional to the surface area, organisms with a high surface/volume ratio demonstrate high concentration factors for adsorbing radionuclides. The surface/volume ratio for representatives of aquatic biota gradually decreases in the order “bacteria–phytoplankton–zooplankton–crustaceous–fish”, see Fig. 5a. The direct correlation between the specific surface of organisms and 144 Ce concentration factor due to passive adsorption is shown in Fig. 5b; in contrast, the 137 Cs concentration factor due to bioassimilation does not correlate with specific surface of organisms. It should be noted, that heavy radionuclides may be adsorbed not only on the external surface of organisms, but also on internal surfaces of organs, such as gills, intestine, kidney, etc. So, two main mechanisms are responsible for radionuclide accumulation by an organism: active bioassimilation, with incorporation of the radionuclide into the tissues; and passive adsorption on external and internal surfaces of organism. In many cases both bioassimilation and passive sorption contribute to the radionuclide uptake by aquatic biota. Usually anthropogenic radionucludes enter aquatic media in the form of non-organic compounds. Later on, their physico-chemical forms are transformed gradually, depending on mineralization, dissolved organic substances, pH and other characteristics of the aquatic media. In particular, the solubility of radionuclides usually decreases with increasing alkalinity reaction of water; high mineralization promotes the precipitation of radionuclides along with other elements. The specific feature of dissolved natural organic compounds is its ability to form soluble complexes with some elements, especially of the metal group, the resulting biological activity of such complexes may differ considerably from the activity of the initial non-organic form of the radionuclide. The physico-chemical transformation of the radionuclide in aquatic systems causes a specific “ageing” process, when the form of the isotope changes gradually to that of the natural form of stable analogous element. The difficulties in interpretation of the results of radioecological experiments are in many cases caused by chemical non-equilibrium between the radionuclide and aquatic media.

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Fig. 5. Radionuclide concentration factors for groups of marine organisms with different values of specific surface of organisms. (a) 144 Ce; (b) 137 Cs; (c) the typical values of specific surface of organisms, cm2 gramm−1 fresh mass. All values are given in log10 scale. From Patin (1971).

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Fig. 6. The typical form of the dynamic curve of radionuclide accumulation by aquatic organism: accumulation of 45 Ca by algae Cystoseira barbata. From Polykarpov (1970).

The typical dynamics in time of radionuclide accumulation by aquatic organisms is shown in Fig. 6. The simple mathematical approximation to the experimental accumulation curve is the following (Polykarpov, 1966, 1970):

 CF(t) = CFeq 1 − α e−λ1 t − (1 − α) e−λ2 t ,

(2)

where CF(t) – dynamic concentration factor [Bq kg−1 d.w.]/[Bq kg−1 water]; CFeq – equilibrium concentration factor [Bq kg−1 d.w.]/[Bq kg−1 water]; α, λ1 , λ2 (α < 1; λ1 > 0, λ2 > 0) – empirical parameters of sorption and bioaccumulation. In comparison, sorption is a more rapid process than bioassimilation, so the parameter λ1 , which determines the sorption process is greater than parameter λ2 , determining bioassimilation. The relative importance of sorption vs. bioassimilation varies, depends on the radionuclide and radioecological situation; in some cases it is possible to neglect one of these processes. Equation (2) refers to an ideal organism with uniform distribution of the radionuclide within the body; actual accumulation process may be rather complicated including uptake and distribution of the radionuclide in different organs and tissues. Nevertheless, at the population level equation (2) is a useful approximation of radioecological processes. 1.4. Migration of radionuclides in aquatic food chains Direct uptake of radionuclides from aquatic media is a major source of contamination of aquatic plants, plankton and filtrating organisms. Many other organisms, however, uptake radionuclides not directly from water, but from contaminated diet. Radionuclide transfer via

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food chains is a complicated branch of aquatic radioecology. At present, the following statements can be made, as a generalization of available experimental information (Nelson & Evans, 1969; Cougthrey & Thorne, 1983; Cougthrey et al., 1983): • radionuclides, which are transformed into insoluble form within organisms, such as 90 Sr, 45 Ca are hardly transferred via food chains; in most cases concentration factors decreased from organisms of lower trophic levels (plants, molluscs) to fish and aquatic mammals. • Radionuclides, which are accumulated in soft tissues of organisms as analogues of major bioelements (137 Cs, 40 K, 32 P) are easily transferred via food chains. Concentration factors for such radionuclides may increase in food chains from lower to higher trophic levels. • Radionuclides which are accumulated in organisms due to adsorption, have a general decrease in concentration factors in aquatic food chains, such as “phytoplankton – aquatic invertebrates – fish”; however, this effect is not caused by trophic relations, but simply by the fact, that larger organisms have lower surface/volume ratio comparing with smaller organisms. Concentration factors for these radionuclides may be estimated from data on sorption on a unit surface area. • In general, the transfer of a radionuclide in aquatic food chain is tracing the transfer of stable analogous element. It is possible to predict the radionuclide transfer via particular food chain based on data on elemental chemical composition of aquatic organisms, see Table 1.

2. Construction of radioecological models based on the equations of mathematical ecology Ecological models have been intensively developed since the middle of the 1920s, following the theoretical investigations of Lotka (1925) and Volterra (1931). Presently, mathematical ecology is a well developed branch of science, which combines theoretical achievements with great experience in simulating complicated ecosystems of natural water bodies (Jorgensen et al., 1978; DeAngelis, 1992; Tilman, 1982; Alekseev et al., 1992). Ecological models describe the transfer of stable elements in aquatic ecosystems, based on non-linear differential equations of ecological processes, as well as interactions of aquatic biota with aquatic media. The purpose of a radioecological model is simulation of a radionuclide transfer in an ecosystem. Considering the fact, that radionuclides are no more than radioactive tracers of corresponding stable elements, it is evident, that ecological models, which are designed to describe the transfer of stable elements in ecosystems, should be well adapted for their radioactive tracers. However, traditionally radioecological models are developed on a pure empirical basis as a tool for analyzing the results of radioecological experiments; linear compartment theory is often used as a basis for modelling (Till & Meyer, 1983). Some modern radioecological models include a combination of population description and empirical radioecological data (Heling, 1997; Garnier-Laplace et al., 1997). The theoretical aspects of modeling the radionuclide migration in ecosystems based on ecological models are discussed in publications (Kryshev & Sazykina, 1986, 1990; Sazykina, 1994, 2000), in this chapter the general methodology of integrating radioecological and eco-

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logical models is presented with consideration of simple examples of radionuclide transfer in model aquatic ecosystems. 2.1. General ecological equations for population growth and trophic interactions In general, ecological models are based on the equations for biomass dynamics of populations or ecological groups of species. The ecological equations describe the transfer of chemical elements (bioelements) from the environment to the food chains and back to the environment (ecological cycles of the elements). The typical form of an ecological equation for a population in aquatic ecosystem is as follows: dM = −BIOLOSS(. . . , t) + ASSIMFOOD(. . . , t) − PREDATION(. . . , t), M dt BIOLOSS(. . . , t) = MORTALITY(. . . , t) + METAB(. . . , t),

(3)

ASSIMFOOD(. . . , t) = BIOSYNT(. . . , t) + METAB(. . . , t), where: M(t) – biomass of the population of the given type of organisms (units of mass per unit of volume of water in a water body); BIOLOSS(. . . , t) – daily loss of biomass under metabolic processes METAB and mortality MORTALITY (per unit of the biomass value); ASSIMFOOD(. . . , t) – daily amount of matter bioassimilated from organism’s diet (per unit of the biomass value): ASSIMFOOD = γ ∗ FOOD, where FOOD is the daily diet of the organisms, γ -bioassimilation coefficient; ASSIMFOOD is used to support the metabolic processes and for biosynthesis of new biomass; BIOSYNT(. . . , t) – rate of biosynthesis of new biomass (per unit of the biomass value); METAB(. . . , t) – daily amount of elements used to service the metabolic needs of the existing biomass – equivalent amount of elements is taken from the diet to tissues and lost to the environment from the tissues of the organisms (per unit of the biomass value); PREDATION – biomass losses per day from predation by other organisms (per unit of the prey biomass value). The terms ‘BIOLOSS’, ‘ASSIMFOOD’, ‘PREDATION’ and others actually involve complex formulae, describing the dependence of biological processes on environmental factors and on other organisms. The complex formulae for the biological processes are described elsewhere (DeAngelis, 1992; Alekseev et al., 1992). Here the dependence on the set of parameters is shown as (. . . , t). In the following paragraphs some examples of formulae, describing ecological parameters in simple ecosystems are presented. 2.2. Transfer of a pair “stable element and its radioactive tracer” in a population of aquatic organisms Two basic assumptions justify the integration of radiological and ecological models. These basic assumptions are actually a generalization of the existing experimental data, and may be formulated in the following way:

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• Living organisms consume radioactive substances as ordinary chemical compounds without distinguishing them by their radioactive nature. If stable and radioactive isotopes of a particular element have the same chemical form, they are indistinguishable to organisms. A radioactive isotope should always be considered in tandem with its stable, chemically similar element-analogue, which is present in the aquatic media and within the organisms, or with a sum of such analogues. Stable and radioactive isotopes with identical or similar chemical properties form a common “pool” of the element. The share of a radionuclide in the transfer process is equal to the relative share of the radionuclide in the common pool of this element (radionuclide + stable analogues). • The elemental composition of biomass for each biological species is unique and, on average, constant (Liebig, 1847; Vernadsky, 1929a, b). To produce one unit of new biomass, an organism consumes strictly specified amounts of particular elements from the environment. • Of fundamental importance are two mechanisms of radioisotope accumulation by organisms: bioassimilation, with incorporation of an isotope from diet and aquatic media into the tissues of organisms; and adsorption at the surfaces of organisms. First of all, consider the transfer of a particular chemical element “A” without a radioactive tracer in a population of aquatic organisms. The specific concentration of the chemical element “A” in the population biomass is constant and equal to QA 1 (g/kg d.w.). For simplicity it is assumed, that the element “A” cannot be replaced in the biomass by any other chemically similar element. The concentration of the element “A” in the diet of population is QA 0 (g/kg d.w.). The organisms bioassimilate the element “A” from their diet, and use it to construct their bodies, and also for metabolic needs. So, the organisms concentrate the element “A” from the A diet with concentration factor QA 1 /Q0 . Let us consider the transfer of the stable element “A” between the biomass of population and the environment. The element “A” is assumed to be a non-limiting element for the population, so its content in the diet is sufficient to cover the needs of the organisms. The use of the element to produce the new biomass is equal to QA 1 · BIOSYNT; the use of the element to serve the metabolic needs is equal to εA · QA · METAB, where the coefficient εA indicates the specific activity of 1 a given element in the metabolic process. The losses of element “A” from the natural death of A organisms and predation are equal to QA 1 · MORTALITY and Q1 · PREDATION. The resulting equation for transfer of the stable element “A” in the population has the following form: d(QA 1 · M) A = −QA 1 · MORTALITY(. . . , t) − εA · Q1 · METAB(. . . , t) M dt 

A + QA 1 · BIOSYNT(. . . , t) + εA Q1 · METAB(. . . , t) − QA 1 · PREDATION(. . . , t).

(4)

Let us assume, that a radioactive isotope of element “A” is added to the diet (in the same chemical form as the stable isotope) with mass concentration q (g/kg d.w.) and corresponding activity X (Bq/kg d.w.). Then the total concentration (pool) of element “A” in the diet will be (QA 0 + q).

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The fraction of the radionuclide participating in elemental uptake from bioassimilated diet is equal to its fraction in the total pool of the element in the diet: Fraction =

X X q = A ≈ A, QA + q Q + X · K Q 0 0 0

where K is the conversion coefficient between mass and activity of the radionuclide. For the most important cases, q  QA 0. The rate of radionuclide removal from the biomass is proportional to the radionuclide concentration γ (t) (Bq kg−1 d.w.), already accumulated in the biomass. Thus, equation (4) is transformed to describe the transfer of a radioactive tracer in the biomass of population: d(M · y) = −λr · y − y · MORTALITY(. . . , t) − εA · y · METAB(. . . , t) M dt  X A + A QA 1 · BIOSYNT(. . . , t) + εA · Q1 · METAB(. . . , t) Q0 − y · PREDATION(. . . , t),

(5)

where: y(t) – the specific activity of radionuclide in the biomass, Bq kg−1 d.w.; λr – the radioactive decay constant. Taking into account that d(M · y) dM dy =y +M · dt dt dt and using equation (3) for dM/dt, equation (5) is transformed to the differential equation for specific activity y(t):    QA dy 1 = BIOSYNT(. . . , t) + εA · METAB(. . . , t) · · X − y − λr · y. dt QA 0

(6)

Equation (6) is the basic equation for calculation of the dynamics of radionuclide transfer between the population and the environment. 2.3. Accumulation of a radioactive tracer by a stationary and growing population Applying the general methodology to a simple case of a single population of planktonic algae in a isolated aquatic microcosm, we assume that the distribution of organisms and elements is space-uniform, so all variables refer to the units of volume. One chemical element “B” is considered to be the limiting nutrient for the algal population, according to the Liebig’s “law of minimum”. The parameter M means the total reserve (pool) of the limiting chemical element “B” in the ecosystem, and S – its concentration in water. It is assumed that the limiting element

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recycles in the ecosystem. The set of simplifying equations for the biomass dynamics is the following: dM1 = M1 (−ε + βS), dt

(7a)

QLIM 1 M + S = M = const,

(7b)

where: M is the biomass of algal population; M is total amount of limiting nutrient “B” in the system; S is concentration of the limiting nutrient in the environment; ε is the rate of biomass losses due to mortality and metabolism; ε = εmort + W ; β are the coefficient of new biomass synthesis by algae; is concentration of limiting bioelement “B” in biomass. QLIM 1 Comparing equations (7a), (7b) with the general form of the ecological equation (3), one can see the following relationships between the equation members: BIOLOSS(. . . , t) = ε = εmort + W,

METAB = W,

MORTALITY = εmort ,

ASSIMFOOD(. . . , t) = β S = BIOSYNT + W. The radioecological equation (6) for specific activity of radioactive tracer of element “A” in the algal biomass is transformed to the form    QA dy 1 = (BIOSYNT + εA · W ) · · X − y − λr · y. dt QA 0

(8)

Several ecological and radioecological cases are considered below, using the set of equations (7a), (7b) and (8). 2.3.1. Case 1 Population is maintained in ecological equilibrium state, where M(t) = const; BIOSYNT = εmort ; the concentration of a radionuclide in the water is held constant X(t) = X(0), y(0) = 0, λr ∼ = 0. In this case, equation (8) has an analytical solution, yi (t) = X(0) ·

 QA 1 QA 0

 1 − exp −(εmort + εA · W ) · t ,

(9)

A where QA 1 /Q0 is the concentration factor (CF) for the element A and its radioactive tracer. Formula (9) is the simplest expression of the dynamics of radionuclide accumulation in the biomass of organisms. This relation, with an empirical value of CF and the parameter of exponential decrease, is commonly used to approximate the results of radioecological experiments and in compartmental radioecological models (Polykarpov, 1966; Whicker & Schultz, 1982). The advantage of the ecological approach is that the parameters of radionuclide transfer are

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Fig. 7. Model predictions of radionuclide transfer between non-growing population and water in a closed microcosm. X(t), y(t) are activity concentrations in water and the biomass respectively. Radionuclide was added to the microcosm at time t = 0 with initial concentration (activity) X(0) = X0 .

described as ecological parameters and functions, which can be obtained from ordinary biological data before the execution of the radioecological experiment. 2.3.2. Case 2 The population is maintained in ecological equilibrium state, where M(t) = const; BIOSYNT = εmort ; a single pulse of radionuclide is added to the ecosystem with initial concentration X(0). In the isolated ecosystem the total content of radionuclide is equal to its initial value with consideration of radioactive decay: X(t) + yM(t) = X(0) exp(−λr t). Equation (8) for this case is transformed to the following: dy = (εmort + εA · W ) · dt



  (X(0) exp(−λr t) − yM) − y − λr · y. QA 0 QA 1

(10)

The calculated dynamics of radionuclide activities in population biomass and water in an isolated aquatic microcosm are shown in Fig. 7. 2.3.3. Case 3 A single pulse of radionuclide is added to the growing population. The dynamics of population biomass is expressed by logistic curve, which is obtained by integrating equations (7a), (7b):   dM(t) = −(εmort + W ) + β M − QLIM ·M , 1 M dt M(0) < M /QLIM 1 , Mmax =

(βM − εmort − W ) , QLIM 1

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Fig. 8. Model predictions of radionuclide transfer between growing population and water in a closed microcosm. M(t) is the biomass of population; X(t), y(t) are activity concentrations in water and the biomass respectively. Radionuclide was added to the microcosm at time t = 0 with initial concentration (activity) X(0) = X0 .

M(t) =

M(0)Mmax . M(0) + (Mmax − M(0)) exp(−βMmax t)

(11)

The dynamics of a radionuclide distribution between the growing biomass and aquatic media in the isolated microcosm are shown on Fig. 8. The predictive capacity of the model for the growing population can be demonstrated, using as an example data from a simple experiment on accumulation of radioisotopes of the iron group (Fe, Mn, Co and Ni) by the green algae Platymonas viridis (marine flagellata), published in (Gromov & Starodubtsev, 1974). The main reason for choosing the experiment with Platymonas for model testing, was the availability of biological information about the concentrations of stable elements in the medium and in the biomass; unfortunately in most publications of radioecological experiments such data are not described. The algal population was cultivated in 1 liter flasks in the standard Goldberg medium, with microelements and radioisotopes being present in the form of chlorides (Gromov & Starodubtsev, 1974). The concentrations of the microelements in the standard Goldberg medium are given in Table 4. The data on the elemental chemical composition of Platymonas viridis were found in the literature (Gribovskaya et al., 1980), see Table 4. During the course of the experiment (80 hours), the relative intake of each radionuclide from the aquatic medium by the growing algal population was measured twice per day. The results are given in Fig. 9. Taking into account the short duration of the experiment (80 hours), the losses of activity due to radioactive decay were considered to be negligible. Also radionuclide elimination from the biomass was assumed to be small in comparison with the bioassimilation process. The growth parameters were taken from the description of the experiment: M(0) = 22 mg l−1 ; Mmax = 34 mg l−1 ; β = 0.005 ((mg/l)hour)−1. The dynamic curves for uptake of each radionuclide, calculated from equations (8), (11), are given in Fig. 9 as solid lines, whereas experimental values are shown by specific symbols with

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Table 4 The concentrations of the microelements (iron group) in the standard Goldberg medium, used for cultivation of Platymonas viridis, the elemental chemical composition of the algal biomass, and initial concentrations of radioactive isotopes added to the medium Stable element

Standard Goldberg medium (modification of Kabanova) (mg l−1 )

Radionuclides, added to the Goldberg medium (mg l−1 ) (Gromov & Starodubtsev, 1974)

Shares (%) of the microelements in the ovendried weight of Platymonas viridis (Gribovskaya et al., 1980)

Fe Mn Ni Co

0.056 (FeCl3 ) 0.055 (MnCl2 ) ∼ 0.001 (NiCl2 ) 0.059 (CoCl2 )

(2–4) × 10−3 (55 FeCl3 ) 0.02 × 10−3 (54 MnCl2 ) (0.5–1) × 10−2 (63 NiCl2 ) (0.5–1) × 10−3 (60 CoCl2 )

(116.2 ± 13.7) × 10−3 (45.4 ± 4.2) × 10−3 (0.31 ± 0.03) × 10−3 no data

The ash content in the Platymonas viridis biomass is 25%.

Fig. 9. Uptake of radionuclides of the iron group by the alga Platymonas viridis. Solid lines show the predicted curves of accumulation; the experimental data (with uncertainty intervals) are taken from the paper (Gromov & Starodubtsev, 1974).

uncertainty intervals (Kryshev & Sazykina, 1986; Sazykina, 2000). As seen from Fig. 9, the theoretical predictions, calculated on the basis of independent, non-radiological information, are in a good agreement with the results of the radioecological experiment. This simple example, on the radionuclide accumulation by Platymonas, shows the ability of the ecological approach to predict the results of a radioecological experiment. It is extremely important that the predictions were made on an absolutely independent basis, and no empirical radioecological data were used for the predictions. 2.4. Modeling the radionuclide transfer in aquatic food chain substrate – prey – predator Radionuclide transfer via food chains is of great theoretical and practical interest. Consider as an example a basic food chain, consisting of prey (one species of planktonic algae), which is

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feeding on non-living substrate (dissolved nutrients), and predator (one species of zooplankton organisms), feeding on prey population. Both populations are uniformly distributed in an isolated microcosm with recycling of limiting nutrients. The set of simplified equations for the biomass dynamics is the following: dM1 = M1 (−ε1 − W1 + βS − γ12 M2 ), dt dM2 = M2 (−ε2 − W2 + γ21 M1 ), dt LIM QLIM 1 M1 + Q2 M2 + S = M = const,

(12a) (12b) (12c)

where M1 is the biomass of prey population; M2 is the biomass of predator population; M is the total amount of limiting nutrient in the system; S is concentration of the limiting nutrient in the aquatic media; ε1 , ε2 are the rates of biomass losses due to natural mortality; W1 , W2 are the rates of biomass losses due metabolism; β is the coefficient of biomass synthesis for prey on limiting nutrient; γ12 , γ21 are the specific consumption and bioassimilation rates for predator, feeding on prey M1 . Accumulation of the radioactive tracer of a non-limiting element “A” may be described, using equation (8) consecutively for populations of prey and predator. The resulting set of equations for specific radionuclide activities y1 and y2 in biomasses of prey and predator are as follows:    QA dy1 1 = (βS − W1 + εA,1 · W1 ) · · X − y1 − λr · y1 , (13a) dt QA 0    QA dy2 2 = (γ21 M1 − W2 + εA,2 · W2 ) · · y − y (13b) 1 2 − λr · y2 , dt QA 1 A where QA 1 , Q2 are concentrations of chemical element “A” (without radioactive tracer) in the biomass of prey and predator correspondingly. Similar to the model, described in Section 2.3, the radioecological model for the “prey– predator” ecosystem makes it possible to consider different radioecological cases: radionuclide transfer by stationary or growing populations; single pulse of radionuclide to the system; chronic contamination of the aquatic media, etc. The results of computer calculations of radionuclide migration in the isolated “substrate– prey–predator” system after a single pulse of radionuclide are shown in Fig. 10. The graphs clearly demonstrate the “effect of trophic levels”, where the maximum contamination of the predator is reached noticeably later than the peak of the prey contamination. The “effect of trophic levels” was observed experimentally and described in publications (Pendleton et al., 1965). The radioecological model (equations (13a), (13b)) predicts this effect theoretically.

2.5. Modeling the processes of radionuclide adsorption on aquatic biota Adsorption of radionuclide on surfaces of aquatic organisms provides in some cases an important contribution to the radioactive contamination of aquatic biota. In the radioecologi-

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Fig. 10. Model predictions of radionuclide transfer in the food chain “prey–predator” in a closed aquatic microcosm. X(t), y1 (t), y2 (t), are activity concentrations in water, prey biomass, and predator biomass respectively. Populations of prey and predator are assumed to be in steady state. Radionuclide was added to water of the microcosm at time t = 0 with initial concentration (activity) X(0) = X0 .

cal model adsorption/desorption are considered as physical processes, which are independent from bioassimilation. The adsorption/desorption of a radionuclide occur in parallel with the stable analogue element. The activity of radionuclide on the surface of the organism is described by the following equation   X dzr = qsorp · − qdes · zr , dt Qstab

(14)

zr is the activity adsorbed on the unit of surface of organism (Bq cm−2 ); qsorp, qdes are parameters of sorption and desorption of radionuclide; Qstab is the concentration of stable analogue element in aquatic media. The results of model calculations on the radionuclide distribution in a system “one nongrowing population in a closed microcosm” with consideration of both bioassimilation and adsorption processes are presented in Fig. 11.

3. Conclusions The unified methodology for modelling radionuclide migration on the basis of ecological models makes it possible to simulate radionuclide transfer in the ecosystem as an ordinary pollutant, taking into account the synergism in the interactions between the ecological and

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Fig. 11. Model predictions of radionuclide transfer between non-growing population and water with consideration of radionuclide adsorption on organisms’ surface. X(t), y(t) are activity concentrations in water and in the biomass respectively; zr (t) is surface activity of organisms. Radionuclide was added to the closed microcosm at time t = 0 with initial concentration (activity) X(0) = X0 .

radioecological processes. The method enables one to calculate the non-equilibrium, dynamic processes of radionuclide transfer for all important components of the ecosystem taking into consideration the dynamics of the biomass and the variability in the ecological parameters. For practical purposes of modelling the radioecological situation in real water bodies, a complex simulation model, called “ECOMOD”, was developed. Modifications of the ECOMOD model were used for various cases of radioecological assessment (Sazykina, 1995, 1998, 2000; Kryshev & Ryabov, 2000); the case study “Chernobyl Cooling Pond” is presented in more detail in Chapter 13, this volume.

References Alekseev, V. V., Kryshev, I. I. & Sazykina, T. G. (1992). Physical and Mathematical Modelling of Ecosystems (367 pp.). St.-Petersburg, Russia: Hydrometeoizdat (in Russian). Bowen, H. J. M. (1979). Environmental chemistry of the elements (228 p.). Acad. Press. Cougthrey, P. J. & Thorne, M. C. (Eds) (1983). Radionuclide distribution and transport in terrestial and aquatic ecosystems. A critical review of data (Vol. 1, 496 p.; Vol. 2, 500 p.), Rotterdam. Cougthrey, P. J., Thorne, M. C. & Jackson, D. (Eds) (1983). Vol. 3, 372 p. DeAngelis, D. L. (1992). Dynamics of Nutrient Cycling and Food Webs (270 pp.). Population and Community Biology Series N. 9. London: Chapman & Hall. IAEA (1975). Design of radiotracer experiments in marine biological systems. Tech. rep. ser. No. 167. Vienna: IAEA. Garnier-Laplace, J., Vray, F. & Baudin, J. P. (1997). A dynamic model for radionuclide transfer from water to freshwater fish. Water, Air, and Soil Pollution, 98, 141–166.

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Gribovskaya, I. F., Novosyolova, O. I., Gladchenko, N. A. & Ustyugova, T. T. (1980). Mineral composition of some species of green and blue-green algae. In Parametric Regulation of the Microalgal Biosynthesis (pp. 36–43). Novosibirsk: Nauka (in Russian). Gromov, V. V. & Spitsin, V. N. (1975). Artificial radionuclides in seawater (224 pp.). Moscow: Atomizdat (in Russian). Gromov, V. V. & Starodubtsev, E. G. (1974). Assimilation of the iron-group elements by phytoplankton. Oceanology (Okeanologiya), 14 (6), 1006–1012 (in Russian). Gutknecht, J. (1965). Uptake and retention of 137 Cs and 65 Zn by seaweeds. Limnology and Oceanography, 10, 58–66. Heling, R. (1997). LAKECO: Modeling the transfer of radionuclides in a lake ecosystem. Radiation Protection Dosimetry, 73 (1–4), 191–194. Hiyama, Y. & Shimizu, M. (1964). On the concentration factors of radioactive J, Co, Fe and Ru in marine organisms. Rec. Oceanogr. Works Jap., 7 (2), 43–77. Jorgensen, S. J., Friis, H. B., Henriken, J. & Mejer, H. F. (1978). Handbook of Environmental Data and Ecological Parameters (1162 pp.). Vaerlse, Denmark: ISEM. Kryshev, A. I. & Ryabov, I. N. (2000). A dynamic model of 137 Cs accumulation by fish of different age classes. Journal of Environmental Radioactivity, 50, 221–233. Kryshev, I. I. & Sazykina, T. G. (1986). Mathematical Modeling of Radionuclide Migration in Aquatic Ecosystems (149 pp.). Moscow, Russia: Energoatomizdat (in Russian). Kryshev, I. I. & Sazykina, T. G. (1990). Simulation models of ecosystem’s dynamics under anthropogenic impact of thermal and nuclear power plants (184 pp.). Moscow, Russia: Energoatomizdat (in Russian). Lehninger, A. L. (1982). Principles of Biochemistry. Worth Publishers, Inc. Liebig, J. (1847). Chemistry in its application to agriculture and physiology, 14nd edn. London. Lotka, A. J. (1925). Elements of Physical biology. Baltimore: Williams & Wilkins. Reprinted in 1956 as Elements of Mathematical Biology. New York: Dover. Lowman, F. G. & Ting, R. J. (1973). The state of cobalt in seawater and its uptake by marine organisms and sediment. In Radioactive contamination of the marine environment (786 p.). Vienna: IAEA. Nelson, J. D. & Evans, F. G. (Eds) (1969). Radionuclides in Ecosystems. Proc. Symp. Radioecology, USAEC. Patin, S. A. (1971). On the subject of universal physico-chemical regularity in differentiation of radioactive and stable nuclides in the open ocean ecosystem. In Problems of marine radioecology. Trudi AtlantNIRO, Kaliningrad, 44, 45–52. Pendleton, R. C., Mavs, C. W., Lloid, R. D. & Church, B. W. (1965). A trophic level effect on 137 Cs concentration. Health Physics (2), 1503. Polykarpov, G. G. (1966). Radioecology of Aquatic Organisms (314 pp.). Amsterdam: North-Holland, New York: Reinhold Book Div. Polykarpov, G. G. (Ed.) (1970). Marine Chemoradioecology (226 pp.). Kiev: Naukova Dumka (in Russian). Sazykina, T. G. (1994). Modelling the Transfer of Fission Products in Aquatic Ecosystems. Proceedings of the Fourth International Conference on the Chemistry and Migration Behaviour of Actinides and Fission Products in the Geosphere (pp. 727–731). Charleston, SC, USA, December 12–17, 1993. Sazykina, T. G. (1995). A Radioecological Model of Radionuclide Bioaccumulation in the Ecosystems of the Barents and Kara Seas. In P. Strand & A. C. Osteras (Eds), Environmental Radioactivity in the Arctic (pp. 159–162). Norway. Sazykina, T. G. (1998). Long-distance radionuclide transfer in the Arctic seas related to fish migrations. Radiation Protection Dosimetry, 75 (1–4), 219–222. Sazykina, T. G. (2000). ECOMOD – an ecological approach to radioecological modelling. Journal of Environmental Radioactivity, 50, 207–220. Scott, R. (1954). A study of Cs uptake by marine algae. Medical and physiological applications. In 2nd radioisotope conf., Oxford, 1954, Vol. 1 (pp. 373–380). Till, J. E. & Meyer, H. R. (Eds) (1983). Radiological Assessment. NUREC/CR-3332; ORNL-5968. USA. Tilman, D. (1982). Resource Competition and Community Structure (296 pp.). Monographs in Population Biology. Princeton, New Jersey: Princeton University Press. Timofeeva, N. A. & Koulikov, N. V. (1968). Role of freshwater plants in accumulation of 90 Sr and its distribution among the components of a water body. In Behaviour of radionuclides in the model systems of terrestrial and

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freshwater biogeocenoses. Proceedings of the Institute of Ecology of Plants and Animals, Ural Branch of the Academy of Sciences USSR, issue 61, pp. 65–72. Sverdlovsk. Timofeev-Resovsky, N. V., Timofeev-Resovsky, E. A., Milutina, G. A. & Getsova, A. B. (1960). Factors of concentration by freshwater organisms of radioactive isotopes of 16 elements and their influence under complexing agent EDTA. Docladi of the Academy of Sciences of the USSR, 132 (5), 1191–1194 (in Russian). Vernadsky, V. I. (1929a). La biosphere. Paris. Vernadsky, V. I. (1929b). On the accumulation of radium by living organisms. Dokladi AN USSR (Reports of the Academy of Sciences of the USSR) (2), 33–34 (in Russian). Volterra, V. (1931). Theorie mathematique de la lutte pour la vie (214 pp.). Paris. Whicker, F. W. & Schultz, V. (1982). Radioecology: nuclear energy and the environment. Vols 1, 2. Boca Raton, Florida: CRC Press.

MODELLING RADIOACTIVITY IN THE ENVIRONMENT E. Marian Scott (Editor) © 2003 Elsevier Science Ltd. All rights reserved

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Chapter 8

Modelling the dispersion of radionuclides in estuarine environments: the Odiel river estuary R. Periáñez, J. M. Abril Dpto. Física Aplicada I, E.U. Ingeniería Técnica Agrícola, Universidad de Sevilla, Ctra. Utrera km 1, 41013-Sevilla, Spain

1. Introduction An estuary is defined as a semi-enclosed coastal body of water which has a free connection with the open sea and in which sea water is mixed with freshwater. Estuaries form the transition from rivers to the sea and present a separate environment, influenced by conditions in the river and in the coastal area. Huge industrial facilities are often located along estuaries, as well as human settlements. Thus, it may be relevant to study contamination in estuaries and to develop models that simulate the dispersion of pollutants so as to assess contamination levels following potential accidental releases from the industries. In particular, this is valid for the case of radioactive contamination, which can be released to the aquatic environment not only from nuclear related industries. A numerical model to simulate the dispersion of natural radionuclides (226 Ra, 238 U, 232 Th) in an estuary of southwest Spain is presented in this chapter. These radionuclides are released from a phosphate fertilizer processing complex. 1.1. Physical features of an estuary: selecting a model The model used to simulate the dispersion of radionuclides (or any pollutant in general) in an estuarine system will be defined by the time scale in which we are interested (usually interest will focus in simulating tide and wind dispersion during several days or weeks) and by the physical characteristics of the particular estuary: from shape and topography to water circulation. The different types of estuarine circulation are presented in Fig. 1 (from Postma, 1980). In the absence of tides (Fig. 1a), the river water flows over the sea water due to its lower density. Mixing between fresh and salt water is limited and mainly occurs because of surface waves and tidal currents. Most of the suspended matter supplied by the river settles in the area of contact between the two water masses due to the reduction in the flow velocity which 223

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Fig. 1. Circulation in estuaries. Isolines represent salinity, vectors represent the residual currents and points the suspended particles. (a) Non tidal estuary, (b) partially mixed estuary, (c) fully mixed estuary.

usually occurs here. In tidal estuaries, two cases can be distinguished: partially mixed and fully mixed estuaries. In partially mixed estuaries (Fig. 1b) vertical mixing is incomplete and strong vertical density gradients occur. This mixing of the fresh water discharged by the river with the incoming sea water is enhanced by turbulence induced by the tidal current. As a result, there is an entrainment of salt water into the surface seaward flow of the river, and the sea water carried out to sea in this way is replaced by a net residual flow at the bottom, directed up the estuary. The residual current in the surface is outwards because of the river water flowing out, while in the bottom it is inwards to compensate for the outflow of sea water. The place where upstream residual bottom flow and the downstream river flow converge is called the null point. In most estuaries the null point is associated with a turbidity maximum, where the suspended matter load may be as high as 10 kg/m3 . This area of maximum turbidity is due to the transport of sediments by the converging currents and in some cases to the flocculation of river clays because of the sudden increase of salinity. The null point and the turbidity maximum move along the estuary with spring and neap tides and with the periods of high and low river discharge: more inward during spring tides and low river discharge, and outwards during neap tides and high river discharge. In fully mixed estuaries (Fig. 1c), vertical mixing

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is virtually complete and the density gradients are horizontal. This occurs where the tides dominate the river outflow. The particular circulation of the estuary of interest will define the model to be used. If the estuary is fully mixed, a two dimensional depth-averaged model may be enough since there are no vertical gradients of salinity and suspended matter to be resolved. In the cases of nontidal and partially mixed estuaries, vertical gradients are present, so that the vertical direction must be resolved too, and a fully three dimensional model is required. However, even if the estuary is fully mixed, a three dimensional model may be required to simulate the dispersion of pollutants depending on the water depth and the magnitude of the tidal currents (which will control the magnitude of the vertical diffusion coefficient). Indeed, Prandle et al. (1993) have shown that vertical structure can be of primary concern when typical simulated times are of the order of one month or less. Moreover, the vertical variability is important for such times, even in shallow waters (depth ∼ 50 m), if tidal mixing keeps the vertical diffusion coefficient smaller than 10−3 m2 /s, a value that can be considered representative of a relatively strong tidal action (Prandle et al., 1993). In the case of narrow estuaries, another option may be to develop a two dimensional XZ model in the longitudinal and vertical directions, thus averaged in the transverse direction, if this is not relevant. This kind of model is often applied in fjords. 1.2. The Odiel river estuary The Odiel river, in southwest Spain, forms a fully mixed estuary, with M2 being the main tidal constituent. It is shallow (maximum water depth is ∼ 10 m) and the flow of the Odiel river is low, typically ranging from 4 m3 /s (dry season) to 100 m3 /s (wet season). A map of the estuary is presented in Fig. 2. It is surrounded by a marsh area and an industrial complex, located at the south of the town of Huelva. The Tinto river joins the Odiel at the Punta del Sebo, and they flow together from this point to the Atlantic Ocean. Due to the low water discharge, mixing between sea and fresh water occurs north of Huelva. Thus, the water of the estuary in the area close to the industrial complex is virtually sea water. This can be seen in Fig. 3, where salinities measured along the estuary (from Periáñez, 1995) are presented. A phosphate fertilizer processing plant is located in the industrial complex. This plant released part of its wastes directly to the Odiel river. It is well known (Heijde et al., 1988; Laiche & Scott, 1991) that such wastes contain significant amounts of natural radionuclides (U, Th, Ra and their daughters) due to the fact that the phosphate rock, from which phosphoric acid is obtained, may contain U concentrations ranging from 50 to 300 ppm (Laiche & Scott, 1991). During the wet processing of phosphate rock for phosphoric acid production, for instance, 86% of U and 70% of Th present in the rock appear in the phosphoric acid itself, while 80% of Ra follows the so called phosphogypsum. This is an impure Ca sulfate that is removed as a precipitate during the process. Phosphogypsum is often discharged directly into estuaries, giving rise to a clear radioactive impact. U, Th and Ra isotopes have been measured in water, suspended matter and sediment samples collected from the Odiel river (Periáñez & García-León, 1993; Periáñez et al., 1994b; Martínez-Aguirre et al., 1994a, b; Periáñez & Martínez-Aguirre, 1997a). These measurements have revealed a clear radioactive impact in the estuary due to the operation of the fertilizer complex. For instance, 670 mBq/l of 226 Ra, 131 mBq/l of 238 U and 5 mBq/l of 232 Th have been detected in waters collected close to the outlet of the complex. However, measurements

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Fig. 2. Map of the Odiel estuary showing the sampling points (O1 to O7) and the fertilizer complex. The rectangular box is the grid used in the two dimensional model.

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Fig. 3. Water salinity along the Odiel (g/l) for samples collected during high and low water.

have pointed out the complex behaviour and dynamics of radionuclides in the estuary. Indeed, water samples were collected during high and low water, and radionuclide concentrations were measured in water and suspended matter for both tidal states. A qualitative description of results is not an easy task since there is significant variability in activity concentrations and distribution coefficients, kd , depending upon the sampling point and the tidal state. This is a logical result since we are dealing with an open system in which different effects are taking place at the same time: non constant input from the source, absorption–desorption reactions, deposition and erosion of the sediment, and water and suspended matter movements due to tides. Thus, the objective is to develop a model to describe the dispersion of these radionuclides in the estuary, including the distribution between the liquid and solid phases (suspended matter and bottom sediments) as well as the effect of tidal oscillations. Two modelling approaches have been applied. First, the general dispersion patterns of radionuclides in the estuary have been studied. To do this, a 1D dispersion model for conservative radionuclides has been developed. The movement of contaminant patches, due to tidal oscillations, discharged in the estuary from the fertilizer complex can be followed with this model. However, this model can only be applied to radionuclides that remain in solution, and thus, can be used to give us a general idea of the way that advective dispersion takes place. Later, a detailed 2D depth averaged model as been developed to study the dispersion of non conservative radionuclides (including the interactions with the solid phases) in the area close to the fertilizer complex. The 1D modelling approach is presented in the next section. Then, the 2D model is described. 2. General dispersion patterns for dissolved radionuclides: 1D approach The 1D model has been developed for both the Odiel and Tinto rivers, since tide propagation can be considered as a one-dimensional phenomenon. The marsh area extending on the

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right bank of the Odiel forms a complex multi-channeled system where little bathymetric or hydrodynamic data exists. Thus, its main connection to the Odiel river, the Canal del Burro (see map of Fig. 2), will be treated as an open boundary and recorded water elevations at this point will be used as boundary conditions. The end of the Juan Carlos I quay will be another open boundary, the upper tidal limits in the Odiel and Tinto rivers are the other two natural boundaries limiting our study area. The objective of this 1D model consists of studying the basic features of the dispersion of dissolved radionuclides. One of its main aspects is the accurate calculation of hydraulic parameters at every possible water level. The model also runs under real tidal conditions: time series of recorded elevations, instead of simple harmonic constituents, are used for boundary conditions. The effects of atmospheric forcing and the longitudinal water surface gradients have also been included. 2.1. 1D-hydrodynamic and dispersion model for the Odiel and Tinto estuary: constitutive equations The model presented by Abril & Abdel-Aal (2000) and Abril et al. (2000), developed to study radionuclide dispersion in the Suez Canal, has been adapted to the estuary. Only the basic features will be presented here. Let us consider a control volume in an open channel, defined by the reference crosssectional area A and the differential length dx along the longitudinal x direction. The free surface is dS = B dx, B being the width of the channel at the cross-sectional area. The water elevation, z, measured from the bottom of the channel, varies with time and position along the channel (x). A, B and dS are functions of the location and of z(x, t). The continuity equation can be written as: dS

∂z ∂Q + dx − QT = 0, ∂t ∂x

(1)

where Q is the water flow through A (in m3 /s) and QT represents the flow from tributary waters at the position given by the spatial coordinate x. A detailed treatment of the momentum equation can be found in Irving (1978). For our control volume, and considering the previous definitions, the equation is written as follows:   ∂Q(t) ∂ Q2 ∂ + + Ag (z + N) = Ffri − Fwin . ∂t ∂x A ∂x

(2)

N is the elevation of the bed of the channel, referred to a fixed datum level, and its spatial derivative gives us the slope of the channel producing the gravitational forcing. Terms in the right side of the equation represent wind forcing, Fwin , and friction, Ffri . The first has been adapted from Pugh (1987) and the second from French (1985). As discussed in Abril & Abdel-Aal (2000), this mathematical formulation can be applied to channelled, shallow and well-mixed estuaries. Thus horizontal and vertical stratification is not considered.

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The mass balance applied to a conservative tracer, with concentration C(x, t), gives the following equation:   1 ∂ ∂C Q ∂C ∂C + = ADx + Sources, ∂t A ∂x A ∂x ∂x

(3)

where a source and/or sink term is included. Dx is a diffusion coefficient parameterized as a function of the water velocity (shear effect) and the turbulent dispersion coefficient, Dtur : Dx = Dtur + ksh (Q/A)2 .

(4)

The hydrodynamic equations are solved using the numerical scheme presented in Prandle (1974). To avoid problems with numerical diffusion, a second order accuracy approximation to the upwind scheme has been applied to solve the advection term of equation (3). More specifically, a three-point formula, as presented in Kowalik & Murty (1993), has been used. 2.2. 1D-Hydrodynamics: input parameters and results To solve the equations, the studied area is divided in 53 sectors, each 1 km length. 31 sectors correspond to the Odiel, numbered upstream with sector 1 being the open boundary at the end of Juan Carlos I quay. The Tinto branch has 21 additional sectors, numbered consecutively from sector 16, the Odiel sector where the junction with the Tinto is located. Finally, 1 additional sector represents the entrance of the Canal del Burro, connecting with the Odiel at sector 17. Eight equispaced cross-sections are digitized, from marine and topographic charts, for each sector. A computer code was developed to produce the required hydraulic parameters as a function of the water elevation above or below each local mean sea level. Tidal records are required to provide both the maximum tidal amplitude and appropriate boundary conditions. Tidal tables are available for sectors 5 and 21 in the Odiel (IHM, 1992). The Harbor Authority provided the tidal records and current measurements from a campaign contracted in 1989 with INTECSA. The data cover sectors 1, 5, 14 and C. Burro (elevations), and 5, 12, 14 and 18 (current magnitudes); all the sectors lie in the Odiel. These records correspond to some 24 hours of measurements carried out during three different tidal situations (spring, medium and neap tides). No information was available for the Tinto river. Thus, currents were measured at an intermediate sector during a complete tidal cycle (Laissaoui, 1999). This information is used for model calibration purposes. Incoming water fluxes at the tidal limit of the Tinto and Odiel rivers were available from the Hydrographic Confederation of Guadiana. Wind velocity and direction is also known from the National Institute of Meteorology. The slope of the channel bed and the details for friction were unknown. Thus they are obtained from a trial and error exercise during the model calibration (Laissaoui, 1999). Recorded water elevations at sector 1 (Odiel) and the Canal del Burro sector are used as boundary conditions. The hydrodynamic module was run for a situation of medium tides. A comparison between computed and recorded tidal series is presented in Fig. 4. Measured and computed current amplitudes are presented in Fig. 5. Further calibration for the Tinto river was possible from the

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Fig. 4. Comparison between computed and recorded water elevations at several locations along the Odiel river in a situation of medium tides.

measured (for medium tide conditions) time series of velocities. Without any more changes, the hydrodynamic module was run for spring and neap tide conditions, producing reasonable agreement with the available data (Laissaoui, 1999). The model also showed its applicability under extreme conditions of spring tides and heavy rain. 2.3. Computing the path for the centroid of a cloud of a conservative tracer Some numerical experiments have been carried out to study the fate of instantaneous discharges of any conservative pollutant released at Foret (a fertilizer plant placed at sector 20 in the Odiel river). The displacement of the centroid along the Odiel river has been evaluated from the computed instantaneous velocities at the location where the centroid is at each

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Fig. 5. Comparison between measured (points) and computed (lines) current amplitudes along the Odiel river (medium tides). Positive and negative values means upstream and downstream currents, respectively.

Fig. 6. Computed paths for an instantaneous release of a conservative tracer at Foret location. The figure shows the required time to reach sector 1 in the Odiel river as a function of different tidal and weather conditions (see text).

time step (eulerian transport, see Abril & Abdel-Aal, 2000). The aim is to evaluate the time required for the centroid to reach the open sea (crossing sector 1 in Odiel). The position of the centroid versus the time after release, for different tidal and weather conditions, is shown in Fig. 6. The results correspond to medium, neap and spring tide situations, respectively, with river inputs of 3.1 (Odiel) and 107 m3 /s (Tinto). Spring tide (2) corresponds to spring tides and higher river inputs (1070 m3 /s at the Odiel, a typical value after heavy rain episodes). In all cases, the releases take place at the beginning of the ebb, which is the optimum situation for a faster cleaning of the system. It can be seen how for

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medium tides the plume oscillates for several tidal cycles with some small downstream drift (the increase in oscillations is due to the neap-spring cycle). Oscillations are higher for spring tides, also with a higher drift current when a larger water flux comes from the upper river after heavy rain episodes. The plume leaves the estuary when the centroid reaches the zero level, although some contamination returns in the following high tide episode. From this, it can be concluded that Juan Carlos I quay (at the right bank of the Odiel river, from sectors 12 to 1) is hindering the natural cleaning of the waters. Thus, it is contributing to the production of higher levels of contamination within the estuarine area. 2.4. 1D-dispersion: some global aspects Some numerical experiments have been carried out with the dispersion model. Concentrations at the open boundaries are evaluated from an analytical fit to the tail of a Gaussian distribution, as in Abril & Abdel-Aal (2000). From our previous work in the southern Suez Canal (Abril et al., 2000), and from field tracing experiments in several Spanish rivers (Plata, 1985), ktur in equation (4) has been fixed as 1 m2 s−1 , and a value of ksh = 5 was adopted. The model shows reasonable behaviour for the spatial and temporal evolution of the plume. For medium tide conditions, the tidal displacement of the centroid is some 10 km. Concentrations decrease roughly following a law of inverse proportionality to the square root of time (as expected from a Gaussian Model), but this law is broken at the Tinto and Burro connections. The plume can penetrate into the Tinto river only with the flood. It hardly reaches 10 km upstream, leaving this river with the ebb. Thus, concentrations in the Tinto river have to be related to those appearing downstream at the confluence of both rivers. During the ebb, a fraction of the pollutants extending over the upper Odiel river can leave the system through the Burro canal. Releases carried out during the flood can lead to a significant contamination of the upper marsh area of the Odiel river. In Fig. 7, an example of the temporal and spatial evolution of a conservative tracer plume is presented (situation of medium tides, initial plume of 1 km length with an homogeneous concentration of 1000 arbitrary units, located at sector 22 at the high water). In a second modelling exercise we simulated the dispersion of a continuous discharge of 226 Ra, taking place at a constant rate of 1.1 × 104 Bq/s (a value suggested by Bolivar, 1995). The simulation extends over several days along a neap-spring-neap tidal cycle. At each sector, the time evolution of concentrations shows tidal oscillations over an asymptotic-like pattern of increase (see Fig. 8). The spatial maps of these concentrations give a picture of the estuary areas being more affected by the discharges. This is plotted in Fig. 9, during high water after 5 days. The distribution pattern shows the expected structure in which we can recognize higher concentrations around the source site in the Odiel, and a decreasing pattern from the junction to the upper tidal limit in the Tinto. Upstream from Foret, concentrations decrease due to the vicinity of the upper tidal limit and the fresh water inflow. A sensitivity test for boundary conditions is presented in Laissaoui (1999). As discussed in this reference, measured concentrations at different tidal and seasonal conditions show a similar pattern, but they are higher by a factor 4 in samples from the summer of 1990 (excluding a point sampled just at the pipe-line outlet). This factor is only 3 in samples from the spring of 1991, and is unity after the wash out effect of a week of heavy rains (also in samples from 1991). One possible explanation is that the real source term should be higher than that estimated by Bolivar (1995).

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Fig. 7. An example of the spatial and temporal evolution of a conservative tracer plume in the Odiel an Tinto rivers (see text for an explanation of initial conditions and for the numbering of sectors). Concentrations are given in arbitrary units.

Also, non continuous discharges can add a more complex structure to the observed patterns. Finally, 226 Ra is not a totally conservative radionuclide. Thus, interactions with suspended matter and bottom sediments should be considered for a more realistic simulation.

3. A 2D model for non conservative radionuclides The first models which were developed to simulate the dispersion of non conservative radionuclides in aquatic systems were averaged box models in which the exchanges between the liquid and solid phases were described in terms of equilibrium distribution coefficients kd . However, in open systems and when studying dispersion in small coastal areas and estuaries

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Fig. 8. Computed time evolution of 226 Ra concentration in sector 21 of the Odiel river. The simulation corresponds to a continuous discharge taking place at a constant rate and at Foret location (sector 20 in the Odiel).

Fig. 9. Computed 226 Ra concentration obtained after 5 days of simulation during high water along the Odiel (solid line) and Tinto (dashed line) rivers for a continuous discharge at a constant rate.

the equilibrium approach will not hold due to the small time steps required in calculations: equilibrium will not be reached in each time step. In these cases a kinetic approach is more appropriate. Thus, the equilibrium will be the dynamic balance between two opposite chemical reactions (Nyffeler et al., 1984). The system under study is divided into a number of grid cells. Four phases are present in each grid cell: water, suspended matter and two sediment grain size fractions. One grid cell is shown in Fig. 10. Radionuclides can be dissolved or associated with suspended matter. As

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Fig. 10. Grid cell in which the radionuclide transfers between the four phases considered in the model are represented.

tides produce a continuous movement of water, radionuclides in these phases will be transported from one grid cell to another by advection and diffusion processes. Only particles with a diameter < 62.5 μm will be considered, as usual, to be present in the water column as suspended matter: larger particles will rapidly sink to the bottom and their horizontal movement is negligible. In sediments, two grain size fractions are considered, particles with a diameter < 62.5 μm (small grain size fraction) and particles with a diameter > 62.5 μm (large grain size fraction). Only the small grain size fraction particles can be resuspended and incorporated into the water column as suspended matter. On the other hand, when suspended matter particles are deposited on the bed sediment, they are incorporated into the small grain size sediment fraction. The dissolved phase is in contact with the other three phases, thus absorption-desorption reactions take place, which are described in terms of the kinetic transfer coefficients k1 (transfer from water to the solid phases) and k2 (desorption from the solid phases). Finally, external sources of radionuclides may exist in the grid cell. It seems clear that the hydrodynamic and the suspended matter dynamic of the estuary must be solved to compute the advective transport of radionuclides and to obtain the suspended matter concentration and the erosion and deposition rates for each point of the estuary and for each instant of time. 3.1. Hydrodynamics The 2D shallow water depth averaged hydrodynamic equations are (Pugh, 1987):

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∂ ∂ ∂z + [(D + z)u] + [(D + z)v] = 0, ∂t ∂x ∂y √ ∂u ∂u ∂z u u2 + v 2 ∂u +u +v +g − v + K = 0, ∂t ∂x ∂y ∂x D+z √ ∂v ∂v ∂v ∂z v u2 + v 2 +u +v +g + u + K = 0, ∂t ∂x ∂y ∂y D+z

(5) (6) (7)

where u and v are the depth averaged water velocities along the x and y axis, D is the depth of water below the mean sea level, z is the displacement of the water surface above the mean sea level measured upwards,  is the Coriolis parameter ( = 2w sin β, where w is the earth rotational angular velocity and β is latitude), g is acceleration due to gravity and K is the bed friction coefficient. The effect of wind is included by adding the terms −

ρa CD |W |W cos θ, ρw D + z

(8)

−

ρa CD |W |W sin θ ρw D + z

(9)

to the left-hand side of equations (6) and (7), respectively. ρa and ρw are the densities of air and water, W is the wind speed and θ is the direction to which wind blows measured anticlockwise from east. CD is a drag coefficient. An acceptable value is (Pugh, 1987): CD = (0.63 + 0.066W ) × 10−3 ,

(10)

if 2.5 < W < 21 with W measured in m/s 10 m above the sea surface. The part of the estuary that is covered by the 2D model is presented in Fig. 11, where water depths are also shown. A spatial discretization of the estuary is carried out to solve the equations by finite differences. Thus, the estuary is divided into a number of grid cells with sizes x = y = 100 m (see Fig. 11). The time step is fixed as t = 6 s. In this way the CFL (Kowalick & Murty, 1993) condition is satisfied. In particular, the explicit finite difference scheme described in Flather & Heaps (1975) has been applied. Water elevations were specified along the southern open boundary of the computational domain and a radiation condition was applied at the northern boundary. The bed friction coefficient was fixed at K = 0.0025 after a calibration process. The output of the model has been compared with observations in situations of medium and neap tides. This is presented in detail in Periáñez et al. (1994a). As an example, a corange chart for the M2 tide is presented in Fig. 12, where it can be seen that the amplitude of the tide increases as it goes inwards along the estuary. The time evolution, over several tidal cycles, of water elevations and currents at a point close to the southern boundary of the grid is shown in Fig. 13. It can be seen that currents in the transverse direction of the estuary (u component) are weaker than currents in the longitudinal direction (v component), so that the water movement is essentially directed along the axis of the estuary.

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Fig. 11. Map of the part of the estuary covered by the 2D model (depths are given in m). The grid used for computations is also shown. Each unit in the x and y axis is 1 m.

3.2. Suspended matter dynamics It is assumed that each grid cell contains a certain amount of suspended matter, m, in g/m3 with an homogeneous distribution in the water column. This is realistic since, as commented above, the estuary is shallow and mixing between fresh and salt water takes place upstream of the modelled area. Also, the flow of the river is low and a fast dispersion of fresh water into a much larger volume of salt water occurs (Borrego & Pendon, 1988), as usual in tidal estuaries of small rivers (Eisma, 1993). As usual in suspended matter studies, we will consider that only particles with a diameter < 62.5 μm can remain in the water column as suspended matter (Belderson, 1964; Clarke, 1995). Indeed, Eisma (1981) has pointed out that for all practical purposes, muds can be regarded as synonymous with suspended matter. The suspended matter equation is (Periáñez et al., 1996a):     ∂ ∂(H m) ∂(uH m) ∂(vH m) ∂m ∂m ∂ + + = H Kx + H Ky ∂t ∂x ∂y ∂x ∂x ∂y ∂y     q q + vr ρs f 103 − ws m 1 − −1 , qcd qce

(11)

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Fig. 12. Computed corange chart for the M2 tide. Tidal amplitudes are given in m. x and y axis denote the grid cell number, thus each unit corresponds to 100 m.

where H = D + z is the total water depth, and Kx and Ky are the diffusion coefficients in the x and y directions, respectively. The last two terms √ represent deposition and erosion: ws is the settling velocity of suspended particles, q = u2 + v 2 is the current magnitude, vr is a resuspension velocity from the sediment, ρs is the dry matter density of the sediment (in kg/m3 ) and f gives the fraction of small particles in the sediment. qcd and qce are threshold currents for deposition and erosion, respectively. Thus, it is considered that deposition of particles can occur only if the current is smaller than qcd . Similarly, there is erosion of the sediment only if the current is larger than a critical value qce . If any of these conditions is not satisfied, the corresponding term is set to zero.

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Fig. 13. Time evolution of computed water elevations (m) and components u and v of the current (m/s) in a point located close to the southern boundary of the grid for the M2 tide.

The description of Prandle (1984) was used for the diffusion coefficients: Kx = β1 |u|q,

(12)

Ky = β2 |v|q,

(13)

where β1 and β2 are numeric factors to be calibrated for each specific site. The settling velocity of suspended particles increases, at low concentrations (smaller than 1000 ppm), as the suspended matter concentration increases due to flocculation of particles. A standard formula to represent this process is (Pejrup, 1988; Mehta, 1989; Clarke, 1995): ws = a1 ma2 , where a1 and a2 have to be obtained from observations or from model calibration.

(14)

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The solution of the suspended matter equation provides the suspended matter concentration at each position of the grid and at each instant of time. In the case of the Odiel river, β1 = β2 = 1000 s (Periáñez et al., 1994a, 1996a); a1 = 1.74 × 10−9 and a2 = 1.6, which are the values found by Mehta (1989); qcd = 0.18 m/s (Eisma, 1993) and vr = 9.6 × 10−7 m/s (Periáñez et al., 1996a). The erosion threshold velocity depends on the roughness length of the sediment, z0 . From data in Pugh (1987), an analytical function relating qce and z0 has been constructed: qce = 0.441e−1.117z0

0.231

,

(15)

if qce is expressed in m/s and z0 in cm. Values of f , ρs and z0 over the estuary have been obtained from measurements (Universidad de Sevilla, 1991). The suspended matter equation has also been solved using finite differences. The first order upwind scheme was used for the advective terms and a second order explicit scheme for the diffusion ones. Both are presented in Kowalick & Murty (1993). It is known that the upwind scheme introduces numerical diffusion. However, it has been proved (Periáñez et al., 1994a) that this is negligible in the present model due to the small size of the grid cells. The boundary condition in the northern boundary of the grid consists of making the suspended matter concentration in the last row equal to that of the previous row. Along the southern boundary, suspended matter concentrations were specified for each time step from observations: a sampling campaign was performed so that water samples were collected every 20 min during a complete tidal cycle in both the southern and northern borders of the grid. Suspended matter concentrations were measured using a calibrated nephelometer (Periáñez, 1995); results can be seen in Fig. 14. The time evolution of computed suspended matter concentrations in three grid cells can be seen in Fig. 15. As we move towards the north of the grid, oscillations in suspended matter decrease. Indeed, the suspended matter concentration in the cell near the northern border is almost constant. This behaviour has been observed in the Odiel estuary: Fig. 14 shows that concentrations at the northern border are quite constant, with a mean value 28 ± 4 ppm. The computed mean value is 24.4 ppm, in good agreement with observations. This behaviour is due to the fact that water velocities decrease as we move towards the north (Periáñez et al., 1994a), so resuspension becomes negligible and deposition must be essentially constant along the tidal cycle. Thus, oscillations in suspended matter concentrations vanish. On the other hand, it can be seen in Fig. 15 that there are two peaks in concentrations for each tidal cycle in the south of the grid: during flood and ebb. In these situations currents are stronger and erosion of the sediment occurs. During high and low water, deposition takes place and the suspended load reaches a minimum. A double peak in m is due to the combination of the quarterly diurnal signal of resuspended material with the semi diurnal signal of advected material (Williams et al., 1998). Suspended matter concentration maps, when water level is increasing and decreasing, can be seen in Fig. 16. The decrease in concentrations as we move towards the north can be clearly seen here. Also, the model can give information about the net sedimentation rates in the estuary (see Periáñez et al., 1996a for details).

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Fig. 14. Suspended matter concentrations (ppm) measured in the northern (a) and southern (b) boundaries of the grid over a tidal cycle. Water elevations are given with respect an arbitrary reference.

3.3. Radionuclide equations As commented above, transfers of radionuclides between the liquid and solid phases will be described in terms of kinetic transfer coefficients. A coefficient k1 governs the transfer from water to the solid phases and a coefficient k2 governs the inverse process. The absorption process is a surface phenomenon that will depend on the surface of particles per water volume unit into the grid cell. This quantity is denoted as the exchange surface (Periáñez et al., 1996b). Thus, the kinetic transfer coefficient k1 will be proportional to the exchange surface: k1 = χ1 (Sm + Ss + Sl ),

(16)

where the proportionality factor χ1 has the dimensions of a velocity and is denoted as the exchange velocity (Periáñez et al., 1996b). It is related to the molecular velocity of dissolved radionuclides, which at the same time is related to the temperature of water. Sm , Ss and Sl are,

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Fig. 15. Time evolution of the suspended matter concentration in three grid cells: near the southern border (solid line), in the middle of the grid (dashed line) and near the northern border (dashed dotted line). The moments at which high and low water occur are also shown.

respectively, the exchange surfaces for suspended matter, the small and the large grain size fractions of sediments. As a first approach, assuming spherical particles and a step function for the grain size distribution of particles, it can be easily shown (Periáñez et al., 1996b) that: Sm =

3m , ρR

(17)

where ρ and R are density and mean radius of suspended matter particles, respectively. The water-sediment interface can be considered as a high suspended matter environment (Li et al., 1984). Thus, the exchange surfaces for both sediment fractions are (Periáñez et al., 1996b): 3Lf φ , RH 3L(1 − f )φ  , Sl = Rl H Ss =

(18) (19)

where L is the average mixing depth (the distance to which the dissolved phase penetrates the sediment), Rl is the mean radius of particles of the large grain size sediment fraction and φ and φ  are correction factors that take into account that not all the surface of the sediment

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(a)

243

(b)

Fig. 16. Suspended matter concentration (ppm) maps when water level is increasing (a) and decreasing (b). Each unit in the x and y axis is 1 m.

particle is in contact with water, since it will be partially hidden by other particles. Thus, the total k1 coefficient can be written as:   3m 3Lf φ 3L(1 − f )φ  + + k 1 = χ1 . (20) ρR RH Rl H It is interesting to note that k1 is proportional to the suspended matter concentration. Indeed, some laboratory experiments (Benes et al., 1992; Benes & Cernik, 1992) have shown that a direct relation between both exists. Equation (20) is the analytical form of such a relation. The equation that gives the time evolution of specific activity in the dissolved phase, Cd (Bq/m3 ), is: ∂(Cd H ) = (adv + dif) − k1 Cd H + k2 Cs mH ∂t

 + k2 Lρs 103 f φAs + (1 − f )φ  Al ,

(21)

where (adv + dif) means advective plus diffusive transport of radionuclides, Cs (Bq/g), As (Bq/g) and Al (Bq/g) are, respectively, specific activities in suspended matter, small and large

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grain size fractions of sediments. The external source of radionuclides should be added to this equation at the points where it exists. The transfer coefficient k1 is given by equation (20). The equation for the time evolution of specific activity in suspended matter particles is: ∂(mCs H ) = (adv + dif) + k1 Cd H − k2 mCs H + (er − dep), ∂t

(22)

where k1 is given by the first term of equation (20) and the deposition and erosion terms are given by:   q , dep = ws mCs 1 − qcd   q er = vr ρs f As 103 −1 . qce

(23) (24)

Again, a source term should be included in this equation if an external input of radionuclides fixed to solid particles exists. The small grain size fraction of the sediment exchanges radionuclides with the dissolved phase and with suspended matter, through erosion and deposition. The corresponding equation is: Cd H −3 ∂As = k1 10 − k2 As φ + (dep − er), ∂t Lρs f

(25)

where k1 is given by the second term in equation (20) and the erosion and deposition terms are:   vr As q er = −1 , (26) L qce   ws mCs −3 q 10 dep = 1− . (27) Lρs f qcd Finally, the large grain size fraction of the sediment exchanges radionuclides only with the dissolved phase: ∂Al Cd H = k1 10−3 − k2 Al φ  , ∂t Lρs (1 − f )

(28)

where k1 is now given by the third term in equation (20). Specific activity in the total sediment will be: Atotal = f As + (1 − f )Al .

(29)

This description of the dispersion of non conservative radionuclides has been applied to simulate the transport of 226 Ra, 238 U and 232 Th in the Odiel estuary, as will be shown below.

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However, it has been applied to other environments to show the universality of the description. Thus, it has been applied to simulate the dispersion of 137 Cs and 239,240Pu in the English Channel (Periáñez, 2000a) and to simulate the dispersion of 239,240Pu in the eastern Irish Sea (Periáñez, 1999, 2000b). In this latter case, the formulation has been extended to a three dimensional description. A very similar description has also been used by other authors to simulate the dispersion of radionuclides in the sea (Aldridge, 1998; Margvelashvily et al., 1997). 3.4. kd description Suspended matter–water distribution coefficients are experimentally obtained as the specific activity in suspended matter divided by the specific activity in water. Absorption–desorption reactions take place only over the surface layer of suspended particles. The central part of particles cannot interact with the liquid phase. Specific activity in suspended matter, Cs , that is calculated by equation (22), refers to specific activity in this surface layer, that is the activity in the surface layer of suspended matter particles divided by the total mass of suspended particles. However, we are interested in modelling Ra, U and Th dispersion, which are natural radionuclides and thus, will also be present in the central part of particles. These radionuclides are considered when an experimental determination of kd is carried out. They will be included in the model by assuming that there is a constant specific activity, for each radionuclide, in the central part of particles (supposed to be constant since radionuclides are locked in particles). Cs is then corrected adding to its value, such specific activity in the central part of particles. This specific activity is estimated from field information. The corrected radionuclide concentration in suspended matter is used to compute kd values and is the presented concentration when results corresponding to specific activity in suspended matter are shown. 3.5. Dispersion of 226 Ra, 238 U and 232 Th in the Odiel estuary 3.5.1. Parameters for the model The different parameters involved in the description of the dispersion of radionuclides have to be obtained from field measurements or from model calibration. The density of particles in suspension has been taken as ρ = 2600 kg/m3 , which is the established value for the density of soil particles (Baver et al., 1992). As commented above, it is considered that suspended matter and the small grain size fraction of the sediment are composed of particles with a diameter < 62.5 μm. Thus, the mean radius of such particles is fixed as R = 15 μm. The mean radius of the large grain size fraction has been estimated (Periáñez et al., 1996c) from measurements (Universidad de Sevilla, 1991): Rl = 440 μm. After a calibration exercise, the mixing depth and the correction factors for the sediments were selected as L = 0.1 m, φ = 0.01 and φ  = 0.005. Some sensitivity tests to study the model response to changes in these parameters can be seen in Periáñez et al. (1996c). In the case of 226 Ra, the coefficients χ1 and k2 , which govern the transfers between the dissolved and solid phases, were obtained from laboratory experiments (Laissaoui et al., 1998). They consist of tracing, at the laboratory, unfiltered Odiel estuary water with a 133 Ba solution, a γ emitter whose chemical behaviour is similar to that of Ra. Conditions at the laboratory were as close as possible to natural conditions (temperature, pH, movement of water). The

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time increase of the 133 Ba activity in suspended matter allows the calculation of both coefficients from a numerical fitting to the analytical solution of the corresponding differential equations. Details can be seen in the above mentioned reference. Results are: χ1 = 0.55 × 10−7 m/s and k2 = 8.17 × 10−6 s−1 . Nevertheless, these coefficients may change slightly from one season to another due to changes in water conditions (salinity and temperature). Thus, some sensitivity tests to study the model response to these changes are presented in Periáñez et al. (1996c). In previous work (Nyffeler et al., 1984), kinetic transfer coefficients were measured for a wide set of elements. It was found that there was very small variation in k2 (less than an order of magnitude) for elements with very different geochemical behaviour (Th, Hg, Pa, Zn, Sb, Cs, Se, Cd, Ba, Sr, Na). This small variation opens the possibility, as suggested by Nyffeler et al. (1984), of estimating χ1 without laboratory experiments. Indeed, it has been shown (Periáñez et al., 1996c) that the following relation between χ1 , k2 and the suspended matter–water distribution coefficient, kd , holds: kd =

χ1 3 . k2 ρR

(30)

Thus, the χ1 values for U and Th can be obtained from the k2 of Ba (obtained from the laboratory experiments carried out with Odiel waters) and the corresponding kd of U and Th measured in the estuary. The mean value of the U distribution coefficient in the estuary is in agreement with the value presented in IAEA (1985) (see Periáñez & Martínez-Aguirre, 1997a): 1 l/g. In the case of Th, measured kd s are lower than the average value given in IAEA (1985). The mean kd value in the estuary, for Th, is 13.7 l/g (Periáñez & Martínez-Aguirre, 1997a). From equation (30) and the corresponding kd values, it is finally obtained that for U, χ1 = 1.06 × 10−7 m/s, and for Th the exchange velocity is χ1 = 1.46 × 10−6 m/s. The exchange velocity of Th is an order of magnitude larger than that of U, revealing a larger affinity of Th to be fixed to the solid phases. These exchange velocities, although not exact values, seem to be good estimates since the model produces, as will be seen, results that are, in general, in agreement with experimental data. 3.5.2. Results As the rate of change of activity in the sediment is slow, there are no significant changes during the typical simulated times (some days). Consequently, we use experimental values of specific activities in the sediments as initial conditions. These have been taken from Martínez-Aguirre et al. (1994a). Samples in the Odiel estuary were collected, as commented above, during high and low water, with a time difference of 18 hours between them (19th and 20th July, 1990). Generally speaking, activity peaks should correspond with discharges performed shortly before or during sampling, while, when a flat distribution is measured, it should correspond with the background that remains in the river after the earlier homogenization of discharges. Activities released to the estuary from the fertilizer complex are not known. Thus, the magnitudes of the activity inputs were changed, by trial and error, until model output was in agreement with observations. Thus, to reproduce the sampling conditions a first activity input was introduced in tidal cycle 1 and lasted 3.3 h. High water concentrations were obtained 2 cycles later (cycle 3).

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A second activity input started in cycle 4 and lasted 9 h. Low water concentrations were obtained from this cycle. In this way, 18 h elapsed between the time-points in which concentrations for high and low water are obtained. The initial conditions applied to the dispersion model consisted of making Cd = Cs = 0 over the grid. However, in the Odiel estuary a significant background activity exists. In the case of 226 Ra, the first activity inputs were 9.0 × 107 and 9.9 × 106 Bq per time step for the dissolved phase and the suspended matter, respectively. During the second input, they were, respectively, 2.0 × 104 and 2.5 × 103 Bq per time step. Thus, the first input is larger than the second because it is used to create such a background, and 2 cycles are allowed to elapse so as to distribute the released radionuclides along the estuary. It has been shown (Periáñez et al., 1994a) that this method of creating the background does not influence the results. Wind speed and direction corresponding to the date of sampling were used in the model (Periáñez et al., 1994a). As can be seen in Fig. 17, there is a good agreement between observed and computed specific activities in both phases (water and suspended matter) and for both tidal states. Computed concentrations correspond to longitudinal sections of the estuary following the points where samples were collected. Sediments are not included in the discussion because there are no apparent changes in their specific activities during a time period of the order of some days. kd distribution coefficients between suspended matter and water have also been calculated with the model, and are presented in Fig. 18 for high and low water together with their observed values. The general behaviour of distribution coefficients along the estuary is reproduced by the model. The time sequence of activity inputs in the cases of 238 U and 232 Th is the same as that of 226 Ra, since specific activities for these radionuclides were obtained from the same set of samples. In the case of U, the first inputs were 1.4 × 106 and 1.1 × 102 mg of U per time step for the dissolved phase and suspended matter, respectively. For the second input, the corresponding rates were 1.3 × 106 and 1.0 × 102 mg per time step. It can be seen in Fig. 19 that there is a good agreement between computed and observed concentrations of U in both phases (however, dissolved U in high water could not be measured and thus, the model output could not be compared with observations in this case). Measured and computed distribution coefficients for the low water samples are presented in Fig. 20. The model reproduces the general behaviour of the kd along the estuary. In the case of Th, the first input rates were 3.3 × 104 and 2.6 × 102 mg per time step for the dissolved phase and suspended matter respectively. Rates were, respectively, 1.9 × 104 and 1.5 × 102 mg per time step for the second input. Experimental and model results are presented in Fig. 21. The general behaviour of Th is well reproduced by the model. It is interesting to note that the model is able to reproduce spatial variations in concentrations of three orders of magnitude. As in the case of U, results for the dissolved phase in the high water samples could not be compared with observations. Computed and measured kd for the low water samples are shown in Fig. 22. The Th/U mass ratios have also been investigated with the model. They are presented in Fig. 23. An excess of U with respect to Th in suspended matter reveals an external source of U contaminated particles to the estuary (Periáñez & Martínez-Aguirre, 1997a; Periáñez et al., 1996d). However, close to the fertilizer complex there is an excess of Th with respect to U, which corresponds to the most contaminated area. Indeed more Th than U has been discharged

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Fig. 17. Model results for high (A) and low (B) water samples. 226 Ra concentrations in water are given in mBq/l and in suspended matter in mBq/g. Points correspond to the measured concentrations are lines are the model results. The x axis is the position in the grid (grid cell number), thus each unit corresponds to 100 m.

to the estuary. From the second input rates (to obtain low water levels) Th/U = 2.4 in solution and Th/U = 1.5 in suspended matter. As a consequence, there is an excess of Th in both phases close to the source. In contrast, there is an excess of U in the rest of the estuary and in both phases. It seems that the dissolved Th that is released to the estuary is quickly fixed to

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Fig. 18. Computed and measured 226 Ra distribution coefficients (l/g) for the high (A) and low (B) water samples.

the solid phase (bottom sediments and suspended matter particles). These particles, together with Th contaminated particles released from the source, are deposited on the estuary bed near the fertilizer complex. This could explain the U excess that is obtained in the rest of the estuary from both measurements and computations. This hypothesis is also supported by

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Fig. 19. Model results for U dispersion, for the low (A) and high (B) water samples. U concentrations in water are given in μg/l, and in suspended matter in μg/g.

the fact that an excess of Th has been measured in sediments collected close to the complex (Martínez-Aguirre et al., 1994a). The model sensitivity to the different parameters, the way of creating the background and the magnitude of the source term have also been studied. These can be seen in Periáñez et al. (1994a, 1996c).

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Fig. 20. Computed and measured distribution coefficients (l/g) for U during low water.

3.5.3. Predictive studies Some predictive studies concerning the process of cleaning of the estuary have been carried out with the model. 226 Ra was used for the numerical experiments. A simulation over 10 tidal cycles was performed, during which there were no activity discharges to the estuary. Concentrations in the sediments were those measured by Martínez-Aguirre et al. (1994a) and arbitrary, although realistic, concentrations were assumed as initial conditions for water and suspended matter. These should simulate the 226 Ra background remaining in the estuary after activity discharges stop. In a first trial they were 50 mBq/l in water and 25 mBq/g in the surface layer of suspended matter particles. After the simulation, it was verified that there were no apparent changes in sediment specific activities over such a time period. Thus, the process of cleaning of the sediment should be studied with a model based upon residual circulation, that allows larger time steps and, consequently, simulations over longer time scales (years) can be carried out (Periáñez & Martínez-Aguirre, 1997b). The time evolution of computed specific activities in water and the surface layer of suspended matter particles, and distribution coefficients in a compartment of the central part of the grid can be seen in Fig. 24. Concentrations in both phases, as well as the distribution coefficient, begin to oscillate at the same frequency as tidal oscillations. It is interesting to note that specific activities increase in both phases although there are no external inputs to the estuary. This activity comes from the sediments, that now act as a source of radionuclides as they are slowly cleaned. Similar effects have been found in other modelling work (Hofer & Bayer, 1997). These results have been obtained with arbitrary initial concentrations in water and suspended matter. The results of the same simulation, but starting from different initial conditions are presented in Fig. 25. The new initial conditions are 50 mBq/l and 100 mBq/g for water and

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Fig. 21. Model results for Th dispersion. (A): water and suspended matter during high water. (B): suspended matter during low water. (C): water during low water. Concentrations are given in μg/l and μg/g for water and suspended matter, respectively.

the surface layer of suspended matter particles respectively. As before, specific activities and distribution coefficients oscillate following tides, and concentration in water increases again. However, it is interesting to note that at the end of the simulation, the distribution coefficient has reached the same value as in the previous case. It seems that the distribution coefficient

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Fig. 21. (Continued).

Fig. 22. Computed and measured distribution coefficients (l/g) for Th during low water.

tends to an equilibrium value (slightly altered by tidal oscillations) independently of the initial activities. From equation (30), substituting the corresponding values of χ1 and k2 for 226 Ra, and the values selected for ρ and R, the equilibrium distribution coefficient for 226 Ra is kd = 0.52 l/g.

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Fig. 23. Computed and measured Th/U mass ratios for suspended matter during high water (A), water samples during low water (B) and suspended matter during low water (C).

Nevertheless, 226 Ra is a natural radionuclide that is present in the central part of suspended particles. These radionuclides that are locked in particles cannot be exchanged with the liquid phase (and are not considered in equation (30)), but are considered when an experimental determination of kd is carried out. The model accounts for this by assuming a background 226 Ra activity in the central part of particles equal to 45 mBq/g (Periáñez et al., 1996c),

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Fig. 23. (Continued).

since this is the measured specific activity in suspended matter far upstream of the fertilizer complex. Thus, the contribution of this activity to the kd is 45/80 = 0.56 l/g (80 is specific activity in water, Fig. 25). If this contribution is subtracted from the computed kd , we obtain kd = 1.1 − 0.56 = 0.54 l/g, which is the value that can be compared with the result obtained from equation (30), 0.52 l/g. Therefore, it can be concluded that after the simulation time, the transfers between water and suspended matter have reached equilibrium (slightly altered by tidal oscillations, as mentioned above). Moreover, equilibrium kd , 1.1 l/g, agrees with that defined in current literature. Indeed, it can be seen in IAEA (1985) that 226 Ra kd in coastal waters ranges from 0.5 to 50 l/g, with a mean value of 5 l/g.

4. Conclusions of the chapter The dispersion of radionuclides in an estuarine system has been studied. The Odiel river estuary, located at the southwest of Spain, forms a fully mixed estuary in which M2 is the main tidal constituent. A phosphate fertilizer processing complex releases its wastes to the estuary, thus leading to clearly enhanced levels of natural radionuclides in water, suspended matter and sediments. The physical characteristics of the estuary defines the model to be applied. In this case, a 1D model has been used to study the general aspects of the movement of dissolved radionuclides in the estuary. Also, a 2D depth averaged model has been applied to study, in detail, the dispersion of non conservative radionuclides in the area close to the release point from the complex. The transfers of radionuclides between the liquid and solid phases have been described in terms of kinetic transfer coefficients. The model must solve, together with the

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Fig. 24. (a) Time evolution of 226 Ra concentrations in water, in mBq/l (solid line), and in the surface layer of suspended matter, in mBq/g (dashed line), for a cell in the center of the grid. (b) Time evolution of the distribution coefficient (l/g) in the same grid cell.

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Fig. 25. Same as the previous figure, but starting from different concentrations in water and in the surface layer of suspended matter particles.

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equations whose solutions give the time evolution of specific activities in each of the four phases considered in the model, the hydrodynamic equations and the suspended matter equation, including the erosion and deposition terms. This model has been applied to simulate the dispersion of 226 Ra, 238 U and 232 Th. In general, the model reproduces the behaviour of these radionuclides in the estuary. Computed kd are also, in general, in agreement with measurements. It has also been observed that cleaning of the estuary is a slow process since the sediments act as a source of radionuclides to the water column once that direct discharges from the fertilizer complex have stopped. Finally, it should be pointed out that these kind of models are useful predictive tools that can be used in the assessment of contamination (radioactive, heavy metals, chemicals) following accidental or deliberate releases. Also, due to the predictive possibilities of the models, they can be applied for engineering purposes, to investigate, for instance, the effect of a potential dredging in the movement of a contaminated sediment. Acknowledgements Work supported by ENRESA and contracts DGICYT PB89-0621, EU FI3P-CT92-0035 and DGESIT 1FD97-0900-C02-01. The authors are also indebted to Prof. M. García-León for his careful reading of the manuscript and useful suggestions to improve it.

References Abril, J. M. & Abdel-Aal, M. M. (2000). Marine Radioactivity Studies in the Suez Canal, Part I: Hydrodynamics and Transit Times. Estuarine, Coastal and Shelf Science, 50, 489–502. Abril, J. M., Abdel-Aal, M. M., Al-Gamal, S. A., Abdel-Hay, F. A. & Zahar, H. M. (2000). Marine Radioactivity Studies in the Suez Canal, Part II: Field experiments and a Modelling Study of Dispersion. Estuarine, Coastal and Shelf Science, 50, 503–514. Aldridge, J. N. (1998). A model for prediction of marine radionuclide transport in both particulate and dissolved phases. Radiation Protection Dosimetry, 75, 99–103. Baver, L. D., Gardner, W.H . & Gardner, W. R. (1992). Soil Physics, New York: Wiley. Belderson, R. H. (1964). Holocene sedimentation in the western half of the Irish Sea. Marine Geology, 2, 147–163. Benes, P. & Cernik, M. (1992). Kinetics of radionuclide interaction with suspended solids in modelling the migration of radionuclides in rivers. Effect of the concentration of solids and temperature. Journal of Radioanalytical Nuclear Chemistry, Articles, 159, 187–200. Benes, P., Cernik, M. & Lam-Ramos, P. (1992). Factors affecting interaction of radiocesium with freshwater solids. Contact time, concentration of solids and temperature. Journal of Radioanalytical Nuclear Chemistry, Articles, 159, 201–218. Bolivar, J. P. (1995). Aplicaciones de la espectrometría gamma y alfa al estudio del impacto radiactivo producido por industrias no nucleares. PhD Thesis, University of Seville (in Spanish). Borrego, J. & Pendon, J. G. (1988). Algunos ejemplos de influencia de los procesos antrópicos en el medio sedimentario: la ría de Huelva. Henares, Revista de Geología, 2, 299–305 (in Spanish). Clarke, S. (1995). Advective/diffusion processes in the Firth of Forth. PhD Thesis. Bangor: University of Wales. Eisma, D. (1981). Supply and deposition of suspended matter in the North Sea. Spec. Publs. int. Ass. Sediment, 5, 515–428. Eisma, D. (1993). Suspended matter in the aquatic environment. Berlin: Springer. Flather, R. A. & Heaps, N. S. (1975). Tidal computations for Morecambe Bay. Geophysical Journal of the Royal Astronomical Society, 42, 489–517. French, R. H. (1985). Hidráulica de canales abiertos, México: McGraw Hill (in Spanish).

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Heijde, H. B. van, Klijn, P. J. & Passchier, W. F. (1988). Radiological impacts of the disposal of phosphogypsum. Radiation Protection Dosimetry, 24, 419–423. Hofer, H. & Bayer, A. (1997). Assessment of the dispersion of radionuclides in flowing water using a dynamic model. In G. Desmet, R. J. Blust, R. N. J. Comans, J. A. Fernandez, J. Hilton & A. de Bettencourt (Eds), Freshwater and estuarine radioecology (pp. 489–496). Amsterdam: Elsevier. IAEA (1985). Sediment kd and concentration factors for radionuclides in the marine environment. Technical Reports Series, 247, Vienna. IHM (1992). Anuario de mareas 1993. Cádiz: Instituto Hidrográfico de la Marina, Sección Oceanográfica (in Spanish). Irving, S. H. (1978). Mecánica de Fluidos. México: McGraw Hill (in Spanish). Kowalick, Z. & Murty, T. S. (1993). Numerical modelling of ocean dynamics. Singapore: World Scientific. Laiche, T. P. & Scott, L. M. (1991). A radiological evaluation of phosphogypsum. Health Physics, 60, 691–693. Laissaoui A. (1999). Estudios sobre los procesos que regulan el comportamiento de los radionúclidos en aguas de estuario. Aplicación a la Ría de Huelva. PhD Thesis. University of Seville (in Spanish). Laissaoui A., Abril, J. M., Periáñez, R., García-León, M. & García-Montaño, E. (1998). Kinetic transfer coefficients for radionuclides in estuarine waters: reference values from 133 Ba and effect of salinity and suspended load concentration. Journal of Radionalytical Nuclear Chemistry, Articles, 237, 55–61. Li, Y. H., Burkhardt, L., Buchholtz, M., O’Hara, P. & Santschi, P. H. (1984). Partition of radiotracers between suspended particles and seawater. Geochimica and Cosmochimica Acta, 48, 2011–2019. Margvelashvily, N., Maderick, V. & Zheleznyak, M. (1997). THREETOX: a computer code to simulate three dimensional dispersion of radionuclides in stratified water bodies. Radiation Protection Dosimetry, 73, 177–180. Martínez-Aguirre, A., García-León, M. & Ivanovich, M. (1994a). The distribution of U, Th and 226 Ra derived from the phosphate fertilizer industries on an estuarine system in southwest Spain. Journal of Environmental Radioactivity, 22, 155–177. Martínez-Aguirre, A., García-León, M. & Ivanovich, M. (1994b). U and Th distribution in solution and suspended matter from rivers affected by phosphate rock processing in southwestern Spain. Nuclear Instruments and Methods, A339, 287–293. Mehta, A. J. (1989). On estuarine cohesive sediment suspension behaviour. Journal of Geophysical Research, 94 (C10), 14303–14314. Nyffeler, U. P., Li, Y. H. & Santschi P. H. (1984). A kinetic approach to describe trace-element distribution between particles and solution in natural aquatic systems. Geochimica and Cosmochimica Acta, 48, 1513–1522. Pejrup, M. (1988). Suspended sediment transport across a tidal flat. Marine Geology, 82, 187–198. Periáñez, R. (1995). Un modelo matemático para la simulación de la dispersión de radionúclidos no conservativos en un sistema estuario. Aplicación a la ría de Huelva. PhD Thesis, University of Seville (in Spanish). Periáñez, R. (1999). Three dimensional modelling of the tidal dispersion of non conservative radionuclides in the marine environment. Application to 239,240 Pu dispersion in the eastern Irish Sea. Journal of Marine Systems, 22, 37–51. Periáñez, R. (2000a). Modelling the tidal dispersion of 137 Cs and 239,240 Pu in the English Channel. Journal of Environmental Radioactivity, 49, 259–277. Periáñez, R. (2000b). Modelling the physico-chemical speciation of plutonium in the eastern Irish Sea. Journal of Environmental Radioactivity, 49, 11–34. Periáñez, R. & García-León M. (1993). Ra-isotopes around a phosphate fertilizer complex in an estuarine system at the southwest of Spain. Journal of Radioanalytical and Nuclear Chemistry, Articles, 172, 71–79. Periáñez, R. & Martínez-Aguirre, A. (1997a). Uranium and thorium concentrations in an estuary affected by phosphate fertilizer processing: experimental results and a modelling study. Journal of Environmental Radioactivity, 35, 281–304. Periáñez, R. & Martínez-Aguirre, A. (1997b). A six phases model to simulate the contamination by non conservative radionuclides of sediments, soils and plants in a marsh area. Application to the Odiel Marsh in southwest Spain. Journal of Environmental Radioactivity, 37, 29–54. Periáñez, R., Abril, J.M. & García-León, M. (1994a). A modelling study of 226 Ra dispersion in an estuarine system in southwest Spain. Journal of Environmental Radioactivity, 24, 159–179. Periáñez, R., García-León, M. & Abril, J. M. (1994b). Radium isotopes in suspended matter in an estuarine system at the southwest of Spain. Journal of Radioanalytical and Nuclear Chemistry, Articles, 183, 395–407.

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Periáñez, R., Abril, J. M. & García-León, M. (1996a). Modelling the suspended matter distribution in an estuarine system. Application to the Odiel river in southwest Spain. Ecological Modelling, 87, 169–179. Periáñez, R., Abril, J. M. & García-León, M. (1996b). Modelling the dispersion of non conservative radionuclides in tidal waters. Part 1: conceptual and mathematical model. Journal of Environmental Radioactivity, 31, 127–141. Periáñez, R., Abril, J. M. & García-León, M. (1996c). Modelling the dispersion of non conservative radionuclides in tidal waters. Part 2: application to 226 Ra dispersion in an estuarine system. Journal of Environmental Radioactivity, 31, 253–272. Periáñez, R., Martínez-Aguirre, A. & García-León, M. (1996d). U and Th isotopes in an estuarine system in southwest Spain: tidal and seasonal variations. Applied Radiation and Isotopes, 47, 1121–1125. Plata, A. (1985). Dispersión en ríos. Madrid: CEDEX (in Spanish). Postma, H. (1980). Sediment transport and sedimentation. In E. Olausson & I. Cato (Eds), Chemistry and biogeochemistry of estuaries (pp. 153–186). New York: Wiley. Prandle, D. (1974). A Numerical Model of the Southern North Sea and River Thames. Report 4. Bidston, U.K.: Institute of Oceanographic Sciences. Prandle, D. (1984). A modelling study of the mixing of 137 Cs in the seas of the European continental shelf. Philosophical Transactions of the Royal Society, London, A310, 407–436. Prandle, D., Jago, C. F., Jones, S. E., Purdie, D. A. & Tappin, A. (1993). The influence of horizontal circulation in the supply and distributions of tracers. Philosophical Transactions of the Royal Society, London, A343, 405–421. Pugh, D. T. (1987). Tides, surges and mean sea level. Chichester: Wiley. Universidad de Sevilla (1991). Coeficientes de distribución de radionucleidos. Contract with ENRESA, Final Report (in Spanish). Williams, J. J., Humphery, J. D., Hardcastle, P. J. & Wilson, D. J. (1998). Field observations of hydrodynamic conditions and suspended particulate matter in the southern North Sea. Continental Shelf Research, 18, 1215–1233.

MODELLING RADIOACTIVITY IN THE ENVIRONMENT E. Marian Scott (Editor) © 2003 Elsevier Science Ltd. All rights reserved

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Chapter 9

Predicting transfer of radionuclides: soil–plant–animal Neil Crout a , Nick Beresford b , Arthur Sanchez b a School of Life and Environmental Science, University of Nottingham, Nottingham, NG7 2RD, UK b Centre for Ecology & Hydrology, CEH-Merlewood, Grange-over-Sands, Cumbria, LA11 6JU, UK

1. Introduction Radionuclides released into the environment can enter the human food chain posing a potential hazard to health. The estimation of radionuclide transfer from soil to plant and subsequently from plant to animal is therefore an important aspect of any overall radiological assessment model. In this chapter we shall present some brief case studies which illustrate a number of approaches which can be employed to model these transfers. We will concentrate on those approaches which go beyond ‘traditional’ tools for the prediction of transfer and attempt to put these in the context of existing radiological assessment models. The case studies are divided into three sections: (1) soil–plant transfer, (2) transfer to animals, and (3) spatial models.

2. Predicting the transfer of radionuclides from soil to plants The plant uptake of radionuclides via the root system is traditionally quantified as a concentration ratio, and referred to as the soil-to-plant transfer factor (TF): TF = plant activity concentration (Bq kg−1 )/soil activity concentration (Bq kg−1 ). Although the above definition is fairly straightforward, there are several ways of reporting activity concentrations, which lead to considerable variations in calculated TF values. For example, the plant activity concentration may be expressed in terms of fresh or dry weight of material; some workers measure the edible plant part only (e.g., for agricultural crops used for human or animal consumption), whilst others use the whole plant. The soil activity concentration could likewise be expressed in terms of fresh, air-dry, or oven-dry weight. To reduce uncertainties, the IAEA (1994) recommend using the dry weight basis for TF calculations. 261

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Alternative methods of describing plant transfer, where the activity concentrations of either the plant or soil, or both, are expressed on an area basis, are also used (e.g., Shaw & Bell, 1994). However, the most widely used definition is that on a mass basis. The root uptake of nutrients and other mineral elements (as well as radionuclides) takes place via the soil solution. Plant roots use a variety of mechanisms to create favourable conditions within the adjacent soil for such uptake, and can regulate their uptake of elements depending on whether deficient, sufficient, or excess levels of the elements exist in the soil (e.g., Baker, 1981 in Sheppard & Evenden, 1988a). The processes affecting plant uptake (hence TF) for any radionuclide would thus be controlled by two sets of factors: those associated with the physiological requirements of the plant, and the physico-chemical factors influencing the distribution of radionuclides between the soil solid phase and soil solution. For example, the uptake of 134 Cs, 137 Cs and 90 Sr is influenced not only by the plant’s requirements for the analogue nutrients K and Ca respectively, but also by the processes affecting the mobility of these radionuclides (and nutrients) in the soil system. The uptake of radionuclides with no obvious physiological role (e.g., Pu and other actinides) is likely to be different (e.g., Shaw & Bell, 1994). The strategies for uptake are plant- and element- specific, and therefore difficult to generalize (Sheppard & Evenden, 1988a). Soil/plant relationships are also affected by weather and the plant growth conditions (e.g., Noordijk et al., 1992). Interspecies differences in plant uptake of radiocaesium and radiostrontium have been reported (e.g., Broadley & Willey, 1997; Gerzabek et al., 1998; Negri & Hinchman, 2000, and references therein). For a given element (or radionuclide), the range in TF values among different plant species can vary by over two orders of magnitude, and is comparable to the range of TF values for a single plant species across a range of soil types (e.g., Nisbet & Shaw, 1994; Skarlou et al., 1996; Sheppard et al., 1999). From lysimeter experiments, Gerzabek et al. (1998) have shown that soil characteristics strongly influence 137 Cs uptake, whereas plant specific factors are more important for 60 Co uptake; for 226 Ra uptake, both soil and plant factors are equally important. Transfer factors have been determined from a variety of experimental approaches. Containers of different sizes (pots or lysimeters) have been used either under controlled conditions inside laboratory greenhouses or outdoors; actual measurements of radionuclides in vegetation and soil samples collected in the field, or field plots, have also been used to derive TF values. The source of contamination could also vary (and will influence TF): in pot or lysimeter experiments, the soils are spiked artificially with radionuclides (either added on to the soil surface or mixed uniformly), whilst contamination in the field could be by bomb fallout, accidental or routine discharges from the nuclear industry (e.g., Bell et al., 1988; Sheppard & Evenden, 1988a). Field measurements have been obtained for varying depths of soil, depending on the assumed depth of the plough layer or the rooting zone. Such wide sources of data inevitably contribute to large variations in TF values reported in the literature. The lack of standardization of experimental design prompted the International Union of Radioecology (IUR, 1982) to set up a working group to recommend standard protocols for the determination and consistent reporting of TF values. Among the important criteria include: plant part (whether unwashed or washed to remove adhering soil particles), concentration of radionuclide (whether applied artificially or as a result of field contamination), soil type, depth of soil contamination, soil pH, duration of radionuclide in the soil, and container type. For field measurements, the recommended soil depth is 20 cm for agricultural soils, and 10 cm for

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grasslands. Subsequently, the IUR established a database of reported TF values that meet the experimental criteria (IUR, 1989); this database is updated when new data become available. The large data bank of TF values compiled by the IUR has been used to evaluate the effects of key soil properties on soil-to-plant transfer (e.g., Frissel et al., 1990; Van Bergeijk et al., 1992). Soil-to-plant transfer of radiocaesium has been found to increase with increasing organic matter content, whereas the opposite trend has been observed for radiostrontium; for Pu, organic matter content did not have an effect. Soil pH (range 3.9 to 8.4) did not affect radiocaesium transfer, whilst TF for radiostrontium decreased with increasing pH. The nutrient status of the soil has important effects on radiocaesium and radiostrontium transfer through interactions with the analogues K and Ca. Soil moisture content also appears to have an important effect on Cs and Sr uptake. There is thus a large variability in TF values for various soil types due to the differences in these soil properties. Sheppard and Evenden (1990) assessed the variability of TF values for 23 naturally occurring elements for a single plant species (sweet lowbush blueberry, Vaccinium angustifolium) across a wide geographic range in Canada. The sites sampled provided a broad range of habitat and growth conditions, which were characterized. Statistical analysis of the data indicated that the variation in TF values could be attributed to several sources, including spatial, analytical and natural random variation. The TF values for the various elements were correlated and cluster analysis showed groupings that reflected the periodic table of the elements. Soil–plant transfer is an important link for the entry of radionuclides into the human food chain, and the TF is widely used as a parameter for predicting plant uptake in radiological assessment models. There is a need to ensure that reliable TF values are used in such models; a compilation of generic TF values for a wide range of crops, radionuclides and soils, based largely on the IUR database, has been published (IAEA, 1994). Given the observed lognormal distribution of TF values (Sheppard & Evenden, 1988b; Konshin, 1992), the use of the geometric mean and the geometric standard variation of TFs to cover the range of observed TF values has been recommended for stochastic modelling (Sheppard & Evenden, 1997). Using the IUR database, Frissel (1998) has attempted a soil ecosystem classification system providing reference TF values for Cs and Sr across a range of soil types and nutrient status. Generic TF values provide a useful first approximation in assessment models, but it is important to note that many reported TF values are likely to be site and species specific (Shaw & Bell, 1994). 2.1. Predicting the transfer of radiocaesium to plants using soil characteristics As outlined above the use of simple soil-plant concentration ratios is fraught with difficulty due to the large variability in the observed values. In the case of radiocaesium models have been developed which predict the soil to plant transfer semi-mechanistically on the basis of soil characteristics. Absalom et al. (1999) describes soil to plant transfer of radiocaesium in terms of soil potassium status and specific radiocaesium sorption by clay and the approach was subsequently extended to take into account the effect of non-specific sorption of radiocaesium by organic matter (Absalom et al., 2001). This novel method relates soil clay content, organic matter content, pH and exchangeable K to three key parameters describing radiocaesium bioavailability as summarised schematically in Fig. 1. The soil solution K concentration is estimated as a function of exchangeable K and organic and inorganic cation exchange ca-

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Fig. 1. Flow chart to show the relationship between variables of Absalom et al.’s (Absalom et al., 2001) model. Shaded boxes indicate the input data required by the model.

pacities (estimated from clay and organic matter contents and soil pH). The radiocaesium distribution coefficient (kd ) describes the partitioning between sorbed and solution phases of radiocaesium in the soil and is calculated from specific (clay) and non-specific (humus) K+ competition coefficients and solution K concentration. The radiocaesium concentration factor (Bq kg−1 DW plant/ Bq dm−3 soil solution) represents the ratio of radiocaesium in vegetation to that in soil solution and may be derived as a function of soil solution K concentration (Smolders et al., 1997). Radiocaesium bioavailability in soil declines with time due to the processes of fixation by clay minerals and leaching from the root zone. To simulate these processes, a double exponential equation was used with half-lives of 1 and 10 years (Jacob & Likhtarev, 1996). Fixation

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Fig. 2. Measured versus modelled 137 Cs plant activity (Bq kg−1 log values) for wheat. Results using parameters optimised ryegrass () and wheat () are given; the 1:1 relationship is shown as a solid line.

is assumed to only apply to radiocaesium adsorbed on the clay fraction of the soil (i.e. not on organic matter) and therefore the rate of decline is adjusted by the relative absorption of radiocaesium on clay minerals to organic matter for a given soil. To run the model, inputs are required for soil clay content, exchangeable K, pH, and organic matter content; these are relatively simple soil characteristics to measure or estimate. The validation data used by Absalom et al. (1999) covered a range of agricultural crop types and, for example, in the case of wheat (the crop most commonly represented in the database) (Fig. 2) the soil–plant transfer model was found to account for 89% of the observed variation in radiocaesium activity concentration (N = 89; p < 0.001). Radiocaesium transfer factors to grass vary by almost 5 orders of magnitude and the model outlined is able to account for much of this variation. This is an advantage over the use of single valued TF values. Of course the model could be improved, in particular, quantitative information on soil clay mineralogy might be expected to mechanistically improve the model’s performance. However such data is unlikely to be available under the circumstances in which the model is applied in practice and therefore whether such an extension is worthwhile is open to question. 2.2. Predicting transfer to fungi An objective of radionuclide food chain modelling is to predict the radionuclide intake of an individual or population. This may not require a sophisticated modelling approach, or indeed

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a precisely defined radionuclide concentration of foodstuffs, for example some processes in the environment are so variable that they cannot be readily explained in the terms of a conventional transfer model. In such situations one approach is to understand the temporal and spatial nature of the variation and how this will influence the prediction of radionuclide intake. To illustrate this we shall consider the transfer of radiocaesium to fungi. The high transfer of radiocaesium to the fruiting bodies of fungi is well known. Moreover the great variability in transfer (as characterised by transfer factors) is notorious and not readily explained in terms of accessible environmental or ecological factors, although differences between different nutritional groupings of fungi are generally recognised (i.e. mycorrhizal, saprophytic, parasitic). Gillett & Crout (2000) undertook a comprehensive review of the available data and used the data to derive a pragmatic modelling approach which is expected to be effective for estimating human intake over time scales of a season. The review collated published literature describing 137 Cs transfer to fungi, with the objective of determining factors controlling transfer so that an appropriate modelling approach could be identified. It was originally anticipated that this would be based on empirical relationships to estimable ecological and environmental characteristics. Aggregated radiocaesium transfer ratio’s (Tag ) were derived for fungi species collected within Europe and the CIS and considerable variability was found, with Tag ’s varying between 0.001 m2 kg−1 and 10 m2 kg−1 across all species and over three orders of magnitude for individual species (e.g. Boletus badius). Generally, metainformation (such as habitat and soil attributes) was poorly reported in the literature so that classification of the Tag was limited to nutritional type, which was found to have a significant effect (p < 0.025) in the order mycorrhizal > saprophytic ≈ parasitic. Analysis of the data set suggested that there was no statistical evidence to indicate a decrease in Tag ’s during the first 6–9 years after the Chernobyl accident. The data were used to estimate probability distribution functions for the Tag values for individual species as shown for a sample of the species considered in Table 1. One data set provided good spatial coverage with observations at 120 sites within the Wallonne Region of Southern Belgium (approximately 200 × 150 km). It was proposed that analysis of the spatial correlation of the variation between sites would indicate over what scale it would be important to predict or estimate accurately the fungi fruitbody activity. The hypothesis was that the variation in radiocaesium transfer within a site was as large as that between sites as suggested by the findings of Dahlberg et al. (1997) who studied the variation of 137 Cs activity in individual fruitbodies of Suillus variegatus at 7 sites over areas less than 50 × 50 m in size. They found that most of the variation occurred over very small distances, at scales that would be difficult to predict without very detailed soil and habitat information. In the case of the Wallonne data set this could be tested by describing the spatial correlation in terms of an experimental variogram whereby the difference in transfer between sites is a function of the distance between them (e.g. Burrough, 1997). The result suggested that there is no distance– fungi activity variation relationship, with most of the difference between 137 Cs uptake for a number of species occurring over a relatively small scale (< 5 km). In other words, within the limitation of the data set, it is possible to conclude that the variation within a forest site may be as large as the variation between forests. At first sight these findings might suggest little hope of a plausible predictive approach. However, the spatial analysis indicates that variability of 137 Cs transfer within a sampling location was such that fruitbodies of a given species collected over an area on a scale of at

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Table 1 Statistics describing the distribution parameters for TR’s of individual fungi species and proposed effective TR’s (data collated by Gillett & Crout, 2000) Species

Best-fitting distribution

n

μ

se

σ

se

TReffective (m2 kg−1 )

se

Boletus edulis Cantharellus cibarius Lepista nebularis Lepista nuda Armillaria mellea Cortinarius anomalus Clitocybe clavipes & Cortinarius brunneus Paxillus involutus

log(X)2 log(X − a) log(X) log(X) log(X) normal normal

59 26 33 25 29 30 60

−2.7436 −1.1215 −2.117 −2.246 −3.1491 10.633 4.9863

0.2122 0.4364 0.1989 0.1997 0.1856 1.2614 0.4543

1.6295 0.4395 1.1426 0.9986 0.9994 6.9073 3.5182

0.1501 0.1977 0.1407 0.1413 0.1313 0.8922 0.3213

0.243 0.242 0.232 0.175 0.071 11.513 5.530

0.080 0.162 0.060 0.043 0.016 1.264 0.456

normal

40

1.6907

0.2187

1.3827

0.1547

1.963

1.370

Note: Difference between fitted and observed distribution patterns significant at: 1 P < 0.01; 2 P < 0.05; n = number of TR’s for which distribution was fitted; se = standard error of fitted parameter; TReffective = effective transfer ratio (see text for definition); CI (68%) = 68% confidence interval for the effective TR. Definition of parameters: Normal distribution: mean = μ; variance = σ 2 ; log normal distribution: mean = a + e(μ+

σ2 ) 2 ;

2 2 variance = e(2μ+σ ) (eσ − 1).

least 5 km (and possibly less) would show activities as variable as those collected over a much larger scale. Therefore, a typical mushroom gatherer who, over the course of a mushroom season, may gather mushrooms from a relatively large area (perhaps a number of forest locations or forests) would be expected to collect many fruitbodies with highly variable activities. The majority of the variation in 137 Cs uptake occurs over a spatial scale smaller than that over which the mushroom gatherer collects mushrooms. Provided sufficient numbers of mushrooms are collected using an ‘effective’ Tag for a particular species, derived from the distribution parameters for individual species, would give a reliable estimate of the 137 Cs transfer ratio. The ‘effective’ species Tag (Tag effective ) can be calculated using the fitted distribution parameters (Table 1) using 

∞

Tagi effective =

pdf · Tagi · dTagi,

0

where: Tagi effective = effective transfer ratio for species i (m2 kg−1 DW); pdf = the probability distribution function for the species in question. The effective Tag (i.e. the expectation of Tag ) is applicable if it is assumed that a sample is drawn from the pdfs derived. The effective Tag ’s and estimated confidence intervals (68%) are also given in Table 1.

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From a reductionist point of view Gillett & Crout’s (2000) approach to the problem of variability in radiocaesium transfer to fungi is highly unsatisfactory. There is no attempt to mechanistically account for the factors which control uptake. However given the current state of knowledge it seems improbable that any reductionist approach would be credible, and for the purposes of predicting transfer through the food chain the approach described above is, at least, viable.

3. Transfer to animals Contamination of animal products is often a major contributor to ingested radiation dose from both routine releases and accidents. The primary health effect arising from the Chernobyl accident was the development of thyroid cancers, especially by young children, as a consequence of ingesting milk contaminated with radioiodine (Kazakov et al., 1992; Likhtarev et al., 1994). Outside the former Soviet Union animal derived food-products remain contaminated after the Chernobyl accident and in some countries radiocaesium countermeasures are still being used (Howard et al., 2001). An ability to accurately estimate radionuclide activity concentrations in animal derived food products is therefore essential if radiation doses to man and the requirement for intervention to reduce doses are to be predicted. Ward et al. (1965) first introduced the transfer coefficient as a parameter to describe the transfer of radiocaesium from the diet to the milk of dairy cattle. The transfer coefficient (Fm ) was defined as the equilibrium ratio between the radionuclide activity concentration in milk and the daily radionuclide intake. The transfer coefficient was rapidly adopted as the basis for quantifying transfer to both milk (Fm ) and meat (Ff ) for all radionuclides and single values were recommended for each radionuclide/animal product combination (e.g., Ng et al., 1982; Coughtrey, 1990; IAEA, 1994). Commenting upon this Ward & Johnson (1989) stated that the wide use of Fm appeared to be justified and that factors such as stable element intake, soil intake, milk production rate, metabolic rate and inhalation could be ignored for most situations. However, they did state that these conclusions were based primarily on Cs data for cow milk and questioned the lack of critical evaluation for other species and radionuclides (Ward & Johnson, 1986). Subsequently, it has been demonstrated that many factors affect transfer coefficients. A number of authors have reported variation in radiocaesium transfer coefficients due to some of the above variables. Differences of over two orders of magnitude in transfer coefficients to the milk and meat of ruminants have been reported for different environmental dietary sources, including soils, sediments, milk (for suckling lambs) and fungi (Hansen & Hove, 1991; Howard et al., 1989; Assimakopoulos et al., 1993; Belli et al., 1993a, b). Further variation in transfer coefficients has been explained by metabolic factors including dietary intake rates (Beresford et al., 1998a), lactation (Assimakopoulos et al., 1994) and exercise (Hansen & Hove, 1993). Variation in radiocaesium transfer coefficients was subsequently noted in the International Atomic Energy Agency handbook of transfer parameter values (IAEA, 1994), but no recommendations were given to accommodate this variation within radiological assessments. The validity of transfer coefficients for radionuclides with long biological half-lives (e.g. Pu, Am, Cd) has been questioned, since equilibrium will not be established within the lifetime of the animal (IAEA, 1994). The recommendation of single transfer coefficient values

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for radionuclides which are either themselves, or through a stable analogue, subject to homeostatic control has been shown to be inappropriate (e.g. radiostrontium (Howard et al., 1997), radioiodine (Crout et al., 2000) and tritium (Galeriu et al., in press)). However, relationships can be derived to enable transfer coefficient to be estimated for such radionuclides which allows for the effect of metabolic control (these are discussed in the following sections). A number of assessment models used to predict the consequences of accidental or routine releases of radionuclides utilise transfer coefficients (e.g. Brown & Simmonds, 1995; Müller & Pröhl, 1993). To allow dynamic predictions transfer coefficients are combined with biological half-lives (i.e. the time taken for the radionuclide activity concentration in a tissues, or milk, to reach one half of its initial value). However, such models rarely allow for recycling between different organs. 3.1. Multi-compartment models of radionuclide transfer to animals Models have been developed in which the absorption, excretion and transfer between tissues of radionuclides are described by transfers between model compartments (e.g. Galer et al., 1993; Assimakopoulos et al., 1991; Crout et al., 1996). Figure 3 shows the radiocaesium model presented by Galer et al. (1993) which comprises nine compartments, each representing a particular tissue or excretion pathway of the animal. The model requires the solution of a set of simultaneous first order differential equations with the general form   dNi = aji Nj − aij Ni , dt j =1,9 j =i

j =1,9 j =i

where: Ni is the activity of compartment i (Bq), aij is the rate coefficient for the transfer from compartment i to j (d −1 ). In the case of Galer et al. (1993) the model parameters (i.e. the aij values) were estimated by fitting the model to data from a ‘calibration’ experiment. At the start of this experiment animals were dosed with radiocaesium directly to the rumen and then slaughtered on successive occasions to make measurements of radiocaesium activities in the various compartments. The model accounted for 92.3% of the variation in the data (n = 51) and it was used to predict a number of radioecological characteristics of sheep (such as the transfer coefficient and biological half life) and a reasonable level of agreement with literature values was found. This type of approach is clearly more sophisticated than the use of simple transfer coefficients, but this is not, in itself, an advantage. Whilst the structure of the model (i.e. the arrangement of compartments and interconnections) is based on a simplified representation of biological processes, the parameterisation (i.e. the values of the model rate coefficients) is empirically based on comparisons to observed data. Although Crout et al. (1996) attempt to take into account the different sizes of animals (with limited success) generally such models are implicitly calibrated to a particular situation and take no account of changing metabolic circumstances. These types of models do offer advantages in that they explicitly link the behaviour of radionuclides in the different tissues of an animal. It is common in single tissue analyses to fit a series of exponential terms to describe different components of the dynamics

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Fig. 3. Multi-compartment model of radiocaesium transfer between sheep tissues (Galer et al., 1993).

(e.g. Beresford et al., 1994). These different components may be due to the contribution of radionuclides secreted from other tissues with different turnover times. Multi-compartment models provide a coherent framework within which these dynamic exchanges can be simultaneously represented, allowing for a more rigorous analysis of experimental data. This advantage is clearly apparent for radionuclides with especially long biological half-lives in certain tissues, for instance experimental and modelling studies have demonstrated that the 109 Cd activity concentrations in kidney continue to rise even after the cessation of exposure to a contaminated diet as 109 Cd lost from other organs is taken up and retained by the kidney (Sharma & Street, 1980; Baxter et al., 1982; Beresford et al., 1999). In summary it appears that multi-compartment models per se do not offer significant mechanistic benefits over the use of transfer coefficients, and their greater complexity is a disadvantage in the context of large dose estimation models. However, they are useful tools for the analysis and interpretation of experimental studies, leading to the estimation of parameters such as transfer coefficients and biological turnover times and may be especially relevant in the case of nuclides with long biological half lives. 3.2. Metabolically based models of radionuclide transfer to animals Some radionuclides are isotopes of elements which are metabolically controlled or regulated within animals (e.g. 131 I and tritium (3 H)), or analogues of metabolically controlled elements (e.g. calcium status determines radiostrontium transfer). In these cases the use of conventional single valued transfer coefficients to predict transfer may be misleading as the transfer and/or turnover of the radionuclide will be determined by metabolic factors unrelated to the intake of the radionuclide itself (e.g. calcium intake, iodine status, stage of lactation). To illustrate this

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Fig. 4. Comparison between calcium intake and Fm for strontium adapted from Beresford et al. (1998b) with additional recent data for cattle from Beresford et al. (2000). The lines represent predicted values based upon calcium contents in milk of 1 g kg−1 (typical for cattle) and 2.6 g kg−1 (typical for sheep) respectively.

point consider Fig. 4 which shows a relationship proposed by Beresford et al. (1998b) for the radiostrontium transfer coefficient to the milk of ruminants. Calcium is a regulated element in the body, its concentration in tissues and milk is maintained close to a fixed value. This is achieved through a range of processes most especially the control of calcium absorption from the gastrointestinal tract and the remobilisation of bone calcium. Strontium is a close analogue of calcium and its behaviour is largely controlled by calcium metabolism. Under ‘normal conditions’ (i.e. when the calcium in the diet of an animal meets its requirements for growth, milk production etc.) increases in calcium intake reduce the proportion of the calcium intake absorbed, with a corresponding effect on strontium uptake. This results in the relationship between radiostrontium transfer to milk and an animal’s daily calcium intake presented in Fig. 4. Clearly there is no single transfer coefficient value for radiostrontium transfer to milk. 3.2.1. Metabolically based model of radiostrontium transfer to milk Ideas such as those outlined above have been used to develop a fully dynamic multicompartment model for radiostrontium transfer to goat milk (Crout et al., 1998a, b). This model primarily simulates the transfer of calcium within the animal and is based heavily on the work of Braithwaite and co-workers (Braithwaite et al., 1969; Braithwaite, 1982, 1983). Radiostrontium is treated as a tracer for calcium, subject to discrimination during transfer between compartments. The basic calcium model is shown schematically in Fig. 5. The goat is represented by five compartments, the gastrointestinal tract (GIT), plasma, rapidly exchange-

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Fig. 5. Schematic representation of the Ca transfer model (Crout et al., 2000) where apef , apes , aefp , aesp , apb , agf , and apu = rate coefficients (d−1 ); A = absorption rate of Ca (mg d−1 ); [Ca]milk = the concentration of Ca in the milk (g l−1 ); Y = milk yield (ml d−1 ); FE = rate of fecal endogenous excretion of Ca (mg d−1 ); R = rate of Ca resorption from the bone compartment (mg d−1 ).

able pool (Efast ), slowly exchangeable pool (Eslow ), and bone. Efast represents the calcium distributed in the soft tissues of the goat. Eslow is taken as exchangeable calcium in bone, whereas bone represents skeletal calcium from which resorption occurrs during periods of calcium stress (such as lactation or low Ca intake). Resorption of calcium from bone is an important part of the calcium balance of lactating animals and, therefore needs to be accounted for in the model, especially if the model is to be applicable in circumstances of low calcium intake. Absorption of calcium from the GIT is assumed to be an active process, and in the model the rate is adjusted to maintain a constant Ca concentration in plasma (simulating the homeostatic control mechanisms of the goat). Unknown parameters in this model were estimated by fitting to experimental data obtained by 45 Ca tracer studies. The rate coefficients for radiostrontium transfer between compartments are taken to be equal to those for calcium but modified to include dimensionless discrimination factors for transfers from GIT to plasma, plasma to milk, and plasma to urine. The values of these discrimination factors were estimated by fitting the model to experimental radiostrontium data without further adjustment of the rate coefficients for Ca transfers. The resulting model was in close agreement with the experimental data used for its development and was used to simulate radiostrontium transfer under a number of different calcium metabolism scenarios. These confirmed that daily dietary calcium intake was the factor with the greatest effect on radiostrontium transfer but that other factors influencing overall Ca status, such as milk yield, would also have an effect. This analysis therefore adds value to the

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empirically derived relationship presented by Beresford et al. (1998b) discussed above. The model was also used to quantify the effect of calcium supplementation as a countermeasure to radiostrontium transfer with the conclusion that a useful reduction would be achieved for most ‘normal’ animals. 3.2.2. Metabolically based model of radioiodine transfer to milk Iodine is an essential element which is actively involved in thyroid metabolism. A basic knowledge of iodine metabolism can be used to develop models which can account for changes in radioiodine transfer to animal products due to changes in iodine metabolism. For example Crout et al. (2000) developed models for transfer to goat milk which account for stable iodine dietary supplements (Fig. 6). Radioiodine is assumed to be a tracer for stable iodine, and therefore the model primarily considers the dynamics of stable iodine within the animal. Iodine enters an intake pool, which represents the gut of the animal; this is used to provide a realistic, finite delay component for absorption which is assumed to be complete. Iodine enters the Inorganic compartment, this represents inorganic iodide distributed throughout the extra cellular fluid and body tissues with the exception of the thyroid. Uptake of iodide occurs from the Inorganic compartment to the Thyroid. Within the Thyroid compartment all iodine is organified to form hormones which are secreted into the Hormone compartment, which represents thyroid hormones distributed throughout the body. These are assumed to be excreted directly via the faeces, or catabolised back to the Inorganic compartment. Losses of iodide occur via urine, faeces, and milk. The thyroid acts as a store for iodine, typically containing 70–80% of the total body content of iodine. This iodine content is much larger than the typical daily requirement of iodine for hormone synthesis. The concentration of thyroid hormones in the soft tissues of the body is homeostatically controlled by the uptake of iodide and secretion of hormones by the thyroid.

Fig. 6. Schematic of the Stable Iodine Model (Crout et al., 2000).

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Fig. 7. Model predictions compared to the experimental data for accumulated 131 I output to urine (F - - -), milk (2 —) and faeces (" – –). Experimental points are shown as symbols ±SE (n = 11), model predictions as continuous curves.

To represent this the model assumes that the Hormone compartment must be maintained at a particular value. The iodide uptake rate of the Thyroid compartment is assumed to be equal to the secretion rate of iodine as thyroid hormone, and in turn this is equal to the total loss from the Hormone compartment. Losses from the Hormone compartment are partitioned between losses to the Faeces and catabolisation to the Inorganic compartment. In effect these transfers are zero order processes (i.e., they proceed at a constant rate, irrespective of the iodine concentration). To include a saturation mechanism for the transfer of iodide to milk the rate of secretion of iodide into milk is described by a Michalis-Menten equation. This initially allows secretion rate to increase as the concentration of stable iodine in the iodide compartment increases. However the secretion rate will eventually plateau as the transport mechanism saturates. The model was parameterised by fitting it to data from a series of experiments in which single administrations of radioiodine were given to goats and the subsequent radioiodine concentrations in milk, urine and faeces measured. The cumulative outputs of 131 I in faeces, urine and milk are compared in Fig. 7. The cumulative output to milk was significantly lower during the second period of the study (following the administration of a comparatively large amount of iodide (as KI)) than during the first period. In contrast, outputs of 131 I in both urine and faeces were significantly higher during the second period of the study. The model simulated this effect well. The model has been used to investigate the potential of stable iodine supplementation of the diet as a countermeasure against radioiodine transfer to milk. Figure 8 shows the predicted milk transfer coefficient as a function of stable iodine intake; the range covered exceeds the maximum recommended tolerable intake. Despite this the predicted response is small, suggesting that this is not a viable countermeasure and in any case would lead to milk exceeding recommended maximum stable iodine concentrations for human consumption.

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Fig. 8. Predicted variation in radioiodine transfer coefficient with varying stable iodine intakes in dairy goats; stable iodine intake rates include those representative of those prevailing under normal practice (1–20 mg d−1 ) to the maximum tolerable advised intake for ruminants (50 mg kg−1 dry matter).

3.2.3. Metabolically based models for tritium transfer Crout et al. (1998a, b) suggested a model describing the transfer of 3 H, 14 C and 35 S to sheep tissues based upon a simplified representation of processes such as protein synthesis and degradation, and respiration. As a consequence of the approach, the models for the individual radionuclides shared many common parameter values. Whilst the models satisfactorily described the experimental data on which they were parameterised, their applicability is currently limited to animals under similar management and physiological conditions. However, models for these radionuclides based upon a metabolic understanding have great potential as they could predict the influence of dietary intake, physiological status and form of radionuclide within the diet. The concept has recently been applied to the prediction of transfer coefficients for 3 H in food producing animals by Galeriu et al. (in press). By modelling the hydrogen balance within animals Galeriu et al. (in press) developed an approach whereby transfer coefficients can be estimated for any food-producing animal under different physiological conditions (e.g. lactating, live-weight) for both organically bound tritium (e.g. 3 H incorporated into protein, carbohydrates etc.) and tritiated water. Predictions compare well to the available data and demonstrate the inappropriateness of species specific transfer coefficients, as recommended by many bodies (e.g. IAEA, 1994). However, the authors demonstrated that whilst transfer coefficients vary with physiological parameters such as milk yield and live-weight, concentration ratios (activity concentration in a tissue:activity concentration in the diet) do not and as such represent a more robust radioecological parameter. Variation in radiocaesium transfer to animal tissues as observed in a number of studies (e.g. Beresford et al., 1998a; Hansen & Hove, 1993) could be related to processes determining protein transfer. Consequently there is current development to determine if the 3 H model as

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presented by Galeriu et al. (in press) can be used to predict radiocaesium transfer to animals under a range of physiological conditions. 3.2.4. Summary Whilst the models described above retain an element of empiricism (i.e. some of their parameters are fitted to experimental data) they have a metabolic basis, not only in the arrangement of their compartments, but in the form and parameterisation of the relationships used. They are therefore distinct from the type of multi-compartment model described for radiocaesium. In the case of isotopes of metabolically controlled elements such approaches are required because simple, single valued, transfer coefficients are not appropriate. Whilst some of the models discussed above may be too complex to be directly applicable within dose estimation models they are useful tools to assist in the development of simplified relationships which can be more widely applied (e.g. Fig. 4).

4. Spatial models Radionuclides can be deposited over large geographical areas and the spatial variation in soil characteristics, land use and deposition gives rise to marked spatial distributions of radionuclide transfer to the foodchain. This was well demonstrated in the unexpected and persistent transfer of radiocaesium to vegetation in certain upland regions of England and Wales after the Chernobyl accident. In order to take account of these spatial factors models have been developed (so far limited to radiocaesium) in which transfer to the food chain is predicted using spatially distributed soils and agricultural production/management data. For example, van der Perk et al. (2001) present a GIS based decision support system to assess the transfer of radiocaesium in the food chain in areas of the Ukraine and Belarus contaminated by the Chernobyl accident. Similarly Gillett et al. (2001) present spatial model simulations for various radioecological scenarios in England and Wales. Both of these spatial models utilise the soil– plant transfer models of Absalom et al. (1999, 2001) outlined earlier, because the required soil characteristic inputs are easily measured and therefore often available, or estimable, in existing soil spatial databases for many areas. 4.1. Model description The model presented by Gillett et al. (2001) is schematically represented in Fig. 9. Spatial databases were derived for England and Wales which contain the required soil characteristics, agricultural production statistics, and crop and agricultural management parameters. The databases have a spatial resolution of 5 × 5 km and comprise 5648 grid squares. The soil characteristics are input to the model to enable the radiocaesium activity concentrations in food products to be estimated for each cell. In conjunction with agricultural production data this is then used to calculate the time integrated flux via each food product to the food chain from each cell (Bq yr−1 cell−1 ). The spatial inputs for soil pH, exchangeable K and organic matter were derived from the Geochemical Atlas of England and Wales (McGrath & Loveland, 1992) which is a database of experimentally measured values, taken at pre-determined grid reference points throughout a

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Fig. 9. Schematic representation of the spatial model described by Gillett et al., 2001. Cylinders represent spatial inputs; rectangles represent major sub-models; circles represent principal model outputs.

5 × 5 km grid across England and Wales. Topsoil percent clay content values have been taken from the Soil Survey and Land Research Centre National Soil Inventory database. These data were partly direct measurements (58% of the total cells) and partly derived from soil texture observations (42% of the total cells). The clay content/texture observations were made using the same 5 × 5 km grid as that used for the England and Wales Geochemical Atlas. In order to remove the small scale variation associated with the localised observations in the data base, the data were interpolated using block kriging to derive the mean and variance of soil properties for each 5 × 5 km raster cell. These data were combined to create a database of initial plant transfer factors (Bq kg−1 plant dry weight per Bq kg−1 soil dry weight) predicted immediately after deposition and the ratio of radiocaesium fixing clay sorption to the whole soil radiocaesium sorption (Absalom et al., 2001). This latter value is used to determine the subsequent rate of decline in radiocaesium bioavailability for each cell.

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Radiocaesium deposition from the Chernobyl accident was derived for England and Wales from the Radiocaesium Atlas for Europe (De Cort et al., 1998). Points were selected along the contours of the Atlas which were then interpolated using block kriging to estimate the kriged mean and variance for total radiocaesium deposition for each 5 × 5 km cell in England and Wales. 4.2. Validation Spatial predictions are probably best used to give a general picture, and the quantitative values may only be meaningful when averaged over a number of cells (for example at county level). Therefore, the most appropriate data sets to validate the model predictions would be food product contamination data for small regions as a function of time since deposition. Ideally, the data would be drawn from unbiased sampling across the region of interest. Following the Chernobyl accident widespread monitoring of radiocaesium contamination of food products was undertaken in England and Wales, mainly by the Ministry of Agriculture Fisheries and Food (MAFF). The most heavily monitored food products were sheep meat and cow milk, and the monitoring programme focused on those areas where contamination was found to be greatest, principally in North West England and North Wales. These data are published as values for individual samples, with the date of observation, however the geographical location is only provided as a county or district (Welsh Office, 1986; MAFF, 1987, 1988; Andrews et al., 1988). From the point of view of model verification a major limitation of these data is that the sampling within each county/district may have been biased. Shortly after deposition the sampling was quite widespread, focusing mainly on areas where relatively high deposition was believed to have occurred. In the longer term the monitoring effort became more directed towards areas where contamination of food products remained high. This was particularly the case with sheep as some areas suffered high levels of contamination which required systematic monitoring, thereby introducing significant spatial bias into the data. Consequently the verification work presented below concentrates on comparisons with observed milk activity concentration. Of course milk is a key contributor to dose to the population and therefore an important model output which merits testing irrespective of the sampling considerations. In order to compare these data to the model predictions, monthly mean cow milk activity concentrations were calculated from observations by county/district from all the individual reported values taken within that county or district within a given month. The number of samples on each individual monitoring date varied between 1 to 40. Whilst these data are not ideal for testing the model presented they are the most comprehensive data sets available for these regions. Results are presented for cow milk in Fig. 10 for the counties of Gwynedd and Cumbria which were the most contaminated counties in Wales and England respectively. For both counties the model is able to predict the time course of mean monthly cow milk activity concentration quite effectively, accounting for 56 and 80% of the observed variation for Gwynedd and Cumbria, respectively. Generally, observations fall outside one standard deviation (kriging error) of the predicted monthly mean. However across each county the spatial input data vary significantly and this gives rise to large variation in predicted milk activity. Given the potential spatial bias in sampling, this variation needs to be considered when interpreting these results.

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Fig. 10. Predicted and observed cow milk activities for Cumbria and Gwynedd following the Chernobyl nuclear accident, 1986–88. () Observed monthly mean (±1 standard error); (—) modelled monthly mean; (· · ·) ±1 standard error of the model prediction due to uncertainty in the spatial attributes (i.e. kriging variance); (- - -) 25th and 75th percentiles of model predictions over the region (i.e. variation between raster cells). (1:1 lines: Gwynedd r 2 = 0.56, P < 0.05; Cumbria r 2 = 0.80, P < 0.001).

Nearly all the observations lie within the lower and upper quartile of predictions made for the individual cells of each region. An important aspect of the comparison presented in Fig. 10 is that the predicted data has been summarised by giving all the cells in the region equal weighting. Given the unknown pattern of spatial sampling this is probably the only reasonable approach, however it limits the interpretation of the comparisons presented.

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4.3. Example simulations To illustrate the application of the model it has been used to identify areas which are expected to be vulnerable to radiocaesium under two deposition scenarios, the Chernobyl pattern of deposition and a uniform deposition of 1535 Bq m−2 (occurring at the same time of year). The latter scenario represents the same total radiocaesium deposition for England and Wales as the Chernobyl scenario (as estimated from the deposition database) but distributed uniformly over all raster cells allowing for comparisons to be made between the vulnerability of different geographical areas. In each case vulnerability is judged in terms of food product activity concentration and radiocaesium flux. Figure 11 shows maps of the predicted radiocaesium contamination of cow milk (activity concentration Bq l−1 ) for the two scenarios one year after the deposition event. Under the uniform scenario the activity concentration of milk has declined to < 5 Bq l−1 suggesting a low vulnerability 1 year following the initial fallout. For the Chernobyl scenario the most vulnerable areas in terms of highest concentration activity are identified as North Wales, Cumbria, the Pennines and some parts of SW England. In a few cases the activity concentration of milk is > 5 Bq l−1 although the levels are low compared to the threshold for intervention (1000 Bq l−1 ). These correspond to areas where relatively high deposition is coincident with soils of relatively low pH, clay content and exchangeable K and relatively large soil organic matter content. In the case of uniform deposition the pattern of activity concentration is differ-

Fig. 11. Predicted radiocaesium activity concentration in cow milk (Bq l−1 ) 1 year after: (a) Chernobyl; (b) uniform (1535 Bq m−2 ) deposition.

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Table 2 The cumulative radiocaesium flux (England and Wales, GBq y−1 ) for sheep meat, cereals (animal and human consumption) and cow milk following the Chernobyl deposition event and a uniform deposition event (1535 Bq m−2 ) and its distribution between the nine MAFF regions of England and Wales MAFF region/product

Chernobyl scenario Year 1

Deposition Year 10

Uniform scenario Year 1

Deposition Year 10

Sheep meat, % of total flux Wales North North West South West Yorks & Humberside West Midlands East Midlands East Anglia South East Total flux (GBq y−1 )

50 24 12 11 4 2 1 0 0 18

59 23 7 4 6 0 0 0 0 0.06

47 25 11 9 6 1 1 0 0 32

44 30 3 6 11 3 2 0 2 0.02

Cereals, % of total flux South West North West Yorks & Humberside North East Anglia Wales East Midlands West Midlands South East Total flux (GBq y−1 )

21 18 17 15 15 7 5 4 0 40

10 13 33 29 2 8 3 1 0 0.23

11 2 14 6 18 2 19 9 22 100

8 3 27 28 8 3 9 5 10 0.24

32 23 19 18 4 2 1 0 0 110

26 13 27 24 9 1 2 0 0 0.13

29 20 22 18 7 1 2 0 0 120

10 18 22 18 15 7 5 1 4 0.06

Cow milk, % of total flux North West South West North Wales Yorks & Humberside West Midlands East Midlands East Anglia South East Total flux (GBq y−1 )

ent in the extent of food contamination and areas affected, with only the Pennines and North Wales standing out as areas of relatively high transfer. The cumulative regional flux (GBq y−1 ) for England and Wales via sheep meat, cow milk and cereals is summarised in Table 2 for both the ‘short’ and ‘long’term. Short term is defined as the first year after the deposition event (i.e. May 1986 to April 1987), with the long term defined as the year between May 1995 to April 1996. The proportion of the total flux attributed to each of the 9 MAFF regions within England and Wales is also presented.

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In general, flux is ranked cow milk > cereals > sheep meat. A large proportion of UK cereal production is used as animal feed and for industrial purposes (approximately 60%) greatly reducing the exposure of the population to this source. The relative contributions of the different regions to the total flux for a food product remains approximately constant between the two time periods, suggesting that there are no major regional differences in the rate at which radiocaesium bioavailability declines with time for the specified agricultural products. Whilst there are differences in the rate of decline of bioavailability between individual cells (due to relative differences in the absorption of radiocaesium on clay minerals and organic matter), these differences disappear once averaged over regions. For sheep meat the regions of Wales and the North account for over 70% of the total flux under both scenarios, whilst for cow milk the flux contributions are more evenly spread between South West, North West, North and Wales. Sheep and milk production are biased towards the western regions of England and Wales, whereas the central and eastern regions dominate cereal production. Chernobyl fallout occurred mainly in the north and west and the interaction of these patterns of production and deposition is reflected in the relative magnitude of the total flux for each product between the two deposition scenarios. For the case of cereals there is a marked change in the spatial pattern of flux under a uniform deposition scenario, with the East Midlands and South East contributing more significantly to the total flux. Models of the type presented link the prediction of radiocaesium transfer to the agricultural food chain with spatially distributed data for soils and production. Such models have the potential to identify the regions where food products may be most contaminated, or areas where the combination of production and contamination level combine to maximise the overall input of radiocaesium into the food chain. Spatial models may therefore be useful for identifying regions where monitoring may be necessary and, potentially to give an estimate of the likely effectiveness of any remedial management.

5. Conclusions ‘Traditional’ models of radionuclide transfer to the food chain are based on the use of empirically derived parameters such as transfer coefficients and Tag values. The application of such parameters has the advantage of straightforward model development but makes little use of a mechanistic understanding of transfer processes which can, in some cases, lead to poor predictions. A widely observed result is that transfer parameters (soil–plant; feed–animal product) can be very variable, and this inevitably leads to problems in predictive models. A theme of this chapter has been the development of approaches which seek to explain such variation whether it be from soil to plant (Sections 2.1/2.2) or from feed to animal (Section 3.2). Mostly this has been achieved through the development of models which mechanistically describe the behaviour of the nuclide in the environment (e.g. Section 2.1) or an animal (e.g. Section 3.2) taking into account the effect of stable isotopes and analogues. The extremely variable radiocaesium transfer to fungi is not amenable to mechanistic explanation, at least not using current knowledge, however the nature of the variation has been analysed and it can be taken into account in a sensible way for the purposes of radiological assessment (Section 2.2). Many of the models we have outlined are probably too sophisticated to be incorporated directly into radiological assessment models (e.g. Sections 3.1/3.2), but they can be used to

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develop improved transfer parameter estimates perhaps functionally accounting for relevant external driving factors (e.g. animal physiological status). Moreover it seems possible that a general metabolic framework for describing the dynamics of a range of radionuclides in animals may be possible, such an approach may lead to the development of metabolically based methods of managing animal contamination. In Section 4 we have outlined how semi-mechanistic models for soil to plant transfer of radiocaesium have been integrated with spatial data sets of soil characteristics and agricultural production to produce models which can predict food chain contamination dynamically and spatially. We believe this is an important step forward as it enables areas most likely to be affected by contamination to be indentified. Such models are currently limited to radiocaesium, similar approaches can be envisaged for isotopes of strontium and iodine and should be developed.

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Howard, B. J., Mayes, R. W., Beresford, N. A. & Lamb, C. S. (1989). The transfer of radiocaesium from different environmental sources to ewes and suckling lambs. Health. Phys., 57, 579–586. Howard, B. J., Beresford, N. A., Mayes, R. W., Hansen, H. S., Crout, N. M. J. & Hove, K. (1997). The use of dietary calcium intake of dairy ruminants to predict the transfer coefficient of radiostrontium to milk. Radiation and Env. Biophysics, 36, 39–43. Howard, B. J. Beresford, N. A. & Voigt, G. (2001). Countermeasures for animal products: a review of effectiveness and potential usefulness after an accident. J. Environ. Radioactivity, 56, 115–137. IAEA (1994). Handbook of transfer parameter values for the prediction of radionuclide transfer in temperate environments. Technical Report Series No. 364. Vienna: IAEA. IUR (1982). Report on a Workshop on the Measurement of Soil-to-Plant Transfer Factors for Radionuclides. Part 1. Wageningen: International Union of Radioecology, EURATOM-ITAL. IUR (1989). VIth Report of the Working Group on Soil-to-Plant Transfer Factors. International Union of Radioecology. Jacob, P. & Likhtarev, I. (1996). Pathway analysis and dose distributions. Joint Study Project No. 5, Final report EUR 16541 EN (p. 130). Brussels: European Commission Directorate General XII. Kazakov, V. S., Demidchik, E. P. & Astakhova, L. N. (1992). Thyroid cancer after Chernobyl. Nature, 359, 21. Konshin, O. V. (1992). Transfer of 137 Cs from soil to grass – analysis of possible sources of uncertainty. Health Physics, 63, 307–315. Likhtarev, I. A., Gulko, G. M., Sobolev, B. G., Kairo, I. A., Chepurnoy, N. I., Pröhl, G. & Henrichs, K. (1994). Thyroid dose assessment for the Chernigov region (Ukraine): estimation based on 131 I thyroid measurements and extrapolation of the results to districts without monitoring. Radiat. Envir. Biophys., 33, 149–166. MAFF (1987). Radionuclide levels in food, animals and agricultural products 1987, Post Chernobyl monitoring in England and Wales (p. 63). London: Ministry of Agriculture Fisheries and Food, Welsh Office, HMSO. MAFF (1988). Radionuclide levels in food, animals and agricultural products 1988, Post Chernobyl monitoring in England and Wales. London: Ministry of Agriculture Fisheries and Food, Welsh Office, HMSO. McGrath, S. P. & Loveland, P. J. (1992). Geochemical Atlas of England and Wales (p. 112). London: Blackie Academic & Professional. Müller, H. & Pröhl, G. (1993). Ecosys-87: A dynamicmodel for assessing radiological consequences of nuclear accidents. Health Physics, 64, 232–252. Ng, Y. C., Colsher, C. S. & Thompson, S. E. (1982). Transfer coefficients for assessing the dose from radionuclides in meat and eggs. Livermore: Lawrence Livermore National Laboratory. Negri, M. C. & Hinchman, R. R. (2000). The use of plants for the treatment of radionuclides. In I. Raskin & B. D. Ensley (Eds), Phytoremediation of toxic metals: using plants to clean up the environment (pp. 107–132). John Wiley & Sons, Inc. Nisbet, A. F. & Shaw, S. (1994). Summary of a 5-year lysimeter study on the time-dependent transfer of Cs-137, Sr-90, Pu-239, Pu-240 and Am-241 to crops from 3 contrasting soil types. 2. Distribution between different plantparts. Journal of Environmental Radioactivity, 23, 171–187. Noordijk, H., Bergeijk, K. E. v., Lembrechts, J., & Frissel, M. J. (1992). Impact of ageing and weather conditions on soil-to-plant transfer of radiocaesium and radiostrontium. Journal of Environmental Radioactivity, 15, 277–286. Sharma, R. P. & Street, J. C. (1980). Public health aspects of toxic heavy metals in animal feeds. J. Am. Vet. Med. Assoc., 177, 149–153. Shaw, G. & Bell, J. N. B. (1994). Plants and radionuclides. In M. E. Farago (Ed.), Plants and the chemical elements: biochemistry, uptake, tolerance and toxicity (pp. 179–220). Weinheim: VCH. Sheppard, S. C. & Evenden, W. G. (1988a). Critical Compilation and Review of Plant-Soil Concentration Ratios For Uranium, Thorium and Lead. Journal of Environmental Radioactivity, 8 (3), 255–285. Sheppard, S. C. & Evenden, W. G. (1988b). The assumption of linearity in soil plant concentration ratios: an experimental evaluation. Journal of Environmental Radioactivity, 7, 221–247. Sheppard, S. C. & Evenden, W. G. (1990). Characteristics of plant concentration ratios assessed in a 64-site field survey of 23 elements. Journal of Environmental Radioactivity, 11, 15–36. Sheppard, S. C. & Evenden, W. G. (1997). Variation in transfer factors for stochastic models: Soil-to-plant transfer. Health Physics, 72, 727–733. Sheppard, S. C., Evenden, W. G. & Macdonald, C. R. (1999). Variation among chlorine concentration ratios for native and agronomic plants. Journal of Environmental Radioactivity, 43, 65–76.

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MODELLING RADIOACTIVITY IN THE ENVIRONMENT E. Marian Scott (Editor) © 2003 Elsevier Science Ltd. All rights reserved

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Pathways of anthropogenic radioactivity in the Northern Oceans Ingo H. Harms a , Michael J. Karcher b a Institute for Oceanography, Centre for Marine and Climate Research, University of Hamburg, Germany b Alfred-Wegener-Institute for Polar and Marine Research (AWI), Bremerhaven, Germany

1. Introduction

The following case studies describe the application of hydrodynamic circulation models to transport and dispersion of anthropogenic radioactivity in the North Atlantic and Arctic Ocean. The technical background of these simulations in terms of model physics and computation is given in Chapter 3, this volume.

1.1. Sources for anthropogenic radioactivity in the Arctic The input of anthropogenic radioactivity to the marine environment can be divided basically into two types: marine sources and atmospheric sources. Atmospheric sources usually appear as ‘fallout’, i.e. contaminated rain, stemming from bomb testing or from accidents. Input from these sources can be very diffuse or scattered and might affect surface waters on large spatial scales. The time and space distribution of the input function, which needs to be prescribed in a dispersion model, is difficult to deduce and often remains uncertain. Examples of major atmospheric sources in the Arctic are the ’bomb fallout’ in the 50s and 60s and the Chernobyl fallout 1986 (Table 1). Marine sources discharge directly into the marine environment. They mostly represent point sources such as outlets of reprocessing plants, leaking nuclear waste units or sunken nuclear ship reactors. These input functions can be more precise than atmospheric sources, at least in terms of input location. In some cases, even time series are known or can be predicted. But there are also very diffuse and difficult to determine marine sources such as input through river runoff, the drainage of liquid nuclear waste from a ship, the intrusion of pore water from underground nuclear bomb testing or remobilization of particle-bound radionuclides. 287

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Table 1 Sources of anthropogenic radioactivity in the Nordic Seas/Arctic Ocean (Aarkrog, 1994; JNREG – Joint Norwegian– Russian Expert Group, 1996; Nies et al., 1998) Source

Remarks/period

Estimated activity

Russian reprocessing plant at Chelyabinsk (Krasnoyarsk)

Between 1949 and 1951 Radioactive discharges via Techa – Iset – Tobol – Ob: 90 Sr ca. 11.6%; 137 Cs ca. 12.2%

100 PBq

Global fallout from the atmospheric weapontests

Between 1957 and 1963 Deposition 90 Sr between 0.25 and 0.75 kBq/m2 87 atmospheric tests (1955–1963)

ca.: 4.2 PBq 90 Sr, 6.7 PBq 137 Cs

Local fallout on Novaya Zemlya Sellafield discharges

Total discharges of 137 Cs into the Irish Sea: 41 PBq (decay corrected to 1994: 28 PBq)

Nuclear submarine ‘Komsomolets’ (7 April 1989; 1680 m)

Reactor Weapons: two nuclear torpedos (= 6.9 kg 239 Pu)

Nuclear submarine ‘Kursk’ (6 August 2000; 100 m)

Reactor

Disposal and discharges of liquid and solid radioactive waste by the former USSR or Russia in the Kara and Barents Seas

17 reactors in the Kara Sea at the date of dumping Presently: Actinides Activation products Fission products Solid wastes (1964–1991) Liquid wastes (1960–1992)

Weapons

20 PBq 90 Sr; 30 PBq 137 Cs Arctic Ocean: 15 PBq

2.9 PBq 90 Sr, 3.1 PBq 137 Cs 16 TBq 239 Pu 5.6 PBq 90 Sr, 6.2 PBq 137 Cs ? 90 PBq

0.2–0.9 PBq 4.4-4.8 PBq 4.7 PBq 0.6 PBq 0.9 PBq

1.2. Defining model scenarios Most of the model studies presented here were carried out in a research project that combined experimental field investigations and numerical modelling work (Povinec et al., 1997; Nies et al., 1998, 1999). Results from this project contributed to the International Arctic Seas Assessment Program (IASAP) launched by the International Atomic Energy Agency in 1994 (Sjoeblom & Linsley, 1998; IAEA, 1998, 1999), in particular to the Modelling Task Group of IASAP (Scott et al., 1997, 1998). One of the main goals of this project was to assess the consequences of possible leakage from underwater dump sites for nuclear waste in the Kara Sea. But there are many other previous, present and also potential sources which need to be considered. From Table 1, it is obvious that atmospheric fallout from bomb testing and the Chernobyl accident formed a considerable source for input of anthropogenic radioactivity in the Arctic in previous decades. However, due to the difficulties in defining realistic atmospheric source functions, we place the focus of our model scenarios on three major marine sources (Fig. 1):

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Fig. 1. Map of the Arctic Ocean, showing the Trans Polar (ice) Drift (TPD), the Beaufort Gyre and sources for anthropogenic radioactivity: Sellafield (1), sunken submarine ‘Komsomolets’ (2), sunken submarine ‘Kursk’ (3) and Kara Sea dump sites (4).

• the Sellafield reprocessing plant discharge, • the dumped radioactive waste in the Kara sea, and • the sunken nuclear submarines ‘Komsomolets’ and ‘Kursk’. These sources are all point sources that discharge directly into the water. Differences appear only in terms of applied release functions. The dump site scenarios and submarine scenarios are purely hypothetical, because significant continuous or instantaneous releases from these objects have not been observed so far. The dump site release functions for example were determined according to the recommendations of the IASAP source term working group, who made a forecast of possible release rates from different dumped objects (IAEA, 1997).

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Applying these hypothetical release functions, the model results have to be regarded as a prediction which of course cannot be validated. In contrast to that, the Sellafield reprocessing plant scenario is based on realistic release rates. Since the concentration of 137 Cs in the Nordic Seas has been monitored for the last four decades, the Sellafield scenario allows the possibility to validate the model results and to assess the ‘background’ radioactivity. One special case study is dedicated to the dispersion of radioactive particles through drifting sea ice. These scenarios are also hypothetical and should be regarded as order of magnitude estimation. Information on key parameters of this pathway, like sediment load in sea ice, radioactive concentration in incorporated sediment or uptake rates, are far too scarce to run realistic, i.e. comparable with observations, scenarios. 1.3. Configuration of model experiments Our model experiments used simulated circulation fields from hydrodynamic, 3-D, coupled ice–ocean models as forcing functions for transport modelling of radioactivity in the water column (see for a brief description of models Harms, 1997a; Nies et al., 1998, 1999); Harms & Karcher, 1999; Karcher & Oberhuber, 2002). A description of the application of hydrodynamic circulation models for transport of marine radioactivity, in particular the coupling of circulation and transport models is given in Chapter 3, this volume. Special emphasis was given to the validation of hydrodynamic model results because reproducing the realistic circulation is a very crucial point for transport modelling. The hydrodynamic circulation models were forced with realistic forcing functions like initial temperature and salinity distribution, freshwater input, wind, tides and ocean–ice–atmosphere heat fluxes. The model results were validated against various kinds of observations. This included ‘classical’ hydrographic parameters like temperature, salinity, sea surface elevation, current/transport measurements and ice observations (Harms, 1997a; Harms & Karcher, 1999; Karcher & Oberhuber, 2002) but also radioactive tracer measurements of 129 I and 137 Cs (Karcher et al., 1999). A benchmarking exercise, carried out within IASAP, compared and evaluated results of different Arctic models (Scott et al., 1997). The input from one of the above mentioned sources not only affects the near field in the range of several kilometres but also the far field within a range of hundreds of square kilometres. To assess the wide range of possible dispersion in time and space, some scenarios required the application of a hierarchy of circulation models on different spatial scales. The Kara Sea dump site scenario is an example of this kind of approach. The model simulations presented here describe the dispersion of the isotope 137 Cs because it represents a major fraction of the reprocessing plant discharges and the dump site inventories. It is highly soluble in water and may be considered a dissolved passive tracer with a decay time of 30.2 years. Apart from the radioactive decay which has to be considered in long term simulations, we used a conservative approach for modelling the dispersion of 137 Cs, as described in Chapter 3 ‘Transport modelling of dissolved radioactivity’. Two types of releases were considered: instantaneous release scenarios describe a sudden release of 137 Cs within a relative short time span whereas in gradual release scenarios, a continuous release rate over a relative long period is prescribed. Depending on the type of release and on the time and spatial scales to be considered, different types of transport models were applied. A sudden release from a point source can

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be traced on short time scales, using a Lagrangian particle tracking model. For long range transport following rather low gradual releases, the Eulerian approach based on the classical advection-diffusion transport equation was favoured. Both methods are described in more detail in Chapter 3, this volume. Circulation and transport models were coupled depending on the simulation time and the spatial scale. According to the coupling modes described in Chapter 3, ‘Coupling HD and transport models’, the large scale and long term applications (e.g. ‘The Sellafield discharge’) were coupled in an off-line mode whereas the small or regional scale applications on shorter time scales (e.g. ‘Kara Sea dump sites’) were coupled in an on-line mode. Please see Chapter 3 for more details on these coupling techniques. Due to the complexity and the high computational effort, the time frame of our hydrodynamic simulations was in the range of a few days for very local dispersion scenarios up to several decades on the large Arctic/North Atlantic scale. This is relatively short compared to the half life of certain isotopes and usually shorter than box model scenarios which assess the long term dose risk to humans and the environment. However, the aim of our model exercise was to reconstruct realistically the oceanic dispersion of radioactivity in the past or present and, based on validated ‘hindcast’ scenarios, to forecast the short term consequences of possible future releases from nuclear sources in the North Atlantic and Arctic Ocean. 2. The Sellafield discharge: dispersion of 137 Cs in the North Atlantic and the Arctic Ocean A major marine source for anthropogenic radioactivity in the North Atlantic and Arctic Ocean is the discharge from the reprocessing plants Sellafield, Dounreay and LaHague. In terms of contribution to the Arctic marine environment, the reprocessing plants discharge constitutes a dominant source second only in importance to nuclear weapons fallout. Historically and currently, Sellafield is the most important source among these (Macdonald & Bewers, 1996; Strand et al., 1996). The cumulative and decay corrected total 137 Cs radioactivity released by Sellafield amounts to approx. 30 PBq until 1995. The discharges from the Sellafield plant contribute to the Arctic inventory through northward transport via the Norwegian Coastal Current (NCC) and the Norwegian Atlantic Current (NAC, see Fig. 1). 14 PBq originating in Sellafield are estimated to have entered the arctic regions (Kershaw & Baxter, 1995). Maximum releases from Sellafield occurred in 1975 when approx. 5 PBq 137 Cs was discharged to the Irish Sea. Since then, releases of 137 Cs have been continually reduced and are now of the order of 0.01 PBq per year (MAFF, 1995). 2.1. Experimental design The reasons for running a model simulation of the Sellafield releases are clear. Its great importance for the state of radioactive contamination of the Nordic Seas and the Arctic Ocean has been mentioned above. Thus knowing the fate of Sellafield radioactivity in the Arctic is indispensable for an evaluation of potential contamination by other sources, e.g. dumped radioactive waste (see case study below). A second reason is the rather good coverage of the

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137 Cs

discharged from Sellafield by measurements. This holds especially for the North Sea and the eastern Nordic Seas, to a lesser degree also for the Barents and Kara Seas. The simulation of the 137 Cs discharge is an ideal case for model validation as for example shown by Preller & Cheng (1999). The basis for the following ‘hindcast’ scenario run is the simulation of a realistic field of oceanic flow. For this purpose a coupled ice–ocean model has been set up, covering the Arctic Ocean, the Nordic Seas and the Northern North Atlantic (Karcher & Oberhuber, 2002). It is driven by a climatological atmospheric data set consisting of a seasonal cycle of wind stresses, air temperatures, cloudiness and precipitation. After 35 years the model has reached an almost equilibrated seasonal cycle in the ice and upper ocean compartments. The Sellafield discharge of 137 Cs from 1965 to 1995 is prescribed in the large scale North Atlantic/Arctic Model at the northern boundary of the Irish Sea and traced for 31 years. Additionally, a regional scale model was used to study the time span (1975–1985) when peak concentrations occurred in the Kara Sea. For this purpose, the regional scale model used boundary conditions for the Kara Strait from the large scale North Atlantic/Arctic Model. On both spatial scales, the transport model used previously calculated monthly mean or 10-day mean flow fields from the circulation model of the same area. This technique represents the off-line mode coupling which is described in more detail in Chapter 3. 2.2. Model results and observations The model simulation shows a general cyclonic circulation of the radioactive tracers in the North Sea. Subsequently they follow the path of the Norwegian Coastal Current northward along the Norwegian coastline. Compared to measurements, the model reproduces levels, pathways and transit times in the surface waters very well (Fig. 2) (e.g. Dahlgaard, 1995; Kershaw & Baxter, 1995; Herrmann et al., 1998). This holds even for the North Sea area which has a rather coarse resolution in the current model (Fig. 3). The highest contamination in the Irish Sea, the North Sea and the Norwegian Sea was observed in the late 1970s when the discharges of 137 Cs reached maximum levels of 2–5 PBq/y between 1974 and 1983. These releases resulted in 137 Cs concentrations which exceeded 200 Bq/m3 in the North Sea. Simulated peak concentrations of 137 Cs reach the Barents Sea approx. 6 years after release, showing levels around 30–60 Bq/m3 in the southern and western parts. This is consistent with observations which revealed concentrations of up to 50 Bq/m3 for the south-western part of the Barents Sea (Kershaw & Baxter, 1995). The major fraction of these tracers flows into the northern Kara Sea between Franz-Josef-Land and Novaya Semlya, a minor fraction crosses the Kara Sea entering through the Kara Strait. Here, the peak values are reached in 1983 with levels around 40 Bq/m3 (Fig. 4) in accordance with observations (Povinec, pers. comm.; Crane & Galasso, 1999). The subsequent pathway of contaminants is difficult to trace from observations alone. The model simulation shows that one part of the tracer plume spreads northward, leaving the Kara Sea whereas another part enters the Laptev Sea and the East Siberian Sea. These different paths converge again and feed the Transpolar Drift, which in turn carries the contaminated water towards the Fram Strait. Maximum concentrations in the central Arctic Ocean at the surface reach about 40 Bq/m3 between 1985 and 1990. Observations catch up the contaminant plume again in the early to mid 1990s, when the highest concentrations of 137 Cs are

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Fig. 2. Simulated surface concentrations of 137 Cs in 1979 and 1985 due to the Sellafield discharge.

Fig. 3. Observed surface concentrations of 137 Cs in Bq/m3 in the west European Shelf area 1979 and 1985 (Nies et al., 1999).

observed in the central Arctic around the North Pole (14–19 Bq/m3 ) (Holm et al., 1996). These concentrations are also reproduced by the model simulation. In the Nordic Seas, the first trace of the simulated Sellafield release appears as a subsurface maximum at 150 m depth on the offshore side of the East Greenland Current in 1978. This feature lasts until 1985 with maximum concentrations of 15–20 Bq/m3 . It was first described by Aarkrog et al. (1983) based on measurements made in 1982. The model shows that this phenomenon is due to the contaminated Atlantic Water recirculation south of the Fram Strait. From 1986 on, the model reveals a subsurface maximum at a much shallower depth (approx. 100 m) and much closer to the coast of Greenland. It belongs to the Polar Water which leaves

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Fig. 4. Simulated peak concentrations of 137 Cs in the surface layer of the Kara Sea from 1978 to 1983 due to the Sellafield discharge.

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Fig. 5. Concentrations of 137 Cs due to the Sellafield releases as simulated with a large scale model for the year 1995. Upper panel shows the 400 m depth level with a maximum near the East Siberian and Alaskan continental slope. Lower panel depicts the several subsurface maxima on a section across the Arctic Ocean in the same year.

the Arctic Ocean with the East Greenland Current through the Fram Strait. These model results are consistent with observations made by Dahlgaard (1994). So far, we have only described the situation at the surface. However, in the eastern Barents Sea strong vertical mixing and densification leads to an input of 137 Cs into deeper water masses. These tracers leave the shelf with denser water and feed the Atlantic Water layer at intermediate depths (200–600 m). The contaminant plume circulates with the Atlantic Water layer cyclonically along the topographic slopes of the Arctic Ocean and finally enters the Eurasian and Canadian Basins. In the mid 1990s, it reaches the slope of the Chukchi Sea in the Canadian Basin (Karcher et al., 1997). At this time the maximum concentration in the Atlantic Water Layer of the Arctic Ocean (200–400 m) is about 20 Bq/m3 (Fig. 5, upper panel). A vertical section through the Arctic Ocean from Canada to the Kara Sea cuts through several of those contaminant plumes which recirculate with the cyclonic flow of the Atlantic Water Layer (Fig. 5, lower panel). This model finding supports the assumption of Smith et al.

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(1998) that the observed occurrence of high 129 I in 200–400 m at the slope of the Chukchi Sea in 1994 is a trace of Sellafield and LaHague emissions. As mentioned above, the overall agreement of the model results with observations is good. However, minor differences of 1 or 2 years can be found in the transit time. We attribute those to the missing interannual variability of the oceanic circulation due to the use of a climatological atmospheric forcing. Other sources of differences are the missing bomb fallout, adding a 2–5 Bq/m3 offset to the data of the 1990s and the missing inflow of 137 Cs (Smith et al., 1998) from the Chernobyl accident. The latter source adds roughly 50 Bq/m3 to the concentrations in the Skagerrak in 1986, but declines to (additional) 20 Bq/m3 in 1995 (Nies et al., 1998).

3. Kara Sea dump sites: simulation of realistic release scenarios on three spatial scales In 1992, it became evident that the former Soviet Union had dumped large amounts of radioactive waste in the Arctic Ocean. The waste has been dumped from 1959 on, in liquid and solid form, sealed in barrels or containers, as reactor parts but also as complete ship reactors containing spent fuel (Yablokov et al., 1993). The dump sites for solid low and high level waste are located in the Kara Sea, along the east coast of Novaya Semlya (Fig. 6). Four dump sites are located in small and shallow fjords, in depths less than 50 m. The dumped objects in the Kara Sea include 17 nuclear ship reactors, 7 of them including spent fuel. The total radioactive inventory at the time of dumping was 37 PBq. It decayed until 1994 to approx. 4.7 PBq. The dominant isotopes are 90 Sr, 137 Cs, 63 Ni and 241 Pu. The amount

Fig. 6. Location and topography of the Abrasimov Bay, one of the main dump site locations in the Kara Sea (Harms & Povinec, 1999).

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of 137 Cs is estimated to be around 1 PBq for 1994 (IAEA, 1997, 1998, 1999). Up to now, significant outflow of radioactivity from dumped objects that may affect the Kara Sea has not been observed (Salbu et al., 1997). However, some sediment samples very close to dumped objects showed considerably enhanced levels of radioactivity, indicating some small leakage (JNREG – Joint Norwegian–Russian Expert Group, 1996). 3.1. Experimental design In a ‘first guess’ assessment of the risk emerging from the dump sites, a ‘worst case’ scenario is simulated, which assumes an instantaneous release of the total radioactive inventory of dumped 137 Cs (1 PBq) over the relative short period of three months. Using an Eulerian transport module, as described in Chapter 3, this scenario gives some general information on local maxima and overall concentrations to be expected (Harms, 1997b; Baxter et al., 1998). The alternative method is a particle tracking simulation which is also described in detail in Chapter 3. In our Kara Sea simulations, this method reveals the main pathways and export routes of radionuclides (Fig. 7). A more thorough discussion on strength and weakness of both methods is given in Chapter 3. However, since an instantaneous release of the total inventory from four major dump sites is very unrealistic, we concentrate on more probable releases from individual dump sites. Following recommendations of the ‘source term working group’ within IASAP (IAEA, 1997), we apply a hypothetical gradual release rate of 1 TBq/y. Three different spatial scales of dispersion are treated separately: (1) the local scale which is focused on the bays in the western Kara Sea where the dumping of radioactive waste took place, (2) the regional scale which covers the shelves of the Barents and Kara Seas, and (3) the global scale which covers the North Atlantic and the entire Arctic Ocean. The local scale scenario is started, assuming a leakage at a dumped object in the inner parts of Abrasimov Bay, a Novaya Semlya Fjord. The Abrasimov Bay Model (see Fig. 6) is forced with realistic wind fields, heat fluxes and temperature and salinity distribution. Seasonal variability of circulation, stratification and ice formation is very pronounced. This affects strongly the dispersion of tracers. During most of the year, the tracer concentrations are higher in the upper layers than close to the bottom, indicating an outflow at the surface and a compensatory inflow below (Fig. 8). This pattern is reversed only during spring and early summer, when the wind directions change. The maximum radioactive concentration might exceed 4000 Bq/m3 but depth averaged it remains in the order of 2000–3000 Bq/m3 (Harms, 1997b; Harms & Povinec, 1999). The above scenario is extended on the regional scale towards the Kara Sea. The outflowing tracer concentrations from Abrasimov Bay are prescribed to the regional scale Kara Sea Model. With this approach it is possible to show the resulting evolution of the contaminant plume over a time span of six years (Fig. 9). It is obvious that maximum concentrations in the Kara Sea are reached after 5–6 years, at values around 0.1 Bq/m3 . The suggestion emerging from this scenario is that, except in regions very close to the bay opening, simulated concentrations in the Kara Sea did not exceed 1 Bq/m3 . The final step in this case scenario is the application of the 1 TBq/y 137 Cs release in Abrasimov Bay to the large scale, the Arctic Ocean and North Atlantic. This simulation shows that

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Fig. 7. Particle tracking simulations in the Kara Sea. Main pathways and transit times of a particle cloud, released instantaneously in spring in Abrasimov Bay: (a) at the surface and (b) in 25 m depth.

in the surface mixed layer, the contaminants move poleward until they are picked up by the Transpolar Drift (TPD, see Fig. 1). The tracers need about 7–8 years to reach Fram Strait. After another 4 years they reach the Denmark Strait. The level of radioactivity in the mixed layer decreases on this pathway from 1.0–0.1 Bq/m3 in the Kara Sea to 0.01 Bq/m3 in the East Greenland Current (Fig. 10). On the northern Kara Sea Shelf, a significant portion of the radioactive tracers is mixed down to the bottom. The contaminated dense Atlantic Water leaves the shelf and turns eastward to follow the continental slope in 200–400 m depth. The shallower portion recirculates along the Lomonosov Ridge in the Eurasian Basin and reaches Fram Strait after about

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Fig. 8. Horizontal flow fields and 137 Cs concentrations in Abrasimov Bay (August), at four depth horizons, following a continuous 1 TBq/y release in the inner part of the bay (Harms & Povinec, 1999).

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Fig. 9. Simulated concentrations of 137 Cs, following a continuous 1 TBq/y release in Abrasimov Bay. The picture shows concentrations at the surface over 6 years with continuous release (Nies et al., 1998).

18 years. The deeper portion crosses the Lomonosov Ridge entirely to enter the Canadian Basin. These tracers reach the Canadian Archipelago after 30 years having concentration levels around 10−2 to 10−3 Bq/m3 .

4. Sunken nuclear submarines: hypothetical releases from ‘Komsomolets’ and ‘Kursk’ Sunken nuclear submarines might present a considerable source for artificial radioactivity in Arctic regions. If no remedial measures are undertaken, nuclear reactor parts and also weapons might leak and release radioactivity into the environment. Most prominent examples are the ‘Komsomolets’ which sank in April 1989 south-west of Bear Island in the Norwegian Sea

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Fig. 10. Simulated concentrations of 137 Cs, following a continuous 1 TBq/y release at Kara Sea dump sites. The picture shows concentrations at the surface, 31 years after release (Nies et al., 1998).

Fig. 11. Concentrations of 137 Cs due to a hypothetical release of 2 PBq 137 Cs from the sunken nuclear submarine ‘Komsomolets’ (hexagon) for a depth level of 1000 m (upper panel) and on a section across the Nordic Sea (lower panel). The black lines in (b) represent the interfaces between adjacent layers of constant potential density in the large scale model.

and, more recently, the ‘Kursk’ which sank on August 12, 2000, about 100 km north-east of Murmansk in the Barents Sea. Although the ‘Kursk’ has been recovered and transported

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to Murmansk, the model study of this submarine accident remains very illustrative because numerous nuclear vessels are still in use or waiting for decommissioning near Murmansk and along the coast of the Kola peninsula. The wreck of the ‘Komsomolets’ at 73◦ 43.4 N; 13◦ 15.8 E, in 1680 m depth, contains one nuclear reactor with an estimated inventory of about 1.5 PBq 90 Sr and 2.0 PBq 137 Cs and two torpedoes with nuclear warheads amounting to about 6.9 kg 239 Pu corresponding to about 16 TBq (Yablokov et al., 1993; JNREG – Joint Norwegian–Russian Expert Group, 1996; Høibråten et al., 1997). 4.1. The ‘Komsomolets’ experiment We perform a case study for a potential leakage from the sunken ‘Komsomolets’ using the large scale North Atlantic/Arctic model in the same fashion as in the Sellafield and dumpsite scenarios described before. In a ‘worst case’ study, an instantaneous release of the entire inventory of 137 Cs is assumed. The subsequent dispersion of the radionuclides is simulated for 3 decades (Nies et al., 1999). The distribution of the dispersed radionuclides at a depth of 1000 m is shown in Fig. 11 (upper panel). The entire area of the deep Nordic Seas is contaminated with concentrations between 0.1 and 1 Bq/m3 . A plume of the tracers can be followed into the Eurasian Basin of the Arctic Ocean. It enters through the deep Fram Strait passage and flows eastward along the continental slope with concentrations of 10−2 –10−3 Bq/m3 . However, the radionuclides are dispersed in the vertical direction as well. The mixing between adjacent layers of potential density and the doming of the isopycnal layers in the centre of the deep Greenland Sea due to the cyclonic circulation (Fig. 11, lower panel) carries contaminants upward. Concentrations reach 10−2 Bq/m3 in about 500 m depth. It is probable that in long term runs these processes and the convective mixing in the centre of the Greenland Sea gyre in winter are able to move contaminants from the sunken vessel even to the surface of the ocean. The expected concentrations, however, would be far below the surface levels due to the Sellafield releases or global ‘fallout’. 4.2. The ‘Kursk’ experiment Much less is known about the radioactive inventory which had been on board of the ‘Kursk’. The Norwegian environmental protection group Bellona estimated the possible inventory of soluble radionuclides for the ‘Kursk’ based on known inventories of other submarines. According to their estimate, the inventory amounts to 5.6 × 1015 Bq (PBq) for 90 Sr and 6.2 PBq for 137 Cs (www.bellona.no). According to Russian authorities, the ‘Kursk’ carried at the time of sinking no nuclear torpedoes or missiles. The potential hazard due to a leakage of the ‘Kursk’ reactors, however, had spurred large public concern, mainly because it was positioned in the rather shallow Barents Sea, which is an important fishing ground. Using a coupled ice–ocean model which is driven by daily mean atmospheric forcing from the period 1979–1999, Gerdes et al. (2001) recently simulated a potential instantaneous release of 1 PBq of 137 Cs from the inventory of the sunken vessel. The pathway of the dispersing contaminants exhibits a structure similar to the one revealed by the Sellafield or Kara Sea dump site scenarios shown above. However, due to the application of two multiyear periods with different intensities of ventilation on the Barents Sea Shelf, the simulations revealed two

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Fig. 12. Surface concentrations of 137 Cs 5 years after a hypothetical release of 1 PBq in August l983 from the sunken nuclear submarine ‘Kursk’ (hexagon). The model had been driven with realistic atmospheric forcing fields. This five years period is typical for a weak flushing of the Barents Sea area (Gerdes et al., 2001).

essentially different contaminant patterns. While an intense flushing of the Barents Sea leads to fast advection of the contaminant cloud into the eastern Eurasian Basin, a weak flushing is typically characterised by a more northern pathway of the cloud when leaving the shelf area. This results in much shorter transit times for the contaminants to reach Fram Strait (faster TPD) and to higher concentrations in the northern Nordic Seas. A more thorough discussion on the influence of atmospheric forcing on dispersion results can be found in Chapter 3, in ‘Forcing strategies in dispersion studies’, this volume. In Fig. 12, the concentration at the sea surface, 5 years after release, during a period of typical weak flushing in the Barents Sea area (here: start August 1983) is shown. The centre of the contaminant cloud has spread from the source location in the Barents Sea into the Kara Sea and the Eurasian Basin of the central Arctic Ocean. Maximum concentrations in the Kara

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Sea are slightly above 10 Bq/m3 . When reaching the northern Fram Strait, the signal divides into a major branch moving southward with the East Greenland Current and a minor branch which follows the northern Greenland and Canadian coastlines westward. North of Alaska, this branch feeds into the anticyclonic Beaufort Gyre of the Canadian Basin. Concentrations in the northern East Greenland Current are up to 1 Bq/m3 , in the western Arctic branch 10−1 – 10−2 Bq/m3 .

5. Transport of radioactivity by Arctic sea ice: evaluation of a complex pathway Sea ice plays a special role in the transport of radionuclides in the Arctic. In general, radionuclide concentrations in sea ice itself are very low. Like salt, most of the dissolved radionuclides are excluded from the ice due to brine rejection, a filtration process that occurs when ice is forming. This means sea ice is usually very ‘clean’ or at least less contaminated than the water from which it is formed (Pfirman et al., 1995). However, several radionuclides are particle reactive which mean that they tend to attach to particles or suspended matter in the water column or to the sediment. Arctic sea ice very often carries loads of sediment and if this is contaminated, the ice drift may play an important role in the long-range redistribution of radioactivity (Pfirman et al., 1995, 1997b). 5.1. Observations The ice movement in the Arctic Ocean is dominated by the Transpolar Drift and the Beaufort Gyre (see Fig. 1). The TPD crosses the Arctic Ocean in two branches: the Siberian branch which is closer to the Siberian landmass, and the polar branch on the side facing the North American continent. In particular the Siberian branch is fed by large ice production areas located in the East Siberian, Kara and Laptev Seas. The TPD is responsible for the transport of sediment-laden sea ice from Arctic Shelf areas through Fram Strait towards the Greenland Sea (e.g. Dethleff et al., 1994; Eicken et al., 1997). Through ice melting, the material is released to the marine environment of the Nordic Seas, Europe’s most important fishing ground. Since the North Atlantic ice-melt period coincides with the locally increasing biological activity during spring bloom, the pollutants may easily enter the food web (AMAP, 1998). The mechanisms for the incorporation of sediment into sea ice can be: (1) suspension freezing, i.e. the scavenging of sediment from the water column during frazil ice formation (Reimnitz et al., 1993a; Dethleff et al., 1993), (2) flooding of fast ice by sediment laden river water (Nürnberg et al., 1994), (3) the formation of anchor ice (i.e. freezing of supercooled water at the sea bed) (Reimnitz et al., 1993a; Dethleff et al., 1993), (4) the adfreezing of bottom sediments to grounded or shore-fast ice (Nürnberg et al., 1994), (5) eolian transport by off-shore winds from Siberia (‘Arctic Haze’) (Pfirman et al., 1995). In an overview which evaluates possible ways of entraining sediments into sea ice, Nürnberg et al. (1994) state that suspension freezing is the most important process. There is strong evidence that high sediment loads in sea ice originate from regions with high suspended load in the turbulent water column (Reimnitz et al., 1993b).

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Fig. 13. Scetch of a cross-section through an Arctic flaw-lead (Nies et al., 1998). The figure explains the principle of incorporating sediments in sea ice by erosion, suspension freezing and frazil ice formation.

Suspension freezing is supposed to happen in so called ‘flaw leads’ (Dethleff, 1995). The coastal areas of the Arctic Shelves are characterised by recurrent open water between fast and drifting ice, which are driven by offshore winds (Fig. 13). These ‘flaw leads’ produce large amounts of new ice through the mechanism of leeward advection (Martin & Cavalieri, 1989; Dethleff et al., 1998). Strong heat loss and enhanced ice formation provide a turbulent convectively mixed water column which enhances resuspension of fine grained material. Under these conditions, the sea water becomes supercooled by a fraction of a degree and frazil ice forms. The underwater ice crystals interact and mechanically interlock with sedimentary particles in the water and lift fine-grained particulate matter to the surface. The observed range of concentrations of sediment in Arctic sea-ice is wide: minimum concentrations were reported to be around 800–3000 t/km2 , while maximum concentrations could reach as much as 90 000 t/km2 . Between 7 and 150 million tons of ice-incorporated sediments may annually leave the Arctic Ocean through Fram Strait and move towards the North Atlantic. Considerable portions of the sea-ice sediments sampled in the eastern central Arctic and in Fram Strait may have been entrained in the Laptev Sea and the eastern Kara Sea (Pfirman et al., 1997b). For further reference see Nies et al. (1998) and Dethleff et al. (2000). The sediment usually accumulates at the surface as patches of 10–20 m in diameter or as surface layers of 10–15 cm thickness, mostly at the interface between ice and snow (Nürnberg

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et al., 1994). The highest concentrations of sediments can be found on the surface of multiyear ‘old’ ice (Wollenburg, 1993). Observed values in the central Arctic range between 10 mg/l and 56 000 mg/l (Nürnberg et al., 1994). The accumulation at the ice surface is due to the fact that the ice melts at the top and freezes at the bottom. In multiyear cycles, the sediment ‘migrates’ to the ice surface. Frequently, the particulate matter accumulates in melt ponds (Meese et al., 1995) or small pits of 10–30 cm depth and 3–5 cm in diameter called ‘cryoconite holes’ (Pfirman et al., 1995). These melt ponds result from the decreasing albedo of dirty sediment at the surface which enhances the melting process of the ice. The cryoconite holes are supposed to be an important living space for algae and other micro-organisms in the Arctic (Wollenburg, 1993). Parts of the Kara Sea may be considered as important source regions for the export of radioactive contamination through sediments, incorporated in sea-ice. In our model studies we investigated two source areas: • the Kara Sea dump sites near Novaya Semlya, and • the Siberian rivers Ob and Yenisei. As already described before, several nuclear objects were dumped in shallow bays at Novaya Semlya with less than 50 m depth. It seems very likely that here contaminated sediment might be in contact with the sea ice cover. Other key regions for the incorporation of various contaminants in sea ice are the estuaries of Ob and the Yenisei Rivers which drain huge land areas in Siberia and Russia. The catchment areas include many diffuse sources, among them two nuclear fuel reprocessing plants, Mayak and Tomsk, which enter the drainage areas of the Ob. The outlet of a third reprocessing facility, Krasnoyarsk, goes directly into the Yenisei (AMAP, 1998; NATO, 1998). 5.2. Model experiments The uptake of sediment by sea ice is very complex and only very few model applications concentrate directly on the process of sediment incorporation, e.g. Sherwood (2000) modelled the entrainment of sediments into newly forming lead ice in the southern Kara Sea off the mouths of Ob and Yenisei Rivers. Using a process oriented model, Sherwood (2000) concluded that bottom material and riverine particles may be incorporated in drifting ice from that area. However, simulating the time and space dependant radionuclide concentration in sediment-laden sea ice is almost impossible because of uncertain parameterisations and missing input data. A regional scale model application, described in Harms et al. (2000), takes a different approach which does not focus on the incorporation of sediments but on the dispersion of sediments in sea ice. Using 3-D coupled ice–ocean models and Lagrangian particle tracking models as described in Chapter 3, the authors try to detect at which time and in which areas the incorporation of simulated river particles into newly forming sea ice is most likely. The model results show that sediment incorporation happens in November and December, mostly along the Siberian shores (Fig. 14). Off-shore winds create thin ice or open water leads where thermodynamic ice formation is larger than 50 cm/month (dark shaded areas). These areas coincide partly with high amounts of river tracers which emerge from the Ob or Yenisei during previous spring (Fig. 14, a, b). In these regions, the uptake of sediment or particles from the water column into freezing ice is very probable. Finally, Lagrangian trajectories of the

(b)

(c)

(d)

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Fig. 14. (a) and (b): simulated mean thermodynamic ice formation and river tracer distribution in November and December, for Ob (a) and Yenisei (b). (c) and (d): simulated ice trajectories started in November in the Ob (c) and Yenisei (d) estuary (Harms et al., 2000). 307

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ice velocities from November to May are used in order to trace particle or sediment laden patches in newly formed sea ice (Fig. 14, c, d). These regional scale simulations showed that ice trajectories started close to the Ob and Yenisei mouth are able to leave the Kara Sea within one winter. Large scale drift studies of possibly contaminated sediments, incorporated in Siberian flaw lead ice were presented by Nies et al. (1998), Dethleff et al. (2000) and Harms et al. (2000). Ice drift, thickness and concentration from a 3-D coupled ice–ocean model of the Arctic Ocean and Nordic Seas are used to simulate ice trajectories for several subsequent years from ice formation to ice melt. These studies suggest that trajectories from the northern and central Kara Sea are able to reach the Arctic TPD within one winter cycle. However, trajectories started in the southern Kara Sea usually melt in the following spring before they can leave the formation area. A direct link between the Kara Sea dump sites and the Fram Strait via the TPD is unlikely, at least for the southernmost dump sites. Trajectories started in the central and eastern Kara Sea, however, show a strong tendency to leave north-westward into the Barents Sea or the Arctic Ocean. Some of the eastern trajectories, close to the Ob and Yenisei estuaries, leave the Kara Sea between Franz-Josef-Land and Novaya Semlya before they end up in the Barents Sea. The most eastern trajectories pass north of Franz-Josef-Land and join the TPD. For these trajectories, the transit time from the Kara Sea towards Fram Strait can be very fast, between 1 and 3 years. Figure 15 shows an example for this icedrift from the northern Kara Sea for a three years period from 1979 to 1982, gained with the coupled ice–ocean model (Karcher & Oberhuber, 2002) which has also been used for the ‘Kursk’ experiment presented in Section 4.2. The model findings are supported by sea ice backward trajectories calculated by Pfirman et al. (1997a) and drift buoy data from King et al. (1997). Other model estimates (c.f. Nies et al., 1998; Dethleff et al., 2000) reveal that at maximum 10% of the ice formed in the southern parts leave the Kara Sea towards the central Arctic Ocean before summer melt. The ice formed in the northern parts leaves the Kara Sea with a probability of almost 100%. Colony & Thorndike (1985) estimated a 79% probability for ice formed in the Kara Sea to drift into the Arctic Ocean. These observations show that simulated ice trajectories are quite reliable and that the time and space variability of ice formation and ice melt are well reproduced. Model simulations thus provide very useful information on pathways and transit times of sediments in sea ice. However, realistic export rates of radioactively contaminated sea ice are difficult to deduce because of the huge lack of sedimentological data and uncertain parameterisations. In a simple conservative approach, Harms (1997b) estimated the possible radionuclide export by sea ice from the Kara Sea to the Arctic Ocean to be of the order of 0.03 TBq/y. This calculation assumes an ice volume flux from the Kara Sea into the Arctic Ocean of 150 km3 /y (Pavlov & Pfirman, 1995), a sediment load in sea ice of 3 mg/l (IAEA, 1994) and a radionuclide concentration in sea ice sediment of 70 Bq/kg (Meese et al., 1997). The estimated export rate is at least one order of magnitude lower than the corresponding export rate through the water column but probably still overestimated. A wide range of possible export rates was also presented in a more detailed study by Dethleff at al. (2000). Both publications confirm that, compared to the dissolved transport in the water column, drifting sea ice might be a very fast transport pathway. It might also be quite effective due to the missing dilution/diffusion of radioactive concentration in the ice and the

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Fig. 15. Simulated ice-trajectories started in October 1979 from the north-eastern Kara Sea. Maximum drift 3 years, minimum drift until melt.

possibility of transporting ‘hot spots’, i.e. sediment patches of very high contamination. Integrated over time and space, however, the dominant pathway for export of radioactivity from the Arctic Ocean is the transport in dissolved form in the water column, even if radionuclide concentrations in the water are much lower than in sea ice. The main reason for this is the large water volume transport compared to a much smaller sea ice transport.

6. Concluding remarks 6.1. Summary of model results The general evaluation of our model results is that the radiological situation in the Arctic Ocean and the Nordic Seas is dominated by the Sellafield signal. A comparable large-scale contamination of the Arctic Ocean by radioactive waste from the Kara Sea or sunken nuclear submarines seems unlikely. Compared to the maximum concentrations in the Sellafield scenario, the dump site and submarine scenario levels are in the same range or even lower. In

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particular the realistic gradual release scenarios reveal levels of radioactivity that are at least two or three orders of magnitude lower than maximum Sellafield values. Exceptions, however, are the local and regional scale results from hypothetical ‘worst case’ scenarios carried out for the Kara Sea dump sites or the sunken submarine ‘Kursk’. Simulated instantaneous releases of the entire dumped inventory would cause radionuclide concentration levels known from the Irish Sea in the late seventies or early eighties, when Sellafield had its peak release period. The transport of radioactivity by sea ice seems to be a likely scenario. Our results confirm the possibility of an export of contaminated sediment from the Kara Sea into the central Arctic Ocean and subsequently through Fram Strait into the Greenland Sea. Drifting sea ice offers a rather fast and effective pathway for pollutant export from the Arctic because contaminated particulate matter can be released more or less undiluted, 2–3 years after incorporation. However, compared to the large volume (and hence contaminant) flux in the ocean, the time and space averaged export rates of radioactivity by sea ice are significantly lower than export rates in water. 6.2. Uncertainties The most important question concerning a forecast of consequences of future releases is of course ‘how reliable are the model results?’. Uncertainties might stem from the hydrodynamic model and/or the transport model that was used. Typical uncertainties in transport models concern the simplified characteristics of transported tracers. An example of such an uncertainty is our assumption that 137 Cs is totally dissolved in the water column, and thus transported passively. Although 137 Cs can be regarded as highly soluble, it is to a small extent also particle reactive. The interaction with particles or sediment is even more pronounced with other radionuclides. Together with processes like deposition, erosion/remobilization, this might result in additional sources or sinks. Another uncertainty, common for both, hydrodynamic and transport models, has its origin in the parameterization of diffusion in the momentum and transport equations. Also artificial diffusion, introduced through the numerical computation, is a source for uncertainty. Several methods and algorithms were introduced in the last few years to cope with these problems. The use of ‘state of the art’ models for circulation and transport is thus essential to achieve realistic model results. According to our philosophy, the hydrodynamic flow fields are the most important parts in a combined circulation and transport model study. The application of a very sophisticated transport model is useless if the prescribed flow fields are too simple or not realistic. The reliability of the flow fields depends on the quality of the hydrodynamic model (e.g. parameterisation of unresolved processes, simplified assumptions, numerical errors). Of almost the same importance is the configuration of the model experiment, including aspects like grid resolution and forcing. In most of the studies shown above, the hydrodynamic models were forced with climatological monthly mean atmospheric data which might lead to an underestimation of the simulated variability, especially in the upper ocean. This holds for high-frequency variability below monthly periods as well as for lower frequency interannual variability. Although the simulated circulation patterns and thus the pathways of tracers are realistic in the sense that they

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compare well with the general image from observations, the missing variability will surely affect the results. While the missing high-frequency variability will influence the concentration levels and the spatial extent of the dispersed substances, the low-frequency interannual variability might even result in different pathways e.g. if flow patterns change on the interannual time scale. The case study for the ‘Kursk’ for example (Gerdes et al., 2001) was based on daily mean atmospheric forcing data from a ‘real’ five years period, thus accounting for high-frequency variability up to the daily time scale. The simulated pathways and transit times from the ‘Kursk’ simulation were comparable to the climatological dump-site scenarios which suggests that high-frequency variability has a minor effect on the large scale circulation patterns. Low-frequency variability, however, affects the general flow field, e.g. the intensity of current systems, which in turn influences pathways and transit times of tracers. We do not expect severe changes in the principal circulation pattern on time scale of up to 30 years but there are indications for changes in the surface flow pattern of the central Arctic Ocean, which show up as an oscillation between two modes every 5–10 years. While in one mode, the Beaufort Gyre is small and the TPD runs from Bering Strait across the North Pole to Fram Strait, in the second mode, the Beaufort Gyre is large and the TPD has a position much closer to the Kara- and Barents Sea slope (Proshutinsky & Johnson, 1997). Our large-scale results from Sellafield and dump-site scenarios are based on climatological flow fields similar to the first mode. For the second mode, we would expect shorter transit times from the Siberian shelves to Fram Strait and more enhanced radionuclide concentrations on the Siberian side of the Transpolar Drift. Gerdes et al. (2001) showed in a comparative model study that the two circulation modes affect the flushing of the Barents Sea which had considerable consequences for the ‘Kursk’ scenario results. The ‘Kursk’ study emphasises the need for using forcing fields from different periods and with different characteristics in order to assess the influence of low-frequency (e.g. interannual and decadal) variability. Concluding we can state that the discussed uncertainties might cause differences in dispersion results, but would not alter the general evaluation of the release scenarios described above. Our model studies confirm that the present radiological situation in the Arctic is far from being dramatic. However, the density of nuclear facilities in this part of the world ocean is remarkable and higher than anywhere else. There still is a potential risk of nuclear accidents and failures with considerable consequences for the environment. Acknowledgements The authors want to express their gratitude towards the funding agencies who supported the presented work: part of M. Karcher’s work has been funded by the EC-MAST II programme under grant MAS3-CT96-0070 (VEINS). The results-presented in Section 4 are based on work performed in close co-operation of M. Karcher with R. Gerdes and F. Kauker at the Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany. Most of I. Harms’ and M. Karcher’s work has been funded by the International Atomic Energy Agency (IAEA) and the German Ministry for Education, Research, Science and Technology (BMBF, 02-E-87054).

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IAEA (1997). Predicted radionuclide release from marine reactors dumped in the Kara Seas. Report of the Source Term Working Group of the IASAP. IAEA-TECDOC-938. IAEA (1998). Radiological conditions of the western Kara Sea. Radiological Assessment Report Series. Vienna, Austria: IAEA. IAEA (1999). Radioactivity in the Arctic Seas. Report for the International Arctic Seas Assessment Project IASAP. IAEA-TECDOC-1075 (71 pp.). JNREG – Joint Norwegian–Russian Expert Group (1996). In P. Strand, A. I. Nikitin, B. Lind, B. Salbu & G. C. Christensen (Eds), Dumping of Radioactive Waste and Radioactive Contamination in the Kara Sea. Results from 3 Years of Investigations (1992–1994) in the Kara Sea. Osteras, Norway: NRPA. Karcher, M. J., Harms, I. H., Dethleff, D. & Nies, H. (1997). Transit times and pathways of contaminants in the Arctic Ocean. Proceedings of the ‘AMAP International Symposium on Environmental Pollution of the Arctic’, 1–5 June 1997, Tromsø, Norway, extended abstract volume. Karcher, M. J., Harms, I. H. & Smith, J. N. (1999). Long-range transport of 129 I and 137 Cs in the Nordic Seas and the Arctic Ocean. IAEA-TECDOC-1094 (193 ff). Vienna. Karcher, M. J. & Oberhuber, J. M. (2002). Pathways and modification of the upper and intermediate waters of the Arctic Ocean. J. Geophys. Res., 107 (C6), 10.1029/2000JC000530. Kershaw, P. & Baxter, A. (1995). The transfer of reprocessing wastes from north-west Europe to the Arctic, Deep Sea Res., 42, 1413–1448. King, S. E., Carroll, J., Johnson, J. R., Krosshavn, M., Phillips, G. W., August, R. A., Boisson, F. & Theyssie, J. L. (1997). Transport of caesium in the Kara Sea. Radioprotection – Collogues, Vol. 32, C2, special issue: Radionuclides in the Oceans (RADOC 96-97) Part I: Inventories, Behaviour and Processes (p. 231). 7–11 October 1996. France: Cherbourg-Octeville. MAFF (1995, and previous reports). Radioactivity in Surface and Coastal Waters of the British Isles, 1994. Macdonald, R. W. & Bewers, J. M. (1996). Contaminants in the arctic marine environment: priorities for protection. ICES Journal of Marine Science, 5, 537–563. Martin, S. & Cavalieri, D. J. (1989). Contributions of the Siberian Shelf polynyas to the Arctic Ocean intermediate and deep water. J. Geophys. Res., 94 (C9), 12 725–12 738. Meese, D., Cooper, L., Larsen, I. L., Tucker, W., Reimnitz, E. & Grebmeier, J. (1995). Cesium-137 Contamination in Sea Ice. Proceedings of the Workshop on Monitoring of Nuclear Contamination in the Arctic Seas. Washington: Naval Research Laboratory. Meese, D. A., Reimnitz, E., Tucker III, W. B., Gow, A. W., Bischof, J. & Darby, D. (1997). Evidence for radionuclide transport by sea-ice. Sci. Tot. Environ., 202, 267–278. NATO (1998). NATO/CCMS Pilot Study: Cross-Border Environmental Problems Emanating from Defence-related Installations and Activities, Phase II: 1995–1998, Final Report, Vol. 2: Radioactive Contamination of Rivers and Transport Through Rivers, Deltas and Estuaries to the Sea, Report No. 225. Nies, H., Harms, I. H., Karcher, M. J., Dethleff, D., Bahe, C., Kuhlman, G., Kleine, E., Loewe, P., Oberhuber, J. M., Backhaus, J. O., Matishov, D., Stepanov, A. & Vasiliev, O. F. (1998). Anthropogenic radioactivity in the Nordic Seas and the Arctic Ocean: Results from a joint project. Deutsche Hydrographische Zeitung, 50, (4), 313–343. Nies, H., Karcher, M. J., Bahe, C., Backhaus, J., Harms, L. & Dethleff, D. (1999). Transportmechanismen radioaktiver Substanzen im Arktischen Ozean-Numerische und experimentelle Studien am Beispiel der Barents- und Karasee, Berichte des Bundesamtes für Seeschiffahrt und Hydrographie Nr. 21 (134 pp.). Hamburg und Rostock. Nürnberg, D., Wollenburg, D., Dethleff, D., Eicken, H., Kassens, H., Letzig, T., Reimnitz, E. & Thiede, J. (1994). Sediments in Arctic sea ice – entrainment, transport and release. Mar. Geol., 119, 185–214. Pavlov, V. K. & Pfirman, S. L. (1995). Hydrographie structure and variability of the Kara Sea: Implications for pollutant distribution, Deep Sea Res., 42 (6). Pfirman, S., Eicken, H., Bauch, D. & Weeks, W. F. (1995). The potential transport of pollutants by Arctic sea ice. Sci. Tot. Environ., 159, 129–146. Pfirman, S., Colony, R., Nürnberg, D., Eicken, H. & Rigor, I. (1997a). Reconstructing the origin and trajectory of drifting Arctic sea-ice. J. Geophys. Res., 102 (C6), 12575–12586. Pfirman, S., Kögele, J. W. & Rigor, I. (1997b). Potential for rapid transport of contaminants from the Kara Sea. Sci. Tot. Environ., 202, 111–122. Povinec, P., Osvath, J., Baxter, M. S., Ballestra, S., Carroll, J., Gastaud, J., Harms, I. H., Huynh-Ngoc, L., Liong Wee Kong, L. & Pettersson, H. (1997). Summary of LAEA-MEL’s Investigation of Kara Sea Radioactivity and Radiological Assessment. Marine Pollution Bulletin, 35 (7–12), 235–241.

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Preller, R., & Cheng, A. (1999). Modeling the Transport of Radioactive Contaminants in the Arctic, Marine Pollution Bulletin, 38, 71–91. Proshutinsky, A. Y. & Johnson, M. A. (1997). Two circulation regimes of the wind-driven Arctic Ocean, J. Geophys. Res., 102, 12 493–12 514. Reimnitz, E., Clayton, J. R., Kempema, E. W., Payne, J. R. & Weber, W. S. (1993a). Interaction of rising frazil with suspended particles: tank experiments with applications to nature. Cold Reg. Sci. Technol., 21, 117–135. Reimnitz, E., McCormick, M., McDougall, K. E. & Brouwers, E. (1993b). Sediment-export by ice rafting from a costal polynya, Arctic Alaska. Arctic Alp. Res., 25 (2), 83–98. Salbu, B., Nikitin, A. L., Strand, P., Christensen, G. C., Chumichev, V. B., Lind, B., Fjelldal, H., Bergan, T. D. S., Rudjord, A. L., Sickel, M., Valetova, N. K. & Føyn, L. (1997). Radioactive contamination from dumped nuclear waste in the Kara Sea – Results from the joint Russian-Norwegian Expeditions in 1992–1994. Science of the Total Environment, 202, 185–198. Scott, E. M., Gurbutt, P., Harms, I., Heling, R., Nielsen, S. P., Osvath, I., Preller, R., Sazykina, T., Sjoeblom, K. L. & Wada, A. (1997). Benchmarking of Numerical Models describing the Dispersion of Radionuclides in the Arctic Seas. Science of the Total Environment, 202, 123–134. Scott, E. M., Gurbutt, P., Harms, I., Heling, R., Kinehara, Y., Nielsen, S. P., Osvath, I., Preller, R., Sazykina, T., Sjoeblom, K. L. & Wada, A. (1998). Radiological impact assessment within the IAEA Arctic Assessment Project. Radiation Protection Dosimetry, special issue: Radionuclides in the Oceans (RADOC 96-97), Part II: Distributions, Models and Impacts, 7–11 April 1997. U.K.: Norwich/Lowestoft. Sherwood, C. (2000). Numerical model of frazil-ice and suspended-sediment concentrations, and formation of sediment-laden ice in the Kara Sea, J. Geophys. Res. (in press). Sjoeblom, K.-L. & Linsley, G. S. (1998). International Arctic Seas Assessment Project (IASAP). Radiation Protection Dosimetry, special issue: Radionuclides in the Oceans (RADOC 96-97), Part II: Distributions, Models and Impacts, 7–11 April 1997. U.K.: Norwich/Lowestoft. Smith, J. N., Ellis, K. M. & Kilius, L. R. (1998). 129 I and 137 Cs Tracer measurements in the Arctic Ocean. Deep Sea Res., 45, 959–984. Strand, P., Sickel, M., Aarkrog, A., Bewers, J. M., Tsaturov, Z. & Magnusson, S. (1996). Radioactive Contamination of the Arctic Marine Environment. In Guegueniat et al. (Coord), Radionuclides in the Oceans – Inputs and Inventories (pp. 95–119). France: Inst. de Protection et de Surete Nucleaire. Wollenburg, I. (1993). Sedimenttransport durch das arktische Meereis: Die rezente lithogene und biogene Materialfracht. Berichte zur Polarforschung (Rep. Pol. Res.), 127, 1–159. Yablokov, A. V., Karasev, B. K., Rumyantsev, V. M., Kokeyev, M. E., Petrov, O. I., Lystsov, B. N., Yemelyanenkov, A. F. & Rubtsov, P. M. (1993). “White Book”. Facts and Problems related to radioactive waste disposal in seas adjacent to the territory of the Russian Federation. Moscow: Office of the President of the Russian Federation.

MODELLING RADIOACTIVITY IN THE ENVIRONMENT E. Marian Scott (Editor) © 2003 Elsevier Science Ltd. All rights reserved

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Modelling the behaviour of radiocaesium in forest ecosystems G. Shaw a,1 , R. Avila b,2 , S. Fesenko c,3 , A. Dvornik d,4 , T. Zhuchenko d a Imperial College at Silwood Park, Ascot, Berkshire, SL5 7PY, United Kingdom b Swedish Radiation Protection Institute, SE-171 16 Stockholm, Sweden c Russian Institute of Agricultural Radiology and Agroecology, Kievskoe Street, Obninsk, Kaluga Region, 249020,

Russian Federation d Forest Institute of the Belarussian Academy of Science, 71 Proletarskaya Street, 246654 Gomel, Belarus

1. Introduction 1.1. Radionuclides in forests before the Chernobyl accident The first detailed studies of forest radioecology concerned the behaviour of a range of radionuclides in the contaminated forests of the Eastern Urals Radioactive Tract (EURT), following the Kyshtym accident in 1957 (Sokolov et al., 1993). Measurements made by Tikhomirov (1993) indicated that 144 Ce, 144 Pr, 106 Ru, 106 Rh, 95 Zr and 95 Nb were all present as significant contaminants of birch and pine forests following the accident but, due to its 28 year physical half life, it was 90 Sr which was the most important long-term contaminant of these forests. Even though the initial contribution of 90 Sr to the total activity deposited to forests in 1957 was only 5%, decay of the other shorter-lived radionuclides resulted in 90 Sr contributing 83% of the total activity in 1962 (Tikhomirov et al., 1993). As a result of 12 years of observations of the biogeochemical cycling of 90 Sr within the contaminated forests of the EURT, Prokhorov & Ginsburg (1971) and Alexakhin et al. (1976), later re-described by Alexakhin et al. (1994), proposed a model based on a compartmental representation of radionuclide fluxes within the major components of both birch and pine forests (8 compartments, viz. soil, litter (×2), wood, bark, branches, leaves and forest herbs). This model, using data collected during an initial calibration period of 11 years, was used to predict 90 Sr distributions in the EURT forests over a period of 50 years following the 1 E-mail address: [email protected] 2 E-mail address: [email protected] 3 E-mail address: [email protected] 4 E-mail address: [email protected]

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Kyshtym accident. The prediction indicated that the major ecological repository for 90 Sr over this period was likely to be the soil and litter compartments which were projected to contain between 85 and 90% of the total 90 Sr in the ecosystem by the year 2007 (Alexakhin et al., 1994). The major feature of the model was that fluxes of 90 Sr between compartments were represented as a series of coupled linear differential equations which was solved for an initial pulse input to the system from the atmosphere. Transfer rate coefficients were considered to be constant over the 50 year simulation period. During the 1960s, the principal source of widespread radioactive contamination of the environment was global atmospheric fallout from nuclear weapons tests. A rain forest monitoring project based in Puerto Rico during this period revealed that both 90 Sr and 54 Mn in fallout from weapons tests had become significant contaminants of the rain forest. Jordan et al. (1973) used both radioactive and stable element data collected from this location to conduct a systems analysis of the biogeochemical cycling of both 90 Sr and 54 Mn using a somewhat simpler compartmental model (4 compartments, viz. soil, litter, wood and canopy) than that of Alexakhin et al. (1994). Unlike the EURT forests, the rain forests of Puerto Rico received prolonged deposition of 90 Sr and 54 Mn which had been monitored in detail from 1962 to 1967 and showed a strong annual fluctuation due to the seasonal rainfall patterns at the site. An analytical solution of the model was obtained using a computer program written to predict compartmental radionuclide concentrations for any combination of inputs to the ecosystem, including positive/negative, steady state and exponentially increasing/decreasing inputs. Predictions using both chronic and instantaneous inputs of 90 Sr and 54 Mn indicated that, as in the case of 90 Sr in the EURT forests, the soil was likely to be the major long-term repository for both radionuclides. However, an interesting discrepancy in the predictions of 90 Sr behaviour in the EURT and Puerto Rican forests was that, while litter was an important long-term repository for this radionuclide in the boreal forests of the EURT, it was the least important component of the forest ecosystem in the rain forest. Additionally, accumulation of 90 Sr in wood in the rain forest was more significant than in the boreal forest. These observations indicate the increased importance of litter decomposition and biomass accumulation in forests of tropical regions compared with the cooler temperate and boreal regions. Results from the Puerto Rico study also confirmed that the principal mechanism of loss of radionuclides from contaminated forests was likely to be physical decay and, to a lesser extent, runoff (Jordan & Kline, 1976). Prior to the Chernobyl accident, interest in radiocaesium behaviour and fate in forest ecosystems was limited since the presence of 137 Cs in forest ecosystems was not considered to present a significant radiological problem from the point of view of human exposure. Rather, radiocaesium was seen as a potentially valuable tracer in studies of nutrient and trace element uptake in forests (Auerbach et al., 1964). The first modelling study investigating 137 Cs cycling in a forest ecosystem was carried out within the Oak Ridge Reservation, USA, by Olson (1965) who used data obtained after artificial contamination of trees within a 20 × 25 m area with 17 M Bq of 137 Cs. This study aimed (i) to estimate rates of cumulative transfer and recycling of 137 Cs in components of yellow poplar (Liriodendron tulipifera) trees, (ii) to use linear compartment models to help interpret these rates and (iii) to predict the larger scale dispersal of radionuclides in natural, forested landscapes. It was determined that simple compartment models with constant rate coefficients representing inter-compartmental transfers simulated the observed data during the early period of the experiment. However, it was noted that pe-

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riodically varying rate coefficients would be needed if the models were to simulate seasonal fluctuations in the ecosystem. Olson (1965) concluded that “it is encouraging that a compartment model with biologically meaningful parameters can be constructed at all for a whole forest” and he noted the strong analogy between the fluxes of materials between compartments of an ecological system and “the commonplace models of physical-chemical systems” such as radioactive decay chains. Other models of radionuclide cycling in forest ecosystems which were developed before the Chernobyl accident included those of Croom & Ragsdale (1980) (see Section 5.2), which considered radiocaesium cycling in a deciduous forest in the USA, and Garten et al. (1978), who modelled plutonium dynamics in a deciduous forest, also in the USA. Plutonium is interesting for its general lack of biological mobility and the latter study showed that less than 0.25% of the total inventory of Pu in the forest ecosystem was present in the dominant white ash (Fraxinus americana) and sycamore (Platanus oxidentalis) trees, the great majority being resident in the soil. The methods used to model radionuclide behaviour in forests in all of the studies cited above are essentially the same, involving the compartmentalisation of the ecosystem into discrete and ecologically relevant components (soil, wood, leaves, etc.) and the representation of radionuclide fluxes between these compartments as linear transfer coefficients which are fixed with respect to radionuclide activity and time. 1.2. The significance of radiocaesium in forests after the Chernobyl accident Since shortly after the Chernobyl accident of 1986, it has been recognised that the widespread contamination of forests across Europe and the Former Soviet Union with 137 Cs has resulted in some degree of radiological exposure of both people and other living organisms. Alexakhin et al. (1994) have pointed out that, even in the most severe cases of radioactive contamination of the environment, the only lethal biological injuries observed as a direct result of radiation exposure have been observed in forests. In the case of the Chernobyl accident, the radiation dose (80–100 Gray) received by Scots Pine (Pinus sylvestris) trees within a distance from the reactor of 4.5 km to the west and 8 km to the north was sufficient to kill the trees, with a significant number of dead pine trees actually falling over within three years of exposure (Kozubov et al., 1991). Coniferous forests are thought to be amongst the most radiosensitive of ecosystems, at least in part because of their ability to capture airborne radioactive particles with greater efficiency than herbaceous vegetation. Once contaminated, forests tend to shed their burden of radioactivity slowly. The earliest studies cited in Section 1.1 demonstrated that the main decontamination process in forests is the physical decay of the radionuclide concerned and studies since 1986 have verified that net-export of 137 Cs from forests is of the order of a fraction of 1% per annum (Nylen, 1996). In the case of both 137 Cs and 90 Sr this means that long-term radiological exposure of humans and non-human biota within contaminated forests is probable. Deposition of 137 Cs from the Chernobyl accident affected all the countries of the continent of Europe to some extent. Within the former Soviet Union, Russia, Ukraine and Belarus each possess significant areas of forest which became severely contaminated with 137 Cs in 1986. Among the countries at a greater distance from the Chernobyl Nuclear Power Plant (ChNPP) Finland, Sweden, Norway and Austria received the heaviest contamination. Forests

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are a major natural feature in each of these countries and, as a result, interest in the potential radiological exposures from and long-term ecological consequences of this contamination has been substantial. Radiological exposure of people can occur both within contaminated forests (i.e. in situ) and as a result of the export of contaminated products (i.e. ex situ) either in large quantities, such as timber, or in small quantities, such as in the case of highly contaminated ‘wild’ food products. Some forested areas of central Sweden received 137 Cs deposition levels of between 40 and 100 kBq m−2 . The industries using contaminated timber from forests in these areas, notably the paper and biofuel industries, can, through processing of raw timber, concentrate 137 Cs to a degree which can lead to significant radiological doses to workers within those industries (Ravila & Holm, 1994, 1996). Direct consumption of various foodstuffs taken from the forest can also lead to significant doses. It has been estimated that the ingestion of roe deer meat alone from some Swedish forests can lead to maximum individual ingestion doses of up to 0.7 mSv per annum (Johanson & Bergstrom, 1994). Edible fungi collected in forests have also been shown to be a highly significant exposure pathway, accounting for a possible annual collective dose to the Swedish population of 166 manSv (Johanson & Nikolova, 1996). In the countries of the former Soviet Union the contribution of forest-derived food products to radiological dose from 137 Cs can be very significant. Strand et al. (1996) estimated that, in the Bryansk region of Russia, semi-natural and forest-derived foods represented 61% of the total population ingestion dose in 1995. This contrasts with an ingestion dose contribution of 8% from semi-natural and forest-derived food sources in 1987 and reflects the importance of contaminated forests and associated ecosystems as a potential long-term source of exposure to 137 Cs. With the recognition that, following the Chernobyl accident, forests could represent a significant source of human exposure to 137 Cs in several European and former Soviet countries, there has been a considerable interest in the development of models to help understand and predict the long term fate of 137 Cs in forest ecosystems. The next section describes some of the main forest modelling activities which have addressed this goal over the last decade. 1.3. A summary of major recent forest modelling activities In 1988 the International Atomic Energy Agency (IAEA) established a co-ordinated research programme concerned with the ‘VAlidation of environmental Model Predictions’ (VAMP). This programme was designed to use information on the environmental behaviour of radionuclides (mainly radiocaesium) which became available as the result of the vast array of environmental measurements made in countries of the former Soviet Union and the European Union after April 1986. The primary objective of the VAMP programme was to provide a forum for the validation of assessment models by using data on radionuclide transfer which resulted from the Chernobyl release (IAEA, 1993). One of the working groups within VAMP (the terrestrial working group) included a sub-group which began to consider the problem of foodchain transfer in natural and semi-natural ecosystems, including forests. By the end of the programme (1994), although the issue of the development of models of radionuclide transfer in forests had been discussed, there appeared to be no models suitable to be put forward for use in the VAMP programme.

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In 1996 Schell et al. published the FORESTPATH model, which was based on transfer rates obtained from a detailed review of the literature. The model was proposed as a generic representation of primarily 137 Cs behaviour in forests, although it was also intended to be easily adjustable to allow site-specific calculations to be performed. At the same time as this model was proposed, two major projects were initiated which had as one of their objectives the development of models of radiocaesium behaviour in forest ecosystems in Europe. The LANDSCAPE project (Brechnignac et al., 2000) and the SEMINAT project (Belli, 2000) both established field experiments and forest modelling development in parallel, the former with a focus on the specific problems of FennoScandia and the latter with a more generic focus on the forests of the European temperate belt. The models to emerge from these studies were the FORESTLAND and RIFE models, respectively, which are both described in Section 3 of this chapter. In 1996 the BIOMASS Programme on BIOsphere Modelling and ASSessment was launched by the IAEA. This was a follow-on from previous programmes, including VAMP, and was intended to provide an international focal point in the area of biospheric assessment modelling, to improve models and modelling methods by model testing, comparison and other approaches, and to develop international consensus, where appropriate, on biospheric modelling philosophies, approaches, and parameter values. This programme included a specific working group on forest ecosystems, which commenced its activities in 1998 as part of the Biosphere Processes theme. Between 1998 and 2000 the forest working group conducted three model validation studies, two of which involved inter-model comparisons and one of which involved a blind model-data validation (IAEA, 2002). Somewhat surprisingly, a total of 12 models were put forward for these exercises, which indicates the high level of interest and activity in the development of forest radioecology models in the four year period between the end of VAMP and the beginning of BIOMASS.

2. The conceptual framework for forest modelling 2.1. Storages and fluxes of radiocaesium in forests Ford (1999) has said that “A model is a substitute for a real system” and “Models are used when it is easier to work with a substitute than with the actual system”. Perhaps one of the reasons why models of radionuclide behaviour in forests have not been developed to the same extent as agricultural models is the apparent complexity of radionuclide fluxes within forest systems. However, since the beginning of studies on biogeochemical cycling in forest ecosystems it has been recognised that the systems approach can significantly assist our understanding of elemental fluxes in forests, even though it involves simplification of the actual system (Likens & Bormann, 1995). Analyses of the fluxes and storage of isotopes such as 137 Cs in and between discrete ecological compartments can provide useful information for subsequent numerical analysis and modelling of radionuclide fate and persistence (Heal & Horrill, 1983). A general framework commonly adopted in field studies of radiocaesium behaviour in forests is shown in Fig. 1. Deposition of radionuclides from the Chernobyl plume was predominantly dry in the immediate vicinity of the ChNPP itself, whereas further afield (particularly in western Europe and

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Fig. 1. A conceptual framework for the measurement and modelling of radiocaesium storages and fluxes in contaminated forest ecosystems.

Scandinavia) wet deposition resulted in highly localised ‘hot spots’ of contamination. However, as deposition from Chernobyl can be considered to approximate a single ‘pulse’ in the long term the most important consideration in flux modelling of forests is the initial total deposit per unit ground area and the relative interception of this by the tree canopy. Losses of contamination from the tree canopy can be rapid, effected by stemflow, throughfall and the loss of leaves or needles. These processes transfer contamination to the litter layer on the forest floor, after which underlying soil horizons become contaminated at a rate which is controlled by soil migration. Ultimately radionuclides may be lost to the drainage waters flowing from forests, but Tikhomirov et al. (1993) have observed that such losses are trivial in forests of the Chernobyl 30 km zone. More important is the return of radionuclides, especially 137 Cs, to the standing biomass via biological uptake processes which complete the cycle of radionuclide movement within the ecosystem. For relatively long-lived radionuclides such as 137 Cs the primary question of interest is whether the persistence of the contaminant within the forest ecosystem will be controlled by physical decay or by ecological self decontamination processes. If self decontamination of the system as a whole is slow then it is relevant to ask

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whether certain components of the ecosystem will accumulate a greater proportion of the total 137 Cs inventory than others. If so, this has important implications for assessment and aversion of effective doses. 2.2. Compartments of significant radiocaesium accumulation in forests It is well known that the activity concentration of 137 Cs in fungal fruiting bodies is generally high when compared with the activity concentrations of most green plants which make up the forest understorey vegetation (Mascanzoni, 1990). Leading on from this observation it is commonly assumed that ‘mushrooms’ are of considerable significance when modelling radiocaesium behaviour in ecosystems such as forests, in which they are often abundant. This is certainly the case when considering the potential radiological dose which might be received after consuming mushrooms. However, the fundamental equation of mass (or activity) balance states that the total activity, A, of a radionuclide within a discrete ecological compartment is the product of the activity concentration of the radionuclide in that compartment, C, and the mass of the compartment, M A = CM. In the case of fungal fruiting bodies, the activity concentration may be high, but the mass is usually rather small (of the order of a few gram m−2 at most). Conversely, the activity concentration of 137 Cs in the woody tissues of trees is usually low, but the mass per unit ground area is often very large (commonly of the order of 10 to 50 kg m−2 ). By treating individual components of the forest ecosystem such as soil, trees, understorey vegetation, fungi and even game animals as discrete compartments the overall activity of radiocaesium within the system can be calculated by simply summing the contributions of individual compartments. Atot = (C1 M1 ) + (C2 M2 ) + (C3 M3 ) · · · + (Cn Mn ). Such simple screening calculations allow the relative significance to be assessed of specific ecological compartments in the overall storage of radiocaesium in an ecosystem. Based on previously published transfer coefficients and estimates of compartment masses (IAEA, 1994; Belli, 2000) Fig. 2 indicates that the soil and tree compartments account for more than 95% of the total 137 Cs activity within a typical forest with compartments such as fungal fruiting bodies and game animals accounting for less than 0.1% of the inventory. The following section describes the development of three models of radiocaesium behaviour in forest ecosystems following the Chernobyl accident. While the aims and approaches of each model are somewhat different, the basic conceptual framework within which each model addresses the forest ecosystem is the same. It should be pointed out that models such as FORESTLAND place a strong emphasis on predicting temporal trends in such biological endpoints as game animals and edible plants. However, as demonstrated above, animals and plants account for a relatively small percentage of the total radioactivity within the system. Hence the emphasis in the remainder of this chapter is on the modelling of radiocaesium within the major structural components of the forest (i.e. tree and soil, as shown in Fig. 1).

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Fig. 2. Calculated storages (%) of 137 Cs in five notional ecological compartments within the forest ecosystem.

3. Case studies of forest model development after Chernobyl 3.1. The FORESTLAND model 3.1.1. Background Scandinavia, and especially Sweden, received some of the highest levels of 137 Cs deposition following the Chernobyl accident, as described in Section 1.2. 68% of the land surface of Sweden is covered by forests and woodland, which represent one of the country’s most valuable natural economic resources as well as providing seasonal foodstuffs such as game and various other ‘wild foods’. FORESTLAND is a dynamic ecosystem model which was developed collaboratively in Sweden and the Russian Federation to interpret and predict temporal and spatial patterns of the radioactive contamination of forest ecosystems in both countries. The model is focused on 137 Cs migration pathways leading to internal and external radiation doses to man in both the acute and long-term phases following atmospheric deposition of radioactive substances. FORESTLAND has a modular structure and comprises the following modules. • FORBIO: a model of the biomass dynamics of trees and understorey vegetation. • FORACUTE: a dynamic model of the migration of radionuclides in forest ecosystems during the short term following contamination.

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Fig. 3. Organisation of individual modules within FORESTLAND.

• FORGAME: a dynamic model of the long term migration of radionuclides in forest food chains (game animals, in particular). • FORTREE: a model of the long-term migration of radionuclides in forest trees. • FOREXT: a dosimetric model for calculation of gamma dose rates in the forest. A sixth module is presently under development: • FORDOSE: a model for the calculation of internal and external doses to the population arising from contaminated forests. Each of the modules is in effect a stand-alone model which, when operated sequentially, comprise the FORESTLAND model (Fig. 3). A specific feature of the model is that it has been calibrated to represent four basic categories of forest ecosystem. Each category corresponds to either coniferous or deciduous tree types and automorphic (dry) or semi-hydromorphic (boggy) landscape types. A set of model parameters, consisting of best estimate values and associated uncertainties, has been evaluated for each forest category based on measurements carried out on experimental plots of the Bryansk region of the Russian Federation. The first four modules of FORESTLAND are now explained in more detail. 3.1.2. FORBIO – a model of biomass dynamics in forest ecosystems In FORBIO, a simple approach which describes the seasonal and long-term biomass dynamics of trees and understorey vegetation has been applied. The input parameters of FORBIO are growth rate constants, biomass decay rate constants, the maximum biomass attained by each vegetation type, the tree mortality and the times marking the beginning and end of different stages of the seasonal dynamics (start of the vegetative period, time of maximal biomass, end of the vegetative period). Default, generic values of these parameters are presently incorporated in the model. These values are derived from long-term data of biomass dynamics in several types of forests within the Bryansk region of the Russian Federation. For the understorey vegetation and mushrooms, biomass growth is simulated using a logistic model, while an exponential decrease in biomass is assumed during senescence. Differentiation is made between mushrooms occurring in the summer and autumn and between fruits of berries and the whole plant (relevant to animal feeding habits). The growth of individual trees is also described using a logistic model, while an exponential equation is used for the calculation of tree mortality. A linear differential equation, obtained by combining the equations for growth and mortality, is used to simulate the long-term changes in tree biomass density (expressed as kg m−2 ).

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For tree leaves (or needles) a distinction is made between seasonal and long-term biomass dynamics. The yearly values of leaf/needle biomass depend on the age of the tree. It is assumed that the contribution of leaves/needles to the total tree biomass decreases with the age of the trees. The seasonal variation of the leaf (needle) biomass is described using a logistic model during the periods of growth and senescence. The endpoints of the FORBIO module are the biomass densities of different categories of vegetation as a function of time. 3.1.3. FORACUTE – a model of radionuclide migration in forests in the short term following contamination FORACUTE is a dynamic model of the migration of radionuclides in a forest over the initial period of a few years following contamination. The model describes the primary interception of the radionuclides by the above ground vegetation and their subsequent redistribution in the system. Transfer processes like weathering, translocation in the tree and the understorey vegetation, and root uptake from the upper soil litter layer are included in the model. The model also permits evaluation of the dynamics of radionuclide activities in forest products consumed by man, including forest game. To model the interception of radionuclides by the tree and understorey canopies, the tree leaves/needles, the tree bark, the understorey vegetation and the soil litter layer are considered as a set of four successive filters (Fig. 4). Interception by understorey vegetation is calculated using Chamberlain’s (1970) exponential function, in which a key variable is the biomass density of the understorey. For the initial retention of radionuclides by trees, a method similar to that commonly used to evaluate the passage of light through tree crowns is used. It is assumed that the initial retention of dry deposits by trees is proportional to the projected leaf area of

Fig. 4. Conceptual overview of FORACUTE, a model of radionuclide transfer in the canopy and upper soil during the short-term following contamination.

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the tree crowns. This cover can be calculated from the crown closure (relative area of crowns) and a coefficient which depends on the tree species being considered. The values of these parameters can be mapped for different areas, which facilitates use of the model in combination with GIS. For deciduous trees, the initial retention of radionuclides depends on the season in which deposition occurs, reaching a maximum value when the leaf biomass is maximal in the spring and summer. The present version of the FORACUTE model has been calibrated for dry deposition in both coniferous and deciduous forests of the Bryansk region of the Russian Federation. Calibration for wet deposition has not yet been carried out. 3.1.4. FORGAME – a model of the long-term migration of radionuclides in forest food chains FORGAME is a dynamic model designed to predict seasonal and long-term changes of 137 Cs activity concentrations in forest food chains and game in particular. FORGAME comprises a set of 20 coupled differential equations which describe the net accumulation of a radionuclide in the compartments over time. Since the model is focused on forest food chains, radionuclide migration in the tree is described in a simpler way than in FORTREE (Section 3.1.5). For example, the transfer rates corresponding to the processes of root uptake and translocation in trees are described using ordinary rate constants. The soil on the other hand is modelled in more detail (10 compartments) in order to describe the influence of the location of both plant roots and mycelia on radionuclide uptake by understorey vegetation and mushrooms, respectively. Root uptake by the understorey vegetation is described as a function of the root distribution in soil, the available fraction of 137 Cs in soil, the soil–plant transfer factor (TF) and the biomass growth rates. The soil–plant TFs are related only to the available fraction of the radionuclide in the soil and have, therefore, the same values for all soil layers, including the litter. They reflect the capacity of each specific type of vegetation or mushroom to assimilate 137 Cs from the soil. In contrast, TFs commonly reported in the literature are related to the total activity of the radionuclide in the soil or in a specific soil layer. The intake rate of 137 Cs by game (roe deer and moose) is described as a function of the total feed intake, the proportion of different feeds in the daily diet of the animal and the activity concentrations in different feeds. It is assumed that the radiocaesium incorporated by the animal via ingestion is instantaneously distributed throughout the animal’s body and that the elimination rate from muscles (edible meat) is proportional to the activity levels in this part of the animal’s body. 3.1.5. FORTREE – a model of radionuclide migration in trees FORTREE is a dynamic compartment model of the long-term kinetics of 137 Cs activity concentrations in both deciduous and coniferous trees. The endpoints of the model include the seasonal change of activity concentrations in leaves or needles and the longer term change in activity concentration in wood. The conceptual model of FORTREE for coniferous trees is presented in Fig. 5. Five compartments are used to describe the temporal and spatial variation of 137 Cs activities in the soil. The first soil layer (0–10 cm) contains the most physiologically active portion of the root system and is responsible for the major part of root uptake during the first years following radioactive contamination. Each soil layer is divided into an available and an unavailable fraction of 137 Cs in order to account for sorption/desorption processes leading to fixation and remobilisation. It is assumed that the fraction of 137 Cs available for root uptake is the same as the fraction that is available for vertical migration in the soil.

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Fig. 5. Conceptual model of FORTREE for a coniferous forest (only the modifications of the model in Fig. 6 are presented).

Fig. 6. Activity levels in wood for different types of forest soil and ages of trees calculated using FORTREE – deposition 1 kBq/m2 of 137 Cs (1, 2) – automorphic soil, (3, 4) – semi-hydromorphic soil, (1, 3) – initial age 20 years, (2, 4) – initial age 80 years.

The tree is described by two compartments which represent the living part of the wood (liquid and solid phases of the wood xylem), one compartment for the dead wood (structural wood) and one compartment for the leaves of deciduous trees. For coniferous trees, needles

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of different age classes are represented by four discrete compartments. Root uptake by trees is described by a function of water flux (transpiration flow) through the xylem during the vegetative growth period. This flux is primarily due to evaporation from the leaves/needles which is proportional to the leaf/needles biomass. Root uptake rates are therefore expressed as a function of the leaf/needle biomass, which depends on the overall tree biomass and age. Thus the model can be used to evaluate the influence of these factors on the accumulation of 137 Cs by trees. An example of temporal 137 Cs distributions in tree tissues calculated using FORTREE is shown in Fig. 6. The values of parameters corresponding to the processes of translocation in the tree were estimated from experimental data of stable potassium contents in different parts of the tree. The experimental data were collected during a program of long-term observations carried out in the Bryansk region, Russia. A comparison of the model predictions using the parameter values obtained this way with independent experimental observations of 137 Cs concentrations in different parts of the tree shows a good agreement. 3.2. The RIFE model 3.2.1. Basic model description The RIFE (Radionuclides in Forest Ecosystems) model was developed as part of the SEMINAT project (Belli, 2000). It was designed to allow the incorporation and use, in a structured and organised way, of experimental data derived from a series of experimental forest sites which were established in 1996–7 in Austria, Ireland, Italy, Switzerland and the United Kingdom. The data collected from these sites were intended to provide a basis for both understanding and modelling the ecological behaviour of radiocaesium in forests, with the goal of strengthening our ability to assess the long-term consequences of post-Chernobyl contamination of forests in western Europe. RIFE consists of five compartments, each of which represents a discrete radionuclide pool within a major compartment of the forest ecosystem (Fig. 7). These radionuclide pools are interconnected by fluxes which can be related to the physical fluxes of radionuclides within a forest, as shown in Fig. 1. Each flux is treated as a first order rate process which can be represented by a rate coefficient, as described in Table 1. While forest soils are often highly horizonated, RIFE simplifies the representation of the soil profile by aggregating organic and mineral layers into single compartments. Similarly, trees are aggregated into external (surfaces) and internal components, with the latter representing the entire radionuclide pool within the tree taken up by either root or foliar absorption. Instantaneous mass balance calculations have shown convincingly that the tree and soil components of the forest ecosystem account for the majority (>95%) of the total radionuclide inventory within a forest, as originally determined by the early studies of Jordan et al. (1973) and Alexakhin et al. (1994). Thus, the system of five compartments within RIFE is intended to account for the effective total activity balance within the forest ecosystem. As for any system of compartments between which proportional transfers of radionuclides occur, a general equation describing fluxes within the compartmental system can be written as follows.    dQx (t) = Ix + kyx Qy − Qx kxy + kx + λ , dt x=y

x=y

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Fig. 7. Structure of the RIFE model. Table 1 Identification and description of first order rate coefficients used in RIFE Symbol

Description

Dimension

k1 k2 k3 k4 k5 k6 k7 λ Tagg

Foliar absorption rate coefficient Foliar loss rate coefficient Tree root absorption rate coefficient Tree canopy weathering rate coefficient Rate coefficient of migration from soil litter layer Rate coefficient of migration from organic soil horizons Rate coefficient of migration from mineral soil horizons Rate coefficient of radioactive decay Soil–understorey or soil–mushroom transfer coefficient

y−1 y−1 y−1 y−1 y−1 y−1 y−1 y−1 m2 kg−1

where Qx (t) is the total activity of radiocaesium in compartment x at time t (Bq), Ix is the rate of radiocaesium input into compartment x (Bq t −1 ), kyx is the transfer coefficient from compartment y to x (t −1 ), kx is the loss rate coefficient from compartment x (t −1 ) and λ is

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the physical decay rate coefficient of the radionuclide under consideration (t −1 ). In the RIFE model, a system of five differential equations based on the above general activity balance equation, is solved simultaneously using a numerical integration method (2nd order Runge– Kutta). This allows the time-dependent behaviour of radionuclides within the five compartments to be calculated for either simple initial conditions, such as pulse inputs into a single compartment, or for more complex initial conditions, such as time-varying inputs to one or more compartments. Thus, the RIFE model is intended to allow predictions of radionuclide behaviour in forests following contamination from a Chernobyl-type accident, or during a complex history of deposition such as that which may be experienced as a result of continuous releases from nuclear facilities or due to atmospheric weapons tests. The activity balance equations of the RIFE model provide calculations of the total 137 Cs activities in five major ecosystem compartments, expressed in Bq m−2 . However, it is often necessary to report values as activity concentrations in specific compartments, such as wood or soil horizons. A simple conversion from a compartmental inventory to an activity concentrations is possible if the mass of the compartment is known. For the soil compartments, this requires a knowledge of the thickness of the individual soil layer of interest (i.e. litter, organic soil and mineral soil) as well as the bulk density of each layer. For the tree compartment the mass of tree components (wood, bark, branches, needles) at the time of initial contamination is required, but there is also a need to account for subsequent increases in the biomass of each compartment in the years following contamination. This growth increase will clearly be large for trees which are relatively young at the time of contamination, and relatively small for older trees. In the RIFE model, the growth of the tree is calculated using the logistic equation as a simple empirical growth function which can be applied to the whole tree or to any sub-component of the tree. W=

a . 1 + b e−ct

Here, W is the biomass (kg m−2 ), a is the expected maximum yield at maturity (kg m−2 ), c is a growth rate coefficient (y−1 ), t is the tree age (y) and b is a dimensionless coefficient. 3.2.2. Calibration of RIFE In total, data from 7 individual forest sites from the SEMINAT network were used to calibrate RIFE. Data from these sites provided ‘snapshots’ of 137 Cs distributions in forests for single years between 1996 and 1998, depending on the site concerned. Calibrations were made by, first, estimating soil leaching rates (k5 , k6 and k7 ) using a technique previously described by Shaw & Wang (1996). This involves numerical optimisation of a fit of the Batemann equations of sequential exponential decay to an observed profile of activity inventories within defined soil horizons at a specified time after deposition. Estimates of k2 were provided directly by flux measurements at most sites. A best estimate and range of possible values of k4 was obtained from the literature (Bunzl et al., 1989). Finally, k3 , which is almost impossible to measure experimentally in mature trees, was estimated by adjusting this parameter until a best fit was achieved. Estimates of k1 were impossible to obtain from the SEMINAT field measurements or from the literature: the rate was retained within the RIFE model for completeness though, by default, it is set to zero. The potential importance of k1 was explored in a

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Table 2 A summary of the geometric means and ranges of the RIFE rate constants Site

k2 (y−1 )

k3 (y−1 )

k4 (y−1 )

k5 (y−1 m−1 )

k6 (y−1 m−1 )

k7 (y−1 m−1 )

Max Min GM

5.06 2.27 3.10

69.3 5.78 17.8

0.64 0.25 0.45

123.6 18.5 57.6

220.0 84.5 158.7

7702 555.4 3367

Fig. 8. Representation of the dynamics of 137 Cs distribution in a Scots Pine forest (Kruki) in Belarus using the RIFE model after site-specific calibration.

sensitivity analysis and it was found to be significant only when k3 (i.e. root uptake) was very low. A summary of the geometric means and ranges of the RIFE rate constants, based on the SEMINAT sites, is shown in Table 2. One possible concern in calibrating a dynamic model to data obtained for single instants in time is that the rates thus obtained might not be representative of all time periods. A further, more general, concern of using first order rate kinetics to describe the behaviour of an ecosystem such as a forest is that such systems are often intrinsically non-linear in their behaviour.

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A simplified model such as RIFE is intended to provide no more than a good approximation of the system behaviour, though, for the model to be suitable for its purpose, it must be demonstrated that it can reproduce the overall kinetics of radionuclide behaviour in a forest over a significant time period. In order to test this, data were obtained from a heavily contaminated site (Kruki) in one of the exclusion zones in Belarus. These data covered a seven year period between 1992 and 1998, inclusive, and were obtained for a forest plantation of Scots Pine (Pinus sylvestris). The results of the model calibration, shown in Fig. 8, give confidence that the system of linear differential equations, assuming constant rates of transfer between compartments, provides a firm basis on which to model radionuclide dynamics in forests. This calibration also provided another group of estimates of rate coefficients on which to base an uncertainty analysis of RIFE. 3.2.3. Probabilistic uncertainty analysis of soil–tree Tagg values using RIFE Deterministic solutions of the rate equations which describe inter-compartmental fluxes in the RIFE model take no account of uncertainties in the values of rate coefficients or other input parameters. The estimates of rate coefficient values shown in Table 2 can be used to define triangular probability distributions of values for each rate coefficient (with the GM as the most likely value) for use in probabilistic uncertainty analysis. A steady-state analytical solution to the rate equation describing the inventory of activity within each compartment can be obtained. The equation describing the steady state internal tree activity (Q2 ) is as follows. Q2 =

k3 (I1 + I2 ) k1 I1 + , k2 (k1 + k4 ) k2 k6

when

dQ = 0, assuming no radioactive decay (i.e. λ = 0). dt Since steady state in the forest system is never actually likely to be achieved under natural conditions, the steady state inventories and activity concentrations of individual forest compartments are somewhat academic and are not presented here. However, it is possible to predict ratios of these values, in other words transfer coefficients, which are of more general applicability and which can provide an assessment of the accuracy and precision of RIFE predictions using the available calibration data. One of the most fundamental transfer processes in forest ecosystems is soil–tree transfer, often quantified by an aggregated transfer factor or Tagg value. Tagg =

activity concentration in tree (Bq kg−1 ) total activity deposited (Bq m−2 )

(m2 kg−1 ).

The range of tree Tagg values computed probabilistically by RIFE can be compared to those measured at SEMINAT forest sites as a measure of the accuracy and precision of RIFE predictions. Figure 9 shows the frequency distribution of soil–tree Tagg values predicted probabilistically using steady state solutions to the RIFE equations. The distribution is approximately log-normal, as expected from analysis of published soil–tree Tagg values, with a GM of 6.03 × 10−3 m2 kg−1 . The accuracy of this prediction is acceptable in comparison with values reported from independent measurements in Sweden of 1.00 × 10−3 m2 kg−1 (Avila,

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Fig. 9. Frequency distributions of Tagg values predicted for 137 Cs in Pinus sylvestris wood using steady-state solutions for the RIFE model. Table 3 Tree Tagg values measured at SEMINAT sites (all tissues considered) compared with Tagg values predicted using RIFE in probabilistic mode at steady state (GM = geometric mean)

Maximum GM Minimum

Predicted

Measured

5.61 × 10−2 6.03 × 10−3 6.31 × 10−4

1.09 × 10−2 3.18 × 10−3 8.96 × 10−4

1998) and 2.00 × 10−3 to 3.00 × 10−3 m2 kg−1 (Ravila, 1998). For an assessment of the precision of these Tagg predictions, Table 3 provides a comparison of measured and predicted Tagg value minima, maxima and GMs. The close level of agreement between measured and predicted value ranges and GMs gives confidence that the RIFE calibrations, based on data from SEMINAT sites, are realistic in terms of both accuracy and precision. 3.3. The FORESTLIFE model 3.3.1. Background Some 35% of the Republic of Belarus is covered by forests and, as a result, forest products play an important role in the economy of this country. Following the Chernobyl accident,

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approximately one quarter of forest stands within Belarus were contaminated with 137 Cs up to a level exceeding 1 Ci km−2 (37 kBq m−2 ). The contaminated forests are mainly located in two zones, namely the exclusion zone in the south of the country, up to 30 km from the ChNPP, and the so-called evacuation zone, up to 180 km from the ChNPP. For the period 1986 to 2015, the annual losses of economically useful timber as a result of restrictions on the use of these forest resources are estimated to be 280 million $ y−1 (Ipatyev et al., 1999). Reliable predictions of the degree of radioactive contamination of standing timber over the decades following the Chernobyl accident are required for the forecasting of both radiological dose and economic loss. With this in mind, systematic monitoring of radionuclide behaviour in a network of pine and deciduous forest plots was established in 1992 in the Belorussian Polesye territory, which received 137 Cs contamination from 10 to 500 Ci km−2 (370 kBq m−2 –18.5 MBq m−2 ). An emphasis was placed on the investigation of Scots Pine stands since this is the commonest forest type. 137 Cs migration was studied in detail in 10 plots of moss-pine forest (Pinetum-Pleuroziosum) each with trees of uniform age varying from about 30 years old on the youngest plots to 80 years on the oldest plots. Each 50 m × 50 m experimental plot was selected to be representative of the surrounding area and was at least 100 m from any road or forest clearing. Annual samples of litter and soil were taken at regular depth intervals down to a maximum depth of 20 cm. In addition, three ‘model’ trees from each plot were selected and samples of wood, bark, needles and shoots were collected for analysis of 137 Cs activities. 3.3.2. Development of the FORESTLIFE model The main feature of FORESTLIFE which distinguishes it from traditional compartment models is that it treats the tree as a living organism which experiences significant physiological change during its growth and development. Among these are changes in nutrient demand with ageing of the tree, an increase in the potential root uptake of radionuclides as the root system extends throughout the soil profile, an overall increase in biomass with age and a radial distribution of radionuclides throughout the stem as this undergoes secondary growth. Another specific feature of FORESTLIFE is that it is strongly based on empirical observations of 137 Cs distributions within specific forest ecosystems and it requires survey data obtained from at least one year’s observations in order to predict radionuclide migration at any site. The model functions are normalised at the moment of observation. The model is illustrated schematically in Fig. 10. The activity concentration of 137 Cs within a specified compartment of a forest ecosystem at any point in time can be estimated as follows. Ai (t) = Z × Ki × Tagg (t), where: Ai (t) is 137 Cs activity concentration in the ith component of the forest ecosystem (Bq kg−1 ). t is the current time (years), Z is the inventory of 137 Cs in the ecosystem immediately after deposition (Bq m−2 ),

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Fig. 10. Conceptual structure of the FORESTLIFE model.

Ki is the ratio of the activity concentration in the ith component (e.g. needles) to that of wood (Ki is estimated on the basis of a large body of experimental data and typically varies within 10%), Tagg (t) is the aggregated transfer factor for 137 Cs uptake into wood (m2 kg−1 ). Tagg (t) is, itself, a function of both bulk uptake of 137 Cs by the growing wood and radial distribution within the trunk after uptake. The following equation describes the dependency of the bulk uptake of 137 Cs by wood on three key variables. FK(t) = FR(x) × FA(x) × FM(t, x), where: FK(t) is a normalised function of 137 Cs incorporation into wood, x is soil depth (m),

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FR(x) is the normalised vertical distribution of the feeding root system in the soil profile, FA(x) is the normalised vertical distribution of biologically available 137 Cs in the soil profile, FM(t, x) is the vertical distribution of 137 Cs in the soil profile calculated on the basis of a two-component quasi-diffusional model, as follows, dC d2 C =D 2, dt dx where C is the activity concentration and D is the diffusion coefficient for 137 Cs. In order to describe the observed distributions of 137 Cs within forest soils in Belarus ‘fast’ and ‘slow’ diffusion coefficients must be applied which approximate to advection and diffusion components of migration, respectively. The following equation describes the dependency of the accumulation of 137 Cs by wood with regard to its radial distribution, biomass increase and changes of physiological activity of the tree while ageing. FAC(t) = F (FAD(t), FRD(t), FIB(t)), where: FAD(t) is a function of 137 Cs transfer from soil to wood according to the age of the tree, FRD(t) is a function of the radial distribution of 137 Cs in the tree stem, and FIB(t) is a function of the tree biomass increase while ageing (i.e. growth). Finally, the aggregated transfer factor for 137 Cs uptake by trees can be written, incorporating a function, FD(t), to account for physical decay of the radionuclide. Tagg (t) = FK(t) × FAC(t) × FD(t). The information required to perform a simulation using the FORESTLIFE model is as follows: • the total 137 Cs deposition to the soil, • information on the forest type and the associated soil type (to select appropriate site-specific parameter values), and • the age of the stand of trees being considered. Calculations performed using the FORESTLIFE model show that 137 Cs accumulation in pine stands in Belarus is expected to have peaked in the year 2000 (±2 years in relation to the tree age). The values of the maximum 137 Cs activity concentrations in pine trees vary according to their age, and can be expressed as ratios of 1 : 3.3 : 6.5 : 8.1 for trees of 60–80 years, 40– 60 years, 20–40 years and 0–20 years, respectively (Fig. 11). Hence, it should be expected that the highest contamination levels will be found in young pine trees. The half-time for the decrease of 137 Cs activity concentration in wood following peak contamination is calculated by the FORESTLIFE model to be 27 ± 1 year.

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Fig. 11. FORESTLIFE predictions of soil-wood aggregated transfer factors for trees of different age classes compared with independent data from Belarus and the Russian Federation.

4. Validation of the FORESTLAND, RIFE and FORESTLIFE models 4.1. Background As described in Section 1.3, the BIOMASS programme of the IAEA provided a forum for forest model testing and validation. During 1999 the FORESTLAND, RIFE and FORESTLIFE models were put forward, along with 9 other models, for a blind validation exercise using data obtained from a forest plot at Rudnya Povcha in the Zhitomir region of Ukraine, some 130 km to the south west of the ChNPP. This forest had become contaminated with 137 Cs following a single dry deposition event, leaving an initial deposition density of 555 kBq m−2 on 1 May 1986. Information provided to the modelers can be summarised as follows. The dominant tree species at the site is Scots Pine (Pinus sylvestris) with sparse birch (Betula pubescens). The average age of the trees at the time of contamination was 50 years for pine and 25–30 years for birch. The average density of wood biomass at the time of contamination was 29.7 kg m−2 (fresh weight). Full and detailed information was also provided on the growth characteristics of the Scots Pine, and on the species composition and characteristics of the understorey vegetation. The soil at the site was described as a soddy podzolic sandy loam with the characteristics shown in Table 4. Given this fundamental information the modellers were asked to provide predictions of the distribution of 137 Cs within the following endpoints, reported at yearly intervals, over a period of 1986 to 1998.

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Table 4 Litter and soil characteristics at the Rudnya Povcha forest site Soil horizon

Depth (cm)

Bulk density (g cm−3 )

pHH2 O

Cation exchange capacity (meq 100 g−1 )

Organic matter (%)

O1∗ Of∗ Oh∗ Ah E Bh (i) Bi

0–2 2–6 6–8 8–18 18–26 26–36 36–130

0.026 0.050 0.093 1.10 1.30 1.61 1.53

5.7 5.5 5.4 5.2 5.4 5.6 5.7

63.5 48.8 42.3 3.7 1.3 1.7 0.9

83.2 75.4 59.1 4.2 0.9 1.8 0.1

∗ The O1, Of and Oh layers comprise the surface organic layer of the forest soil.

• Wood (Bq kg−1 dry weight) • Annual shoots (Bq kg−1 dry weight) • Needles (Bq kg−1 dry weight, annual average) • Total bark (Bq kg−1 dry weight) ∗ • Soil profile (Bq kg−1 dry weight) • Roe deer (Bq kg−1 fresh weight, muscles, September–October) • Mushrooms (Bq kg−1 dry weight) • Bilberries (Bq kg−1 dry weight). Migration of radiocaesium down the soil profile is represented somewhat differently by different models. In the case of FORESTLAND and RIFE some soil horizons are aggregated to reduce the complexity of the soil profile. On the other hand, the physically continuous representation of soil migration by the FORESTLIFE model allows predictions of activity concentrations at any number of arbitrary soil depths. Hence, the predicted endpoints with regard to soil contamination differ from model to model. This was allowed for within the test scenario, but unfortunately it means that direct comparisons of model performance with respect to soil predictions are difficult. After the model predictions were submitted, they were plotted against time series data obtained from the site on an annual basis from 1991 to 1998. It was only after this procedure had been completed that the modellers were shown the original data from Rudnya Povcha. The results produced by the FORESTLAND, RIFE and FORESTLIFE models are now shown for the tree-associated endpoints and for the soil. While each of these models did provide predictions for the other biological endpoints, these are not presented here. 4.2. Predictions of tree-associated endpoints Figures 12, 13 and 14 show blind predictions of the time trends of 137 Cs contamination in wood, needles and bark of Scots Pine, respectively, at the Rudnya Povcha site. In the case of each of these endpoints the model predictions are quite different. Wood is the most economically important material to originate in the forest and is probably the most important material in terms of ex situ doses in situations such as the pulp and paper industries. The FORESTLIFE model was constructed primarily to forecast wood contamination in forests in

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Fig. 12. A comparison of blind model predictions by the FORESTLAND, RIFE and FORESTLIFE models of 137 Cs activity concentrations in Scots Pine wood at the Rudnya Pochva site, Ukraine.

Belarus, and this model provided the most accurate prediction of the magnitude and dynamics of 137 Cs in wood at the Rudnya Povcha site. The FORESTLAND model simulated appropriate dynamics of 137 Cs in wood, though overestimated the magnitude of contamination. The RIFE model underestimated 137 Cs contamination in wood and predicted a decline in contamination after 1992, after which the 137 Cs activities were still increasing in wood at Rudnya Povcha. What is interesting about these three predictions is that, taken together, they provide a best estimate and upper and lower bound predictions which provide both an accurate prediction of wood contamination at Rudnya Povcha as well as providing a reasonable (perhaps pessimistic) estimate of overall uncertainty during the period of interest. The data obtained on needle contamination with 137 Cs at the test site are more variable than would have been desired to see clear temporal trends during the test period. Again, the FORESTLIFE model provided the most accurate prediction with the predicted curve passing through the mid-point of the monitored data. The FORESTLAND model calculated the correct magnitude of needle contamination, but predicted a general decline in needle contamination from 1990 to 1998 while the data, though highly variable, suggested a general increase in needle contamination during this period. The RIFE model again underestimated needle contamination. The predictions by all three models of bark contamination were more highly clustered and generally closer to the monitored data than for the previous two endpoints. The RIFE and FORESTLIFE models both predicted the data with a good degree of accuracy. They both captured the general decline in bark contamination observed during the 1991 to 1998 period, although the RIFE model also predicted an increase in bark contamination at the beginning

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Fig. 13. A comparison of blind model predictions by the FORESTLAND, RIFE and FORESTLIFE models of 137 Cs activity concentrations in Scots Pine needles at the Rudnya Pochva site, Ukraine.

Fig. 14. A comparison of blind model predictions by the FORESTLAND, RIFE and FORESTLIFE models of 137 Cs activity concentrations in Scots Pine bark at the Rudnya Pochva site, Ukraine.

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of this period. The FORESTLAND model predicted the dynamics of the observed bark contamination well, but underestimated the magnitude of this contamination by a relatively small margin. As a summary of the predictions by all three models of tree-related endpoints it can be said that the FORESTLIFE and FORESTLAND models performed particularly well. As described in Section 3, these were calibrated with data from sites in Belarus and the Bryansk region of Russia, respectively, at which the forest characteristics were similar to those of the Rudnya Povcha site. For these types of site which, in the context of the European continent, were the most severely affected by 137 Cs contamination following the Chernobyl accident, these two models are clearly well developed and both make use of appropriate, though distinct, methods of modelling contamination of the tree and its various components. The relatively poor performance of the RIFE model in making predictions of 137 Cs contamination of trees at the Rudnya Povcha site illustrates the importance of appropriate model calibration and perhaps indicates the limitations of models using parameters which are based on generic site information. 4.3. Predictions of soil-associated endpoints Figures 15 to 18 show blind predictions of the time trends of 137 Cs contamination in various soil horizons at Rudnya Povcha. As described in Section 4.1, since the representation of the soil profile is slightly different in each of the models direct comparison of the predictions of each model is slightly more difficult for the tree-related compartments. However, the figures are drawn so that the performance of individual models can be judged and, where possible, compared to that of the other models. Figure 15 shows results for the soil litter layer and the Of horizon. Both RIFE and FORESTLIFE treat the litter layer as a separate compartment so the predictions of these models should be compared with the data for the Ol layer in isolation. Both the RIFE and FORESTLIFE models provided very similar over-predictions of contamination of litter at Rudnya Povcha by approximately one order of magnitude. The FORESTLAND model aggregates the litter layer with the Of horizon and predicted both the magnitude and dynamics of contamination of this aggregated soil layer with accuracy. Figure 16 shows results for the Of and Oh horizons, which RIFE and FORESTLIFE both treat as an aggregated layer. The RIFE model predicted both the magnitude and dynamics of 137 Cs contamination in this aggregated layer with good accuracy, while the FORESTLIFE model slightly over-predicted contamination of the Of + Oh layers from1991 to 1996. The FORESTLAND model treats the Oh horizon as a separate soil layer, and generally tended to over-predict the monitored data from Rudnya Povcha until 1997 to 1998, when the measured 137 Cs activities increased to a degree which appears out of step with the previous years. The greatest discrepancies in the way the soil is represented by the three models is in the soil mineral horizons which underlie the highly organic layers at the soil surface. FORESTLAND and RIFE both aggregate the uppermost mineral horizon, the Ah horizon as shown in Table 4, into a single compartment. This simplification is justified by the observation after 1986 that the majority of 137 Cs activity within podzolic forest soils is located in the Of and Oh horizons, in which it is hypothesised that biological recycling plays an important role in the recycling and retention of radiocaesium (Olsen et al., 1990). Thus, the underlying mineral soil makes a

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Fig. 15. A comparison of blind model predictions by the FORESTLAND, RIFE and FORESTLIFE models of 137 Cs activity concentrations in the soil litter layer at the Rudnya Pochva site, Ukraine.

Fig. 16. A comparison of blind model predictions by the FORESTLAND, RIFE and FORESTLIFE models of 137 Cs activity concentrations in the soil organic horizons at the Rudnya Pochva site, Ukraine.

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Fig. 17. A comparison of blind model predictions by the FORESTLAND and RIFE models of 137 Cs activity concentrations in the soil Ah horizon at the Rudnya Pochva site, Ukraine.

Fig. 18. A comparison of blind model predictions by the FORESTLIFE model of 137 Cs activity concentrations in the soil Ah horizon at the Rudnya Pochva site, Ukraine.

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relatively small contribution to the overall activity balance of 137 Cs in the soil when compared with the organic layers at the soil surface. At sites with very sandy sub-soils, such as those in the Belorussian Polessie, however, deep penetration of 137 Cs down the soil profile has been observed. Hence, the FORESTLIFE model places an important emphasis on the migration of radiocaesium within the mineral part of the forest soil. Due to these differences between the models the predictions of the FORESTLAND and RIFE models and the FORESTLIFE model are shown separately in Figs. 17 and 18, respectively. Both the FORESTLAND and RIFE models predicted the dynamics and magnitude of 137 Cs within the aggregated Ah horizon with good accuracy (Fig. 17). Two sets of monitored data are shown, each representing the slightly different depth of averaging which each of these models assumed for the Ah horizon. The performance of the FORESTLIFE model is shown in Fig. 18 for three discrete soil depths within the Ah horizon, although in principle predictions for any soil depths can be made using FORESTLIFE. For the 8 to 10 and 12 to 14 cm depths, FORESTLIFE predicted both the magnitude and dynamics of 137 Cs migration at Rudnya Povcha with good accuracy, but under-predicted significantly the 137 Cs activities in the deeper 16 to 18 cm layer. 4.4. Conclusions from the Rudnya Povcha validation study Each of the models described was designed and constructed for distinct purposes, using different approaches in their conceptualisation and parameterisation. However, their overall performance is broadly comparable, with strengths and weaknesses apparent in each model. One of the aims of the IAEA’s BIOMASS programme was to explore the possibility that consensus could be reached in modelling the behaviour of radionuclides in specific ecosystems such as forests, with respect to both modelling approaches and parameter values. The relatively close clustering of the predictions of each of the models examined here indicates that, to a broad extent, this has been achieved for model endpoints related to the major structural components of the forest (i.e. trees and soils). However, perhaps greater challenges are provided by individual biological endpoints such as edible species of fungi and specific game animals. Although the results are not shown here, the models used in the BIOMASS Forest Working Group were generally less successful at providing accurate predictions for these endpoints. 5. Future forest modelling developments and requirements 5.1. Empirical versus mechanistic approaches to forest modelling It could be argued that the approaches currently taken to model radiocaesium behaviour in forests have not changed significantly since the first models were developed in the 1960s. Each of the models described in Section 3 take essentially the same approach as, for instance, the model developed by Olson (1965) and involve the treatment of the forest as a series of discrete compartments between which fluxes can be determined by empirical measurement. One essential feature of the models which have been developed since the Chernobyl accident, however, is that they are based on an increasingly substantial data base, with a large number of measurements having been taken since 1986 in forests exposed to ‘real’ nuclear accident conditions, both in the near-field and in the far-field.

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It is perhaps the representation of inter-compartmental fluxes which provides the major challenge to the forest modeller, since consensus on the division of the forest ecosystem into notional compartments is relatively straightforward to achieve. All the models of radionuclide cycling in forests so far proposed have represented these fluxes in an empirical or semiempirical way. One of the first requirements of any ecosystem model is that it should account for the mass balance or, in the case of radionuclides, the activity balance of radionuclides within the ecosystem. For this reason traditional compartmental models with first order transfer rate coefficients, such as FORESTLAND and RIFE, should be fundamentally robust so long as it is possible to determine reliable values for the rate coefficients. One major difficulty in this parameterisation process is the time scale over which data on radionuclide cycling in forests are available compared with the potentially extended timescales over which forest models are required to make predictions. At the time of writing it is 15 years since a large number of forests across Europe became contaminated with 137 Cs following the Chernobyl release. In plantation forestry a typical crop rotation of Pinus sylvestris might take anywhere between 65 to 200 years, depending on the location and potential value of the crop (Savill et al., 1997). Variability between crop rotation times at similar sites might be as much as 10 to 15 years, the same time period for which data are available on the Chernobyl 137 Cs pulse. So it could be argued that we must continue to make measurements of 137 Cs distribution in forests for at least another 50 years to obtain maximum value from the existing data. One way around this problem of long-term monitoring of forests is to adopt the chronosequence approach, as used in the development of the FORESTLIFE model. As described in Section 3.3, measurements of 137 Cs distributions within Pinus sylvestris stands of varying ages were taken to obtain age-related tree contamination parameters (soil–tree Tagg values). However, while such chronosequences enable tree-related parameters to be obtained it does not assist with the determination of appropriate long-term soil migration parameter values. Since 90 to 95% of radiocaesium is commonly found in the soil compartment this is a serious deficiency. The alternative to empirical parameterisation of forest models is to use more mechanistically based approaches in model development. The major driving force for the migration of solutes in ecosystems is water flow which is strongly controlled by vegetation. Evapotranspiration from forest canopies can drive large upwards vertical fluxes of water in the soil, while an excess of precipitation over evapotranspiration leads to net leaching of solutes down the soil column. Such water movements can be modelled physically so long as reliable site-specific information can be obtained on the non-linear relationships between soil moisture content, water potential and hydraulic conductivity (Bornhoft, 1994). However, such relationships are notoriously difficult to obtain for highly organic soil horizons such as the Of and Oh layers in which the bulk of the 137 Cs inventory is found (Shaw & Wang, 1996). Even if soil water fluxes can be modelled reliably, radionuclide sorption processes, too, need to be accounted for. In the highly organic soil layers of the forest floor the activity of fungal biomass is typically very high and it has been estimated that in excess of 50% of the 137 Cs within these layers may be present in fungal mycelia (Olsen et al., 1990). It is these difficulties in applying physical models which have led to the continuation of the use of more empirically based compartment models for forest radioecology. Another alternative strategy for modelling radionuclide behaviour in forest ecosystems is to use more ecologically and physiologically orientated approaches. Van Voris et al. (1990) published the RADFORET model of the long-term cycling of 137 Cs in forests at Oak Ridge

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and at Savannah River, both in the USA. The principal mechanism underlying the model was the growth of the forest on an individual tree basis, which was modelled using a so-called ‘Gap Model’, as fully described by Shugart (1984). Biomass accumulation and turnover is one of the key forcing functions of material fluxes in forests and a model which accounts for forest growth on the basis of carbon and nitrogen dynamics has been published by Thornley (1991). Similarly, fundamental equations of carbon and nitrogen dynamics which can be applied at both the organism and ecosystem levels have recently been proposed by Ågren & Bosatta (1996). Undoubtedly, models of radionuclide cycling in forest ecosystems could be made more appropriate for long-term and generic simulations if they were based on more fundamental physical, physiological and ecological principles. However, it is clear that a certain level of site-specific parameterisation will always be required and the essential value of data on 137 Cs distributions in real contaminated forests cannot be overstated. 5.2. Uncertainty in forest radioecology models Inherent in any model are both Type A and Type B uncertainties, both of which can have a major effect on model uncertainty (Hoffman & Hofer, 1988). Forest radioecology models are no exception to this and may even be subject to greater Type A and Type B uncertainties than models of radionuclide fate in agricultural ecosystems in which generally more research effort has been invested. Type B uncertainties originate from a lack of knowledge on the part of the modeller of the system being considered. It was stated in Section 5.1 that agreement between modellers on the division of the forest ecosystem into notional compartments should be relatively straightforward to achieve. Much emphasis has been placed recently on the use of a matrix-based method to assist in the structuring and design of models of radionuclide behaviour in forests (Avila & Moberg, 1999). In this method, ecological compartments are identified as diagonal components in the matrix and interactions between these compartments are identified as off-diagonal elements in the positive and negative quadrants of the matrix. This procedure systematises the identification of compartments within the forest, and reciprocal transfers between compartments, in a way which minimises the chance of omitting significant state variables and transfer processes from the model. It also facilitates ‘design by committee’ and the application of expert judgement in evaluating the relative importance of individual compartments and processes. In theory the method could be used to develop an idealised model based on the best current understanding of the forest ecosystem by a group of experts. Deviations from this idealised model are likely to lead to enlargement of Type B uncertainties in individual models, i.e. those uncertainties due to incorrect representation of compartments or processes within a model. One major failing of the matrix method, however, is that it is not able to inform the modeller about the nature of the processes of inter-compartmental transfer of radionuclides. Most forest radioecology models to date have assumed first order transfers between compartments and, if this basic philosophy is accepted, the application of the matrix approach to model design is straightforward. More challenging, however, is to consider transfer processes from a more physical or physiological perspective as discussed in the previous section. In practice, existing models of radionuclide behaviour in forests have been developed on an ad hoc basis and undoubtedly fall short of any ideal representation of storages and fluxes

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of radionuclides within the forest ecosystem. In an analysis of the results of the IAEA’s BIOMASS forest working group Linkov & Burmistrov (2001) determined that model uncertainty contributed about two orders of magnitude while parameter uncertainty contributed about one order of magnitude. The twelve models on which this analysis was based varied widely in complexity. Model complexity is a major issue which every forest modeller must face: in other words, how many ecological compartments should be represented in a forest model? Croom & Ragsdale (1980) addressed this question by developing three models of radiocaesium cycling in a Turkey Oak (Quercus laevis) sand hills ecosystem. The models consisted of three, four or five compartments and it was found that the five compartment model, which included a simple representation of binding and unbinding of bioavailable radiocaesium in the lower soil, reproduced most accurately the measured distributions of 137 Cs originating from global weapons fallout. A similar conclusion was drawn by Schell et al. (1996) who proposed that “labile and fixed compartments must be distinguished for the appropriate description of radionuclide migration”. In reducing Type B uncertainties in models there is likely to be a trade-off between model realism and model complexity. This dictates that there should be some optimal level of model complexity which makes best use of available information on significant storage compartments and transfer processes without introducing unnecessary details which cannot be supported by the available evidence. In every area of environmental modelling the principle of parsimony should be one of the modeller’s tenets (Mackay, 1991). Type A (stochastic) uncertainties are perhaps a more familiar problem which both modellers and experimentalists face together. The major problem confronting the forest modeller is the conversion of often highly variable environmental measurements into useful model parameter values. Monte Carlo methods for the propagation of stochastic uncertainties in parameter values are becoming easier to apply as computer software improves, although most existing models of radionuclides in forests are still deterministic. A simple application of the Monte Carlo method to determine steady-state distributions of 137 Cs in components of a forest ecosystem can be illustrated using the case of the activity balance calculation described in Section 2.2. Figure 2 shows the results of a deterministic assessment of the relative storages of 137 Cs in discrete compartments in a forest based on single estimates of C and M for each compartment. In each of these compartments, however, both C and M are subject to considerable stochastic uncertainty. Figure 19 shows the results of a probabilistic calculation in which the distributions of values of C and M for each compartment have been assumed to be triangular, with the most probable values based on geometric means or best estimates of each parameter obtained from previously published sources (IAEA, 1994; Belli, 2000). This type of screening calculation confirms the combined importance of the soil and trees as the components in which the bulk of radiocaesium is stored. In the case of trees (all tree parts, including woody and leafy tissues) a modal value of 9.6% is indicated, with a range of possible values from 1.5 to 37%. For soil the modal value is 86%, with a range of possible values from 59 to 96%. Despite the generally high activity concentrations of 137 Cs found in fungal fruiting bodies in contaminated forests, the maximum percentage of the total ecosystem activity likely to be found in fungal biomass is 1.75%, with a value of 0.08% most likely. In constructing activity balance models of radionuclides in forest ecosystems the ‘mushroom’ compartment can be neglected justifiably, even in the most extreme cases. The degree of stochastic variability indicated in Fig. 19 can be seen in spatial sampling exercises in relatively small areas of forest which can be considered homogeneous, such as

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Fig. 19. Percentage distributions of 137 Cs in four ecological compartments within a coniferous forest calculated using a Monte Carlo method employing previously published estimates of parameter values.

the sampling plots in the Belorussian Polessie described in Section 3.3. Monte Carlo analyses can be used to account for such local scale geographical variations in parameter values. When regional scale variability is considered, however, systematic variability due to geologic and climatic differences between locations becomes important. This scale of variability has been addressed in models such as FORESTLAND and RIFE in the sense that appropriate groups of model parameter values have been obtained which can be used to perform simulations for different forest sites across Europe, including Scandinavia and the Russian federation. So far, however, no model of radionuclide behaviour in forests has been coupled with a geographic information system to allow it to carry out spatial simulations. This seems a logical next

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Fig. 19. (Continued).

step in the development of forest radioecology models, but the potential problems involved in such a model are non-trivial. Fundamental issues such as the relationship between forest biomass and various combinations of climate and soil type must be addressed. Empirical models have been established which allow predictions of tree growth under different climatic conditions, but these tend to be for relatively restricted geographical areas such as England and Wales (Matthews et al., 1996). Application of such models outside the regions for which they were originally designed is likely to lead to large systematic errors in predicted biomass (D’Onofrio, 1999). Another fundamental problem to be solved is the relationship between the root uptake of radionuclides such as 137 Cs by trees in response to a potentially infinite

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number of combinations of soil, hydrological and climatic conditions. This challenge has been overcome for agricultural plants (Van der Perk et al., 2000) but is likely to prove more difficult for the forest ecosystem.

6. Summary Forests are extensive natural and semi-natural ecosystems which have been subject to radioactive contamination almost since the beginnings of the nuclear power programme in the 1950s. This contamination can lead to radiation exposures of both human and non-human biota both within the forest itself and ex situ when forest-derived products (both edible and non-edible) are exported, often over significant distances. Despite the development of several mathematical models of radionuclide behaviour in boreal, temperate and tropical forests during the 1960s, the potential radiological significance of contamination of forests with radiocaesium after the Chernobyl accident was not immediately recognised. However, since, the mid-1990s significant efforts have been made to develop and test new models of radiocaesium behaviour in forests within the European continent. These models have made use of the growing availability of data on 137 Cs cycling in forests situated in diverse geographical regions ranging from the former Soviet Union to the western European temperate belt and Fennoscandia. Perhaps the major challenge to be met by forest radioecology modellers in the future will be to simulate successfully the developing spatial patterns of forest contamination across these regions following the introduction of the Chernobyl pulse and any future releases of radioactive materials to the environment. Acknowledgements The FORESTLAND and RIFE models were both developed with funding from the European Commission under the 4th Framework Programme. Collaboration between GS, AD and TZ was facilitated by a grant awarded by the Royal Society, London, for collaborative projects between the UK and the former Soviet Union. The Forest Working Group’s activities within the International Atomic Energy Agency’s BIOMASS programme were supported with funding from the Swedish Radiation Protection Institute and United Kingdom Nirex Limited. The validation data from Rudnya Povcha were kindly provided by Alexander Orlov of the Ukrainian Scientific Research Institute of Forestry and Amelioration, Zhitomir, Ukraine. Finally, the validation study would not have been possible without the excellent organisational skills of Ansie Venter, Enviros Quantisci, UK.

References Ågren, G. I. & Bosatta, E. (1996). Theoretical ecosystem ecology: understanding element cycles. Cambridge University Press. Alexakhin, R. M., Ginsburg, L. R., Mednik, I. G. & Prokhorov, V. M. (1994). Model of 90 Sr cycling in a forest biogeocoenosis. Science of the Total Environment, 157, 83–91. Alexakhin, R. M., Prokhorov, V. M., Ginsburg, L. R. & Mednik, I. G. (1976). Model of 90 Sr cycling in a forest biogeocoenosis. Ekologiya, 3, 5–14.

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Chapter 12

Mathematical and numerical approaches in performance assessment for radioactive waste disposal: dealing with uncertainty J. C. Helton Department of Mathematics, Arizona State Univeristy, Tempe, AZ 85287-1804, USA

1. Introduction In the context of this chapter, performance assessment (PA) is used to indicate an analysis intended to answer the following questions about a natural or engineered system (Helton, 1996a; Kaplan & Garrick, 1981): Q1, “What occurrences can take place?”; Q2, “How likely are these occurrences?”; Q3, “What are the consequences of individual occurrences?”; and Q4, “How much confidence exists in the answers to the first three questions (i.e., Q1–Q3)?” Performance assessment for radioactive waste disposal is a topic of wide national and international interest (Ahearne, 1997; Carter & Pigford, 1999; Crowley, 1997; Flynn et al., 1997; Kastenburg & Gratton, 1997; McCombie, 1997; North, 1997, 1999; Rechard, 1999b; Thompson, 1999). As examples, major radioactive waste disposal programs exist in the United States (i.e., the Waste Isolation Pilot Plant (WIPP) (Helton et al., 1999; NRC (National Research Council), 1996) and the Yucca Mountain Project (U.S. DOE (U.S. Department of Energy), 1998), Canada (Atomic Energy of Canada Limited, 1994), Sweden (Swedish Nuclear Power Inspectorate, 1996), and France (Hoorelbeke & Mouroux, 1998). This chapter will be illustrated primarily with results from a PA carried out for the WIPP. However, the ideas under consideration are quite general and apply not only to PAs for radioactive waste disposal facilities but also to PAs for other complex systems such as nuclear power plants, chemical production facilities, and transportation systems. A central focus of this chapter is how uncertainty should be treated in a PA for a radioactive waste disposal facility or some other complex system. In such assessments, uncertainty is often divided into two components: (i) stochastic or aleatory uncertainty, which arises because the system under study can potentially behave in many different ways, and (ii) subjective or epistemic uncertainty, which arises from a lack of knowledge about quantities that are assumed to have fixed values within the computational implementation of the PA (Helton, 1994, 1997; Helton & Burmaster, 1996; Hoffman & Hammonds, 1994; Paté-Cornell, 1996; 353

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Thompson & Graham, 1996). The use of probability to quantify these two classifications of uncertainty can be traced to the beginnings of the formal development of probability in the seventeenth century (Bernstein, 1996; Hacking, 1975). In the context of the previously indicated questions, stochastic uncertainty relates to Q1 and Q2, and subjective uncertainty relates to Q4. A PA can be discussed at an intuitive and informal level. For example, describing a PA as an attempt to answer questions Q1–Q4 is an example of such usage. For many descriptive purposes, such informality is appropriate. However, the actual computational implementation of a PA requires a careful mathematical description of what is to be done; without such a description, it is difficult to produce a coherent analysis. Indeed, given that PAs invariably involve mathematical calculations, an inability to produce a formal description of the mathematical approaches in use suggests an analysis that has not been carefully thought through. Further, meaningful review of a PA also requires a careful mathematical description of the analysis; without such a description, it is difficult to communicate to reviewers what was done in the analysis. Descriptions of the mathematical approaches in use are required on at least three levels: (i) overall structure of analysis, (ii) probabilistic characterization of uncertainty, and (iii) representation of individual physical processes. Three distinct entities typically underlie the mathematical approaches used in a performance assessment: EN1, a probabilistic characterization of what could occur at the site under consideration; EN2, mathematical models for estimating the consequences of what could occur; and EN3, a probabilistic characterization of the uncertainty in the parameters used in the definitions of EN1 and EN2. The term parameter in the definition of EN3 is used broadly enough to include designators for alternative models or model structures. At a formal level, EN1 and EN3 are suitably defined probability spaces, and EN3 often consists of systems of ordinary or partial differential equations used to represent various physical processes. In the context of the previously indicated questions, EN1 is providing the answer to Q1 and Q2; EN2 is providing the answer to Q3; and EN3 is providing the answer to Q4. Careful definition of EN1, EN2 and EN3 leads to a description of the mathematical approaches being used in a PA. However, this description is typically at a formal level (i.e., as integrals involving probability spaces and solutions to ordinary or partial differential equations). The actual production of numerical results also requires both procedures for approximating the mathematical structures used in the description of the PA and computer software for the implementation of these procedures. The quality of these procedures and the implementing software can have a significant effect on the practicability of the analysis and the correctness of the final results. This chapter elaborates on the preceding ideas and is organized as follows. First, owing to its extensive use in the illustration of the ideas under discussion, the WIPP is briefly described (Section 2). Then, the screening of features, events and processes (FEPs) as an initial step in a PA is discussed (Section 3). This is followed by discussions of EN1 (Section 4), EN2 (Section 5), the combining of EN1 and EN2 to produce complementary cumulative distribution functions (CCDFs) (Section 6), and EN3 (Section 7). Next, example results from a PA for the WIPP are presented (Section 8). Finally, the chapter ends with a summary discussion (Section 9). With some editorial modifications, the material contained in Sections 1 and 3–9 is adapted from Helton (2001), and the material contained in Section 2 is adapted from Helton et al. (1999).

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Fig. 1. Cross-sectional view of the WIPP (Fig. 1–9, Vol. 1, WIPP PA (Performance Assessment) Division, 1991–1992; see Section 2.2, Vol. 2, WIPP PA (Performance Assessment) Division, 1992–1993, for detailed stratigraphy).

2. Waste Isolation Pilot Plant (WIPP) The WIPP is under development by the U.S. Department of Energy (DOE) for the deep geologic disposal of transuranic (TRU) waste that has been generated at government defense installations in the United States (Mora, 1999; NRC (National Research Council), 1996; Rechard, 1999a, 2000; U.S. DOE (U.S. Department of Energy), 1980, 1990). The WIPP is located in southeastern New Mexico in an area of low population density approximately 42 km (26 mi) east of Carlsbad. Waste disposal takes place in excavated chambers (i.e., waste disposal panels) in a bedded salt formation, the Salado Formation (Fm), approximately 655 m (2150 ft) below the land surface (Fig. 1). The Salado Fm is contained in the Delaware Basin, which is a large sedimentary basin located in southeastern New Mexico and western Texas (Swift & Corbet, 2000; Chapter 2, U.S. DOE (U.S. Department of Energy), 1996; Section 1.5.3, Vol. 3, WIPP PA (Performance Assessment) Division, 1991–1992). The sequence of events that led to the development of the WIPP by the DOE for the disposal of TRU waste in bedded salt began in 1955 when the Atomic Energy Commission (AEC), part of which later became the DOE, asked the National Academy of Science (NAS) to examine disposal options for radioactive waste (Section 1.5.1, Rechard, 1995; see Rechard, 2000 for more details on the historical development of the WIPP than can be presented here). In response, the NAS reported in 1957 that, while various options and disposal sites were feasible, disposal in bedded salt was the most promising disposal method (NAS/NRC (National Academy of Sciences/National Research Council), 1957). From that point through the early 1970s, Oak Ridge National Laboratory conducted radioactive-waste disposal experiments, most no-

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tably Project Salt Vault in an abandoned salt mine near Lyons, Kansas (Bradshaw & McClain, 1971). Although the AEC considered using the mine as a repository, the discovery of boreholes in the nearby area prompted the AEC to search for more suitable sites (Carter, 1987). At the invitation of New Mexico’s governor, the AEC investigated the Delaware Basin in the Carlsbad area of New Mexico. After an initial examination, a potential site was identified in the 1970s. The site was named the Waste Isolation Pilot Plant (WIPP) in January 1976 (NAS/NRC (National Academy of Sciences/National Research Council), 1984). The regional site-characterization phase of this potential waste disposal site (Powers et al., 1978) ended with the preparation of an Environmental Impact Statement (EIS) in 1980 (U.S. DOE (U.S. Department of Energy), 1980), as required by the National Environmental Policy Act of 1969 (NEPA) (Public Law 91-190, 1970). In response to the EIS, the DOE decided to proceed with a preliminary design phase at this site. During the 1970s, the mission of the WIPP, and thus the design (Weart, 1979), varied between including and not including defense high-level waste (HLW) in addition to TRU waste. However, with passage of the National Security and Military Applications of Nuclear Energy Authorization Act of 1980 (Public Law 96-164, 1979), Congress defined the WIPP as a research and development facility for the storage and disposal of TRU waste, and exempted the WIPP from regulation by the U.S. Nuclear Regulatory Commission. In 1981, the “Stipulated Agreement” and “Consultation and Cooperation Agreement” defined the WIPP’s relationship with the State of New Mexico and stipulated specific geotechnical experiments required by the state (U.S. DOE (U.S. Department of Energy) and State of New Mexico, 1981). After much planning, construction of the WIPP began in 1983 (U.S. DOE (U.S. Department of Energy), 1983a, b). Experiments to characterize the local disposal system followed (Lappin, 1988; Lynch et al., 1991; Tyler et al., 1988). In preparation for the WIPP’s opening, a Supplemental EIS was published in 1990 (U.S. DOE (U.S. Department of Energy), 1990). In the Waste Isolation Pilot Plant Land Withdrawal Act of 1992 (LWA) (Public Law 102579, 1992), Congress defined the process by which WIPP’s compliance with applicable regulations would have to be evaluated and transferred ownership of the WIPP site to the DOE. This act officially marked the transition from the construction and disposal-systemcharacterization phase to the compliance and testing phases, although these phases had begun unofficially in 1985 when the U.S. Environmental Protection Agency (EPA) issued its standard 40 CFR Part 191 (U.S. EPA (U.S. Environmental Protection Agency), 1985) for the management and disposal of spent nuclear fuel, HLW and TRU waste and in 1989 when Sandia National Laboratories (SNL) first began to assess performance of the WIPP using the EPA standard (Bertram-Howery et al., 1989; Lappin et al., 1989; Marietta et al., 1989). Additional PAs were carried out for the WIPP in 1990, 1991 and 1992 (Bertram-Howery et al., 1990; WIPP PA (Performance Assessment) Division, 1991–1992, 1992–1993), with summaries of the 1991 and 1992 PAs available in the journal literature (Helton et al., 1993, 1995, 1996; Helton & Trauth, 1997). The efforts to produce a PA for the WIPP to satisfy the requirements in 40 CFR Part 191 began in 1992, when Congress passed the LWA in which it established several mandates. First, Congress required that the DOE demonstrate compliance to the final disposal standards codified in 40 CFR Part 191, Subparts B and C, prior to opening the WIPP for the disposal of TRU waste (U.S. EPA (U.S. Environmental Protection Agency), 1985, 1993). Second, Congress mandated that the DOE submit an application to the EPA seeking certification of the DOE’s

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compliance demonstration. Third, Congress mandated that the EPA issue certification criteria to judge the adequacy of the DOE’s application. The EPA met this obligation in February 1996 with the issuance of 40 CFR Part 194 (U.S. EPA (U.S. Environmental Protection Agency), 1996a, b). For illustration purposes, this chapter uses a PA carried out at SNL to support the Compliance Certification Application (CCA) made by the DOE to the EPA in October 1996 for the certification of the WIPP for the disposal of TRU waste as mandated in the LWA (Helton et al., 1998a; Helton & Marietta, 2000; U.S. DOE (U.S. Department of Energy), 1996). The indicated CCA resulted in a positive certification decision for the WIPP by the EPA in May 1998 (U.S. DOE (U.S. Department of Energy), 1998). The first disposal of TRU waste at the WIPP took place in March 1999. The WIPP is the first operational facility in the United States for the deep geologic disposal of radioactive waste.

3. Screening of features, events and processes (FEPs) A PA should be based on a well-defined structure in which both the mathematical components of the analysis and their numerical approximations can be unambiguously specified. However, a PA does not start out well-defined. Rather, some type of procedure must be used to decide what is to be incorporated into, and excluded from, the analysis. This process is often referred to as the screening of features, events and processes (FEPs). Once the screening of FEPs is completed, the more formal structure of the analysis as embodied in EN1, EN2 and EN3 can be developed. In practice, most important FEPs will probably be identified without the use of any formal process by the time that the development of a radioactive waste disposal facility has reached the point that a formal PA is required. However, to satisfy public concerns and regulatory requirements, a formal procedure may be called for that provides a systematic development and description of the selection process used to determine what is to be included in the PA. For example, the following requirement appears in the regulations that apply to the WIPP (U.S. EPA (U.S. Environmental Protection Agency), 1996a, p. 5242): §194.32 Scope of performance assessments. . . . (e) Any compliance application(s) shall include information which: (1) identifies all potential processes, events or sequences and combinations of processes and events that may occur during the regulatory time frame and may affect the disposal system; (2) Identifies the processes, events or sequences and combinations of processes and events included in performance assessments; and (3) Documents why any processes, events or sequences and combinations of processes and events identified pursuant to paragraph (e)(1) of this section were not included in performance assessment results provided in any compliance application.

The 1996 WIPP PA provides an example of a real analysis in which an extensive screening of FEPs was carried out. As a starting point, a list of potentially relevant FEPs was assembled from a compilation developed for the Swedish Nuclear Power Inspectorate (SKI) (Stenhouse et al., 1993). The SKI list is based on several FEP lists developed for other waste disposal programs (Andersson et al., 1989; Cranwell et al., 1990; Goodwin et al., 1994; Hodgkinson & Sumerling, 1989; Miller & Chapman, 1992; NAGRA, 1985; Thorne, 1993) and constituted the best documented and most comprehensive starting point for the WIPP PA. WIPP-specific

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Table 1 Natural FEPs and their screening classifications (Table 2, Helton et al., 1999; adapted from Tables 6–3, U.S. DOE (U.S. Department of Energy), 1996) GEOLOGICAL FEPs (Section SCR.1.1, U.S. DOE (U.S. Department of Energy) 1996): 1. Stratigraphy. 1.1 Stratigraphy (UP). 1.2 Brine reservoirs (DP). 2. Tectonics. 2.1 Changes in regional stress (SO-C). 2.2 Regional tectonics (SO-C). 2.3 Regional uplift and subsidence (SO-C). 3. Structural FEPs. 3.1 Deformation. 3.1.1 Salt deformation (SOP, UP near repository). 3.1.2 Diapirism (SO-P). 3.2 Fracture development. 3.2.1 Formation of fractures (SO-P, UP near repository). 3.2.2 Changes in fracture properties (SO-C, UP near repository). 3.3 Fault movement. 3.3.1 Formation of new faults (SO-P). 3.3.2 Fault movement (SO-P). 3.4 Seismic activity. 3.4.1 Seismic activity (UP). 4. Crustal processes. 4.1 Igneous activity. 4.1.1 Volcanic activity (SO-P). 4.1.2 Magmatic activity (SO-C). 4.2 Metamorphism. 4.2.1 Metamorphic activity (SO-P). 5. Geochemical FEPs. 5.1 Dissolution. 5.1.1 Shallow dissolution (UP). 5.1.2 Lateral dissolution (SO-C). 5.1.3 Deep dissolution (SO-P). 5.1.4 Solution chimneys (SO-P). 5.1.5 Breccia pipes (SO-P). 5.1.6 Collapse breccias (SO-P). 5.2 Mineralization. 5.2.1 Fracture infills (SO-C). SUBSURFACE HYDROLOGICAL FEPs (Section SCR.1.2, U.S. DOE (U.S. Department of Energy) 1996): 1. Groundwater characteristics. 1.1 Saturated groundwater flow (UP). 1.2 Unsaturated groundwater flow (UP, SO-C in Culebra). 1.3 Fracture flow (UP). 1.4 Density effects on groundwater flow (SO-C). 1.5 Effects of preferential pathways (UP, UP in Salado and Culebra. 2. Changes in groundwater flow. 2.1 Thermal effects on groundwater flow (SO-C). 2.2 Saline intrusion (SO-P). 2.3 Freshwater intrusion (SO-P). 2.4 Hydrological response to earthquakes (SO-C). 2.5 Natural gas intrusion (SO-P). SUBSURFACE GEOCHEMICAL FEPs (Section SCR.1.3, U.S. DOE (U.S. Department of Energy) 1996): 1. Groundwater geochemistry. 1.1 Groundwater geochemistry (UP). 2. Changes in groundwater chemistry. 2.1 Saline intrusion (SO-C). 2.2 Freshwater intrusion (SO-C). 2.3 Changes in groundwater Eh (SO-C). 2.4 Changes in groundwater pH (SO-C). 2.5 Effects of dissolution (SO-C). GEOMORPHOLOGICAL FEPs (Section SCR.1.4, U.S. DOE (U.S. Department of Energy) 1996): 1. Physiography. 1.1 Physiography (UP). 2. Meteorite impact. 2.1 Impact of a large meteorite (SO-P). 3. Denudation. 3.1 Weathering. 3.1.1 Mechanical weathering (SO-C). 3.1.2 Chemical weathering (SO-C). 3.2 Erosion. 3.2.1 Aeolian erosion (SO-C). 3.2.2 Fluvial erosion (SO-C). 3.2.3 Mass wasting (SO-C). 3.3 Sedimentation. 3.3.1 Aeolian deposition (SO-C). 3.3.2 Fluvial deposition (SO-C). 3.3.3 Lacustrine deposition (SO-C). 3.3.4 Mass Wasting (SO-C). 4. Soil development. 4.1 Soil development (SO-C). SURFACE HYDROLOGICAL FEPs (Section SCR.1.5, U.S. DOE (U.S. Department of Energy) 1996): 1. Fluvial. 1.1 Stream and river flow (SO-C). 2. Lacustrine. 2.1 Surface water bodies (SO-C). 3. Groundwater recharge and discharge. 3.1 Groundwater discharge (UP). 3.2 Groundwater recharge (UP). 3.3 Infiltration (UP, UP for climate change effects). 4. Changes in surface hydrology. 4.1 Changes in groundwater recharge and discharge (UP). 4.2 Lake formation (SO-C). 4.3 River flooding (SO-C). CLIMATIC FEPs (Section SCR.1.6, U.S. DOE (U.S. Department of Energy) 1996): 1. Climate. 1.1 Precipitation (for example, rainfall) (UP). 1.2 Temperature (UP). 2. Climate change. 2.1 Meteorological. 2.1.1 Climate change (UP). 2.2 Glaciation. 2.2.1 Glaciation (SO-P). 2.2.2 Permafrost (SO-P). MARINE FEPs (Section SCR.1.7, U.S. DOE (U.S. Department of Energy) 1996): 1. Seas. 1.1 Seas and oceans (SOC). 1.2 Estuaries (SO-C). 2. Marine sedimentology. 2.1 Coastal erosion (SO-C). 2.2 Marine sediment transport and deposition (SO-C). 3. Sea level changes. 3.1 Sea level changes (SO-C). ECOLOGICAL FEPs (Section SCR.1.8, U.S. DOE (U.S. Department of Energy) 1996): 1. Flora and fauna. 1.1 Plants (SO-C). 1.2 Animals (SO-C). 1.3 Microbes (SO-C, UP for colloidal effects and gas generation). 2. Changes in flora and fauna. 2.1 Natural ecological development (SO-C). Legend: UP ∼ FEPs accounted for in the assessment calculations for undisturbed performance for 40 CFR §191.13 (U.S. EPA (U.S. Environmental Protection Agency) 1993) (as well as 40 CFR §191.15 and Subpart C of 40 CFR Part 191); DP ∼ FEPs accounted for (in addition to all UP FEPs) in the assessment calculations for disturbed performance for 40 CFR §191.13; SO-R ∼ FEPs eliminated from PA calculations on the basis of regulations provided in 40 CFR Part 191 and criteria provided in 40 CFR Part 194 (U.S. EPA (U.S. Environmental Protection Agency) 1996a)) SOC ∼ FEPs eliminated from PA (and compliance assessment) calculations on the basis of consequence; SO-P ∼ FEPs eliminated from PA (and compliance assessment) calculations on the basis of low probability of occurrence.

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FEPs were added to the SKI list, and the combined list was edited to remove redundancy and ambiguity, resulting in a FEPs list (U.S. DOE (U.S. Department of Energy), 1996, Attachment 1, Appendix SCR) that was appropriate for the WIPP (U.S. DOE (U.S. Department of Energy), 1996, Section 6.2.1). This list was then carefully analyzed to identify the FEPs that should be incorporated into the computational structure of the 1996 WIPP PA and also the FEPs that did not require incorporation into this structure. Decisions to remove (i.e., screen out) FEPs from the computational structure used for the 1996 WIPP PA were based on the following criteria: regulatory exclusion (SO-R), low probability (SO-P), and low consequence (SO-C). The three screening criteria derive from specific regulatory requirements (U.S. DOE (U.S. Department of Energy), 1996, Section 6.2.1). The FEPs not screened out were retained for inclusion in the PA and were classified as undisturbed performance (UP) or disturbed performance (DP) FEPs. As an example, a summary of the screening process for natural FEPs is given in Table 1; in addition, waste- and repository-induced FEPs (U.S. DOE (U.S. Department of Energy), 1996, Table 6-46) and human-initiated events and processes (EPs) (U.S. DOE (U.S. Department of Energy), 1996, Table 6-56) were also considered. A detailed description of the screening process is available in Appendix SCR of U.S. DOE (U.S. Department of Energy) (1996). Additional information is also available in Galson et al. (2000). 4. EN1: probabilistic characterization of different futures An important outcome of the FEPs process is an identification of occurrences (e.g., volcanism, faulting, human disruption, . . .) that could take place with sufficient likelihood and consequence to merit inclusion in a PA for the facility under consideration. Typically, these occurrences are constrained to take place over some time interval of interest (e.g., 10 000 yr in the regulations that apply to the WIPP). There are two aspects to the specification of such occurrences: (i) definition of occurrence properties (e.g., time, location, magnitude, . . .) within the time interval of interest, and (ii) assignment of probabilities to possible occurrences within the time interval of interest. The assignment of probabilities is only necessary if some form of probabilistic presentation of results is to be used. If results are presented conditional on individual occurrences, then probabilities are not needed. However, presumably even in this case, some sort of probabilistic selection is taking place as there would be little purpose in analyzing a group of occurrences that had a probability of zero of taking place. For the following discussion, the assumption is made that probabilities are to be assigned to future occurrences. The development of probabilities requires a careful specification of exactly what probabilities are to be assigned to. As an example, suppose only two occurrences, designated o1 and o2 , with the following properties were selected for inclusion in a PA: (i) the occurrences of o1 and o2 are independent, (ii) the occurrences of o1 and o2 follow Poisson processes with rate constants λ1 (yr−1 ) and λ2 (yr−1 ), respectively, and (iii) a time interval [0, T = 1000 yr] is under consideration. There are many possible patterns of occurrence of o1 and o2 over the interval [0, T ]. Each possible pattern can be represented by a vector x = [t11 , t12 , . . . , t1m , t21 , t22 , . . . , t2n ],

(1)

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where: 0  t11  t12  · · ·  t1m  T = 1000 yr, 0  t21  t22  · · ·  t2n  T = 1000 yr, and t1i , i = 1, 2, . . . , m, and t2j , j = 1, 2, . . . , n, are times at which o1 and o2 , respectively, occur. Each vector x in equation (1) represents a possible pattern of occurrences over the time interval [0, T ]. The pattern of occurrences characterized by x is sometimes referred to as a future in radioactive waste disposal because each x is describing one possible future at the facility under consideration. The set S = {x : x a vector of the form in equation (1)}

(2)

corresponds to the universe of all possible futures that could occur over the time interval [0, T ] given the assumption that o1 and o2 are the only two occurrences to be considered. The introduction of x provides an exact mathematical description (i.e., definition) of occurrence properties within the time interval of interest. The vector x in equation (1) is quite simple; in general, more properties than initiation times are required to define individual occurrences in a PA. A more general formulation of the example would be x = [o11 , o12 , . . . , o1m , o21 , o22 , . . . , o2n ],

(3)

where o1i , i = 1, 2, . . . , m, and o2j , j = 1, 2, . . . , n, are vectors of properties associated with the occurrence of o1 and o2 . Further, PAs can involve more than two occurrences of interest. However, x as defined in equation (1) is sufficient as an example. The assignment of probabilities is now considered. Typically, individual futures of the form illustrated by x in equation (1) have a probability of zero. Thus, probabilities are usually used in reference to sets of futures or, equivalently, to subsets E of S in equation (2). For example, E might be defined by   E = x = [t11 , t21 , t22 ] : 0  t11  500 yr, 0  t21  t22  1000 yr .

(4)

Then, under the previously stated assumptions that o1 and o2 are independent and follow Poisson processes with rate constants λ1 and λ2 , the probability p(E ) of E is given by      (500λ1)0 /0! exp(−500λ1) (500λ1 )1 /1! exp(−500λ1 )    × (1000λ2)2 /2! exp(−1000λ2)      (1000λ2)2 /2! exp(−1000λ2) . = (500λ1 )1 /1! exp(−1000λ1)

p(E ) =

(5)

Thus, in this example and more generally in most situations, probability distributions are introduced for the individual elements of x and then used to define probabilities for sets of x. The closed form calculation of such probabilities can be difficult or impossible; often,

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Monte Carlo techniques or other numerical procedures must be used to calculate approximate probabilities. To the extent practicable, descriptions of the various parts of a PA should use widely recognized mathematical constructions. Then, the computational implementation of the PA can be viewed as the calculation of approximations to these constructions. For probability, the fundamental mathematical construction is a probability space, which involves three components: (i) a set S that contains everything that could occur in the particular universe under consideration, (ii) a set S of subsets of S, and (iii) a function p that defines the probability of the individual elements of S (p. 116, Feller, 1971). The preceding three components appear in the previous example. In the terminology of probability theory, S is the sample space; elements x of S are elementary events; elements E of S are events; p is a probability measure; and the triple (S, S, p) is a probability space. In terminology often used in PA for radioactive waste disposal, elements x of S are futures; elements E of S are scenarios; the probabilities p(E ) are scenario probabilities; and questions of “completeness” deal with whether or not S has been defined in a way that incorporates all significant occurrences at the site under consideration. In practice, most probability calculations are based on the distributions assigned to the individual elements of x and do not rely on the formal properties of the probability space (S, S, p). However, the introduction of the concept of a probability space provides a way to formally describe what is being done in a PA and also a way to distinguish between different uses of probability within a PA. As indicated in Section 1, uncertainty about occurrences that may take place in the future is often designated stochastic or aleatory uncertainty. Thus, the probabilistic characterization of different possible futures in a PA involves stochastic uncertainty. For this reason, the subscript ‘st’ is sometimes added to vectors x of the form illustrated in equation (1) (i.e., xst replaces x) to indicate that these vectors are being used in the characterization of stochastic uncertainty. Correspondingly, (Sst , Sst , pst ) is used instead of (S, S, p) to indicate the characterization of stochastic uncertainty. The probability space (Sst , Sst , pst ) is providing a probabilistic characterization of the different futures that could occur at the facility under consideration and formally defines the entity EN1 referred to in Section 1 and the heading to this section. Results from the 1996 WIPP PA are now used for illustration. The FEPs development process for the WIPP identified exploratory drilling for natural resources as the only disruption with sufficient likelihood and consequence for inclusion in the definition of EN1 (App. SCR, U.S. DOE (U.S. Department of Energy), 1996). In addition, regulatory requirements (U.S. EPA (U.S. Environmental Protection Agency), 1996a) specify that the occurrence of potash mining near the site must be included in the analysis. The preceding considerations led to the elements xst of Sst being vectors of the form xst = [t1 , l1 , e1 , b1 , p1 , a1 , t2 , l2 , e2 , b2 , p2 , a2 , . . . , tn , ln , en , bn , pn , an , tmin ]          1st intrusion

2nd intrusion

(6)

nth intrusion

in the 1996 WIPP PA, where n is the number of drilling intrusions in the vicinity of the WIPP, ti is the time (yr) of the i th intrusion, li designates the location of the i th intrusion, ei designates the penetration of an excavated or nonexcavated area by the i th intrusion, bi designates whether or not the i th intrusion penetrates pressurized brine in the Castile Formation, pi designates the plugging procedure used with the i th intrusion (i.e., continuous plug, two discrete

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plugs, three discrete plugs), ai designates the type of waste penetrated by the i th intrusion (i.e., no waste, contact-handled (CH) waste, remotely-handled (RH) waste), and tmin is the time at which potash mining occurs. In the development of (Sst , Sst , pst ), the probabilistic characterization of n, ti , li , and ei derives from the assumption that drilling intrusions occur randomly in time and space (i.e., follow a Poisson process); the probabilistic characterization of bi derives from assessed properties of brine pockets; the probabilistic characterization of ai derives from the properties of the waste to be emplaced at the WIPP; and the probabilistic characterization of pi derives from current drilling practices in the sedimentary basin in which the WIPP is located. A vector notation is used for ai because a given drilling intrusion can penetrate several different types of waste. Further, the probabilistic characterization for tmin follows from regulatory guidance that the occurrence of potash mining should be assumed to occur randomly in time (i.e., follow a Poisson process with a rate constant of λm = 10−4 yr−1 ), with all commercially viable potash reserves being extracted at time tmin . Additional information on the definition of (Sst , Sst , pst ) in the 1996 WIPP PA is available in Helton et al. (2000b).

5. EN2: mathematical models for estimating consequences of individual futures The entity EN2 corresponds to the mathematical models required to estimate various consequences in a PA for a radioactive waste disposal site or some other facility. Conceptually, EN2 can be viewed as a function f of the form y = f(xst ),

(7)

where xst is a particular future, f corresponds to the models used to estimate various outcomes associated with xst , and y is the estimated outcomes. A vector notation is used for f and y because typically a large number of outcomes is estimated for each future xst . For a radioactive waste disposal site, the models denoted by f are often systems of ordinary or partial differential equations used to represent processes such as material deformation, corrosion, fluid flow, radionuclide transport in flowing groundwater, and radionuclide movement and associated human radiation exposure in the surface environment. The models used in a particular PA will depend on many things, including the site under consideration, the resolution of the data available for characterization of the site, the desired resolution in model predictions, and the time and resources available for the analysis. However, most models will consist of three parts: (i) a formal mathematical representation (e.g., a system of nonlinear partial differential equations), (ii) a numerical procedure that produces approximations to the formal mathematical representation (e.g., a finite difference procedure, see Chapter 19, Press et al., 1992), and (iii) a computer program that implements the numerical procedure. Given the almost universal use of numerical approximations and computer programs to implement these approximations, a PA typically involves two extensions of the representation in equation (7): y = ˆf(xst ),

(8)

Mathematical and numerical approaches for radioactive waste disposal

y = F(xst),

363

(9)

where ˆf is used to represent the numerical approximation that replaces f, and F is used to represent the computer program that implements this approximation and actually produces y. For example, if f was based on a system of ordinary differential equations, ˆf might correspond to an Adams–Moulten method, a variable order, variable stepsize Runge–Kutta method or a Gear’s backward differentiation method, and F would correspond to the program that actually implemented the method (Chapter 16, Press et al., 1992). The absence of good numerical procedures and their appropriate implementation can lead to (i) excessive computational times, (ii) erroneous results, and (iii) possibly no results at all. Models used in the 1996 WIPP PA to estimate total normalized radionuclide release to the accessible environment are now used for illustration, where normalized release refers to a single release value derived from the releases of several radionuclides on the basis of the initial radionuclide inventory in the repository and the radiological hazard of the individual radionuclides (Section 2, Helton et al., 1999). In this example, f is real valued and has the form   



f (xst) = fC (xst) + fSP xst , fB (xst ) + fDBR xst, fSP xst , fB (xst) , fB (xst )

   + fMB xst , fB (xst ) + fDL xst , fB (xst) + fS xst , fB (xst ) 

 (10) + fS−T xst,0 , fS−F (xst,0 ), fN−P xst, fB (xst ) , where xst ∼ particular future under consideration, xst,0 ∼ future involving no drilling intrusions but a mining event at the same time tmin as in xst , fC (xst ) ∼ cuttings and cavings release to accessible environment calculated with the CUTTINGS_S program, fB (xst) ∼ two-phase flow results calculated with the BRAGFLO program, fSP [xst , fB (xst)] ∼ spallings release to accessible environment calculated with the CUTTINGS_S program, fDBR {xst , fSP [xst, fB (xst )], fB (xst )} ∼ direct brine release to accessible environment calculated with the BRAGFLO_DBR program, fMB [xst , fB (xst)] ∼ release through anhydrite marker beds to accessible environment calculated with the NUTS program, fDL [xst , fB (xst)] ∼ release through Dewey Lake Red Beds to accessible environment calculated with the NUTS program, fS [xst, fB (xst )] ∼ release to land surface due to brine flow up a plugged borehole with the NUTS or PANEL program as appropriate, fS−F (xst,0 ) ∼ Culebra flow field calculated with the SECOFL2D program, fN−P [xst , fB (xst)] ∼ releases to Culebra calculated with the NUTS or PANEL program as appropriate, fS−T {xst,0 , fS−F (xst,0 ), fN−P [xst, fB (xst )]} ∼ groundwater transport release through Culebra to accessible environment calculated with the SECOTP2D program. Although f is shown as being real valued in equation (10) because a single analysis outcome is under consideration, the 1996 WIPP PA was quite complex and a description of the analysis in full generality would require use of vector representations for all models. As indicated in Table 2, each model actually consists of a formal mathematical structure (typically a system of ordinary or partial differential equations), a numerical procedure for approximating this structure, and an implementing computer program. The 1996 WIPP PA is typical of large PAs in that f does not correspond to a single model, but rather to an ensemble of models that is used to produce results of interest. The model fB for two-phase flow in equation (10) and Table 2 illustrates the ideas associated with equations (7)–(9). This model is based on the following system of two conservation

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Table 2 Summary of Computer Models Used in the 1996 WIPP PA (adapted from Table 1, Helton et al., 1998b) BRAGFLO: Calculates multiphase flow of gas and brine through a porous, heterogeneous reservoir. Uses finite difference procedures to solve system of nonlinear partial differential equations that describes the mass conservation of gas and brine along with appropriate constraint equations, initial conditions and boundary conditions. Additional information: Vaughn et al. (2000). BRAGFLO_DBR: Special configuration of BRAGFLO model used in calculation of dissolved radionuclide releases to the surface (i.e., direct brine releases) at the time of a drilling intrusion. Uses initial value conditions obtained from calculations performed with BRAGFLO and CUTTINGS_S. Additional information: Stockman et al. (2000). CUTTINGS_S: Calculates the quantity of radioactive material brought to the surface in cuttings and cavings and also in spallings generated by an exploratory borehole that penetrates a waste panel, where cuttings designates material removed by the drillbit, cavings designates material eroded into the borehole due to shear stresses resulting from the circular flow of the drilling fluid (i.e., mud), and spallings designates material carried to the borehole at the time of an intrusion due to the flow of gas from the repository to the borehole. Spallings calculation uses initial value conditions obtained from calculations performed with BRAGFLO. Additional information: Berglund et al. (2000). GRASP-INV: Generates transmissivity fields (estimates of transmissivity values) conditioned on measured transmissivity values and calibrated to steady-state and transient pressure data at well locations using an adjoint sensitivity and pilot-point technique. Additional information: LaVenue (1996); LaVenue & RamaRao (1992). NUTS: Solves system of partial differential equations for radionuclide transport in vicinity of repository with finite difference procedures. Uses brine volumes and flows calculated by BRAGFLO as input. Additional information: Stockman et al. (2000). PANEL: Calculates rate of discharge and cumulative discharge of radionuclides from a waste panel through an intruding borehole. Discharge is a function of fluid flow rate, elemental solubility and radionuclide inventory. Uses brine volumes and flows calculated by BRAGFLO as input. Based on closed-form solution of system of linear ordinary differential equations. Additional information: Stockman et al. (2000) SANTOS: Solves quasistatic, large deformation, inelastic response of two-dimensional solids with finite element techniques. Used to determine porosity of waste as a function of time and cumulative gas generation, which is an input to calculations performed with BRAGFLO. Additional information: Stone (1997a, b). SECOFL2D: Calculates single-phase Darcy flow for groundwater flow in two dimensions. The formulation is based on a single partial differential equation for hydraulic head using fully implicit time differencing. Uses transmissivity fields generated by GRASP-INV. Additional information: Ramsey et al. (2000). SECOTP2D: Simulates transport of radionuclides in fractured porous media. Solves two partial differential equations: one provides two-dimensional representation for advective and diffusive radionuclide transport in fractures and the other provides one-dimensional representation for diffusion of radionuclides into rock matrix surrounding the fractures. Equations solved with finite difference procedures. Uses flow fields calculated by SECOFL2D. Additional information: Ramsey et al. (2000).

equations ((11), (12)), two constraint equations ((13), (14)), and three equations of state ((15)– (17)): gas conservation:  ∇•

 αρg Kg krg ∂(φρg Sg ) , (∇pg + ρg g∇h) + αqwg + αqrg = α μg ∂t

(11)

Mathematical and numerical approaches for radioactive waste disposal

brine conservation:   αρb Kb krb ∂(φρb Sb ) ∇• (∇pb + ρb g∇h) + αqwb + αqrb = α , μb ∂t

365

(12)

saturation constraint: Sg + Sb = 1,

(13)

capillary pressure constraint: pC = pg − pb = f (Sb ),

(14)

gas density: ρg determined by Redlich–Kwong–Soave equation of state (see equations (31), (32), Vaughn et al., 2000),

(15)

brine density:

 ρb = ρ0 exp βb (pb − pb0 ) ,

(16)

formation porosity:

 φ = φ0 exp βf (pb − pb0 ) ,

(17)

where g = acceleration due to gravity (m/s2 ), h = vertical distance from a reference location (m), Kl = permeability tensor (m2 ) for fluid l (l = g ∼ gas, l = b ∼ brine), krl = relative permeability (dimensionless) to fluid l, pC = capillary pressure (Pa), pl = pressure of fluid l (Pa), qrl = rate of production (or consumption, if negative) of fluid l due to chemical reaction (kg/m3/s), qwl = rate of injection (or removal, if negative) of fluid l (kg/m3/s), Sl = saturation of fluid l (dimensionless), t = time (s), α = geometry factor (m in present analysis), ρl = density of fluid l (kg/m3), μl = viscosity of fluid l (Pa s), φ = porosity (dimensionless), φ0 = reference (i.e., initial) porosity (dimensionless), pb0 = reference (i.e., initial) brine pressure (Pa) (constant in equation (16) and spatially variable in equation (17)), ρ0 = reference (i.e., initial) brine density (kg/m3 ), βf = pore compressibility (Pa−1 ), βb = brine compressibility (Pa−1 ), and f is defined by the model for capillary pressure in use (see the right hand sides of equations (10), (19) and (20) in Vaughn et al. (2000)). The conservation equations are valid in one (i.e., ∇ = [∂/∂x]), two (i.e., ∇ = [∂/∂x ∂/∂y]) and three (i.e., ∇ = [∂/∂x ∂/∂y ∂/∂z]) dimensions. In the 1996 WIPP PA, the preceding system of equations was used to model twophase fluid flow in a two-dimensional region (Fig. 2), with the result that the spatial scale factor α in equations (11) and (12) had units of meters (m) (see Fig. 2, Vaughn et al., 2000). In general, the individual terms in equations (11)–(17) are functions of location and time (e.g., pg (x, y, t), ρg (x, y, t), krg(x, y, t), . . .) and often other variables as well. A full description of how the individual terms in these equations are defined is beyond the scope of this

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Fig. 2. Computational grid used in BRAGFLO to represent two-phase flow in 1996 WIPP PA subsequent to a drilling intrusion (Fig. 1, Kleijnen & Helton, 1999). Same formulation is used in the absence of a drilling intrusion except that regions 1A, 1B and 1C have the same properties as the regions to either side.

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presentation and is available elsewhere (Vaughn et al., 2000). The system of partial differential equations defined in equations (11)–(17) is too complex to permit a closed form solution. In the 1996 WIPP PA, these equations were solved with finite difference procedures implemented by the BRAGFLO program (Bean et al., 1996; Section 9, Vaughn et al., 2000; WIPP PA (Performance Assessment) Department, 1996) on the computational grid in Fig. 2. In the context of equations (7)–(9), the model for two-phase flow defined by equations (11)–(17) corresponds to f in equation (7); the finite difference procedure used to solve equations (11)– (17) on the computational grid in Fig. 2 corresponds to ˆf in equation (8); and the BRAGFLO program that actually caries out the numerical solution of equations (11)–(17) corresponds to F in equation (9).

6. Combining EN1 and EN2: complementary cumulative distribution functions (CCDFs) In a PA, EN1 (i.e., the probability space (Sst , Sst , pst )) provides a probabilistic characterization of what could happen at the facility under consideration, and EN2 (i.e., the function f indicated in equation (7)) provides an estimate of the consequences associated with individual futures xst . In many PAs, a representation of results is desired that provides a display of both consequence and likelihood. The usual mathematical construction used to display this information is a CCDF, which provides a display of the probability that individual consequence values will be exceeded (Fig. 3).

Fig. 3. Example CCDF: plot of probability prob(c > C) that a consequence with a value larger then C will occur.

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A CCDF can be formally defined by an integral involving EN1 and EN2. Specifically, 

 prob(c > C) = δC f (xst ) dst (xst ) dVst, (18) Sst

where prob(c > C) is the probability that a future xst yielding a consequence value larger than C will occur, dst (xst ) is the density function associated with (Sst , Sst , pst ), dVst is an incremental (i.e., differential) volume from Sst , 

 1 if f (xst ) > C δC f (xst ) = 0 if f (xst )  C and a nonbold representation for f is used because a single consequence (i.e., analysis outcome) is considered in the construction of a CCDF (i.e., f is real valued rather than vector valued in this context). The effect of the indicator function δC is to pick out the subset of Sst for which f (xst ) > C, with the result that the integral yields the desired probability. To be fully general, the preceding integral should be shown as a Lebesgue integral (App. B, Sokolnikoff & Redheffer, 1966); however, notation associated with the more familiar Riemann integral has been used for simplicity. In practice, the integral in equation (18) is too complex to permit a closed-form evaluation. Part of this complexity comes from the properties of (Sst , Sst , pst ) (e.g., see equation (6)) and part of it comes from the complexity of f (e.g., see equation (10)). This complexity typically precludes the use of standard numerical procedures for evaluating integrals. In practice, either Monte Carlo techniques or importance sampling is used in the evaluation of the integral in equation (11). With the Monte Carlo approach, the integral in equation (18) is approximated by prob(c > C) = ˙

nR 



δC f (xst,i ) /nR,

(19)

i=1

where xst,i , i = 1, 2, . . . , nR, is a random sample from Sst generated in consistency with the definition of (Sst , Sst , pst ) (Fig. 4). In practice, generating a random sample from Sst in consistency with the definition of (Sst , Sst , pst ) means generating a sample using the distributions assigned to the individual elements of xst (e.g., see xst in equation (6) and associated discussion). In most PAs, evaluation of f (xst ) is too expensive to permit an evaluation for each sample element xst,i in equation (19). Rather, f (xst) is evaluated for a relatively small number of elements of Sst and then these evaluations are used in the construction (i.e., approximation) of f (xst,i ) for use in the summation in equation (19). Thus, there are actually three numerical approximations involved: (i) the Monte Carlo procedure to approximate the integral in equation (18), (ii) an initial approximation of f (xst ) for selected elements of Sst , and (iii) a subsequent approximation of f (xst,i ) from these initial approximations for use in the summation in equation (19). As an example, the procedures used in the 1996 WIPP PA to estimate the CCDF for

 f (xst ) = fSP xst , fB (xst ) (20)

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Fig. 4. Approximation of a CCDF with a random sample of size 1000.

(i.e., the CCDF for the spallings release calculated by the program CUTTINGS_S) are described in Section 9 of Helton et al. (1999) and also in Section 10 of Berglund et al. (2000). With importance sampling, Sst is divided into strata Sst,i , i = 1, 2, . . . , nS, and a value xst,i , i = 1, 2, . . . , nS, is selected from each strata. Then, the integral in equation (18) is approximated by prob(c > C) = ˙

nS 

 δC f (xst,i ) pst (Sst,i ),

(21)

i=1

where pst (Sst,i ) is the probability of Sst,i . When the xst,i are randomly selected within the corresponding strata, importance sampling is a variance reduction technique for simple random (i.e., Monte Carlo) sampling. Importance sampling involves the same approximation considerations as the Monte Carlo approach described in conjunction with equation (19). The fault tree and event tree techniques (Breeding et al., 1992; Vesely et al., 1981) used extensively in conjunction with PAs for nuclear reactors and other complex engineered systems can be viewed as algorithms for defining an importance sampling procedure over the sample space associated with all possible accidents at these facilities. Several early PAs for the WIPP used an importance sampling procedure in CCDF construction (Helton & Iuzzolino, 1993). Another possible display of the combined effects of EN1 and EN2 is given by the expected value and variance of a consequence of interest. The expected value E(f ) and variance V (f ) of a particular consequence are formally defined by  E(f ) =

Sst

f (xst )dst(xst ) dVst

(22)

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and  V (f ) =

Sst

2 f (xst) − E(f ) dst (xst ) dVst

(23)

and are easily estimated with the results used in the CCDF approximations in equations (19) and (21). However, as a large amount of information is lost in the calculation of E(f ) and V (f ), the combined effects of EN1 and EN2 are better displayed as CCDFs. Additional information related to the ideas discussed in this section in the context of the 1996 WIPP PA is available in Helton et al. (2000a).

7. EN3: probabilistic characterization of parameter uncertainty The entity EN3 is the formal outcome of the data development component of a PA and provides a probabilistic characterization of the uncertainty in the many parameters required in the definitions of EN1 (e.g., occurrence rates, parameters in distributions, . . .) and EN2 (e.g., distribution coefficients, solubilities, . . .). The term parameter in the description of EN3 is used in a sense that is sufficiently broad to include designators for alternative models or model structures. Formally, EN3 is defined by a probability space (Ssu , Ssu , psu ), with the subscript ‘su’ selected to indicate subjective (i.e., epistemic) uncertainty. The sample space Ssu associated with EN3 has the form Ssu = {xsu : xsu is possibly the correct vector of inputs to use with EN1 and EN2 in the calculation of analysis outcomes}. (24) Further, xsu has the form xsu = [x1 , x2 , . . . , xnV ],

(25)

where each element xi , i = 1, 2, . . . , nV , of xsu is quantity required in the formulation of EN1 or EN2 that is assumed to have a fixed value in the context of the particular PA under consideration but with this value being imprecisely known. The basic idea is that the PA and the associated definitions of EN1 and EN2 have been developed to the point that appropriate analysis outcomes would be obtained if the appropriate value of xsu was used as input to the analysis. As an example, the representation for the CCDF in equation (18) would now become  prob(c > C|xsu) =

Sst (xsu )

 δC f (xst, xsu ) dst(xst |xsu ) dVst,

(26)

with the addition of xsu to the notation for various quantities in equation (18) indicating that the values and/or predictions associated with these quantities are dependent on the value assigned to xsu . Thus, different values for xsu lead to different values for the CCDF in equation (18) and other analysis outcomes of interest (Fig. 5). The sample space Ssu contains all

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Fig. 5. Different values of CCDF defined by equation (18) that derive from different values of xsu .

values for xsu consistent with available information about the facility under consideration and thus leads to all possible values for the CCDF defined in equation (18) and other analysis outcomes of interest. The probability space (Ssu , Ssu , psu ) that formally defines EN3 provides a characterization of the uncertainty in the elements of xsu . In practice, (Ssu , Ssu , psu ) is defined by specifying a distribution Dj , j = 1, 2, . . . , nV ,

(27)

for each element xj of xsu . Correlations and various other restrictions involving the possible values of xj can also be specified. The purpose of the Dj is to provide, on the basis of all available information, a quantitative (i.e., probabilistic) description of where the possible values for the xj are located. In contrast, Ssu only provides a complete compilation of the possible values for the xsu and thus for the xj . In some instances, it may be possible to use formal statistical procedures to estimate the Dj . However, typically some type of expert review process is needed to develop the Dj (Bonano & Apostolakis, 1991; Cooke, 1991; Hora & Iman, 1989; Keeney & von Winterfeldt, 1991; Meyer & Booker, 1991; Mosleh et al., 1988; Thorne, 1993). As an example, the 1996 WIPP PA considered nV = 57 uncertain variables, with the result that xsu had the form xsu = [x1 , x2 , . . . , x57 ] = [ANHPRM, BHPRM, . . . , WTAUFAIL].

(28)

Example elements of xsu are given in Table 3. In concept, elements of xsu can affect the definition of EN1 (i.e., the probability space (Sst , Sst , pst )) or the definition of EN2 (i.e., the function

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Table 3 Example elements of xsu in the 1996 WIPP PA (see App. PAR, U.S. DOE (U.S. Department of Energy), 1996, and Table 1, Helton et al., 2000c for a complete listing of the nV = 57 elements of xsu and sources of additional information) ANHPRM – Logarithm of anhydrite permeability (m2 ). Used in BRAGFLO. Distribution: Student’s t-distribution with 5 degrees of freedom. Range: –21.0 to –17.1 (i.e., permeability range is 1 ×10−21 to 1 ×10−17.1 m2 ). Mean, Median: –18.9, –18.9. Correlation : –0.99 rank correlation with ANHCOMP (Bulk compressibility of anhydrite, Pa−1 ). BHPRM – Logarithm of borehole permeability (m2 ). Used in BRAGFLO. Distribution: Uniform. Range: −14 to −11 (i.e., permeability range is 1 × 10−14 to 1 × 10−11 m2 ). Mean, median: −12.5, −12.5. BPCOMP – Logarithm of bulk compressibility of brine pocket (Pa−1 ). Used in BRAGFLO. Distribution: Triangular. Range: –11.3 to –8.00 (i.e., bulk compressibility range is 1 × 10−11.3 to 1 × 10−8 Pa−1 ). Mean, mode: −9.80, −10.0. Correlation: –0.75 rank correlation with BPPRM (Logarithm of brine pocket permeability, m2 ). BPINTPRS – Initial pressure in brine pocket (Pa). Used in BRAGFLO. Distribution: Triangular. Range: 1.11 × 107 to 1.70 × 107 Pa. Mean, mode: 1.36 × 107 Pa, 1.27 × 107 Pa. CFRCSP – Culebra fracture spacing (m). Used in SECOTP2D. Equal to half the distance between fractures. Distribution: Uniform. Range: 0.05 to 0.5 m. Mean, median: 0.275 m, 0.275 m. HALPOR – Initial value for halite porosity (dimensionless). Used in BRAGFLO. Distribution: Piecewise uniform. Range: 1.0 × 10−3 to 3 × 10−2 . Mean, median: 1.28 × 10−2 , 1.00 × 10−2 . HALPRM – Logarithm of halite permeability (m2 ). Used in BRAGFLO. Distribution: Uniform. Range: −24 to −21 (i.e., permeability range is 1 × 10−24 to 1 × 10−21 m2 ). Mean, median: −22.5, −22.5. Correlation: −0.99 rank correlation with HALCOMP (Bulk compressibility of halite, Pa−1 ). WASTWICK – Increase in brine saturation of waste due to capillary forces (dimensionless). Used in BRAGFLO. Distribution: Uniform. Range: 0 to 1. Mean, median: 0.5, 0.5. WGRCOR – Corrosion rate for steel under inundated conditions in the absence of CO2 (m/s). Used in BRAGFLO. Distribution: Uniform. Range: 0 to 1.58 × 10−14 m/s. Mean, median: 7.94 × 10−15 m/s, 7.94 × 10−15 m/s. WMICDFLG – Pointer variable for microbial degradation of cellulose. Used in BRAGFLO. Distribution: Discrete, with 50% 0, 25% 1, 25% 2. WMICDFLG = 0, 1, 2 implies no microbial degradation of cellulose, microbial degradation of only cellulose, microbial degradation of cellulose, plastic and rubber. WPRTDIAM – Waste particle diameter (m). Used in CUTTINGS_S. Distribution: Loguniform. Range: 4.0 × 10−5 to 2.0 × 10−1 m. Mean, median: 2.35 × 10−2 m, 2.80 × 10−2 m. WRBRNSAT – Residual brine saturation in waste (dimensionless). Used in BRAGFLO and BRAGFLO_DBR. Distribution: Uniform. Range: 0 to 0.552. Mean, median: 0.276, 0.276. WTAUFAIL – Shear strength of waste (Pa). Used in CUTTINGS_3. Distribution: Uniform. Range: 0.05 to 10 Pa. Mean, median: 5.03 Pa, 5.03 Pa.

f in equation (7)). In the 1996 WIPP PA, the elements of xsu only affected EN2. However, in a subsequent PA carried out at the request of the EPA, the probability of penetrating pressurized brine in the Castile Formation was treated as being uncertain (see bi in equation (6)) and so the analysis incorporated subjective uncertainty in the definition of (Sst , Sst , pst ) (MacKinnon & Freeze, 1997).

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For a specific future xst and a single analysis outcome f (xst , xsu ), the resultant distribution of possible values for f (xst , xsu ) is given by 

 δC f (xst, xsu ) dsu (xsu) dVsu, (29) prob(c > C|xst) = Ssu

where prob(c > C|xst ) is the probability that f (xst , xsu) will have a value larger than C due to the uncertainty in xsu as characterized by (Ssu , Ssu , psu ) and the remaining terms are defined similarly to those in equations (18) and (26) except that (Ssu , Ssu , psu ) rather than (Sst , Sst , pst ) is now under consideration. In concept, this distribution has the same appearance as in Fig. 3 except that C = f (xst, xsu ) appears on the abscissa and prob(c > C|xst ) appears on the ordinate. In practice, closed form evaluation of prob(c > C|xst ) is not practicable. For example, evaluation of f (xst , xsu) could require the solution of a complex system of nonlinear partial differential equations as is the case for the BRAGFLO model in Table 2. Typically, a sampling-based approach is used to approximate prob(c > C|xst ). For example, a random sample xsu,i , i = 1, 2, . . . , nS, generated in consistency with the definition of (Ssu , Ssu , psu ) (i.e., in practice, in consistency with the distributions in equation (27) and any associated restrictions) leads to the approximation ˙ prob(c > C|xst) =

nS 

 δC f (xst, xsu,i ) /nS.

(30)

i=1

The resultant approximation is similar in appearance to that shown in Fig. 4 except that C = f (xst , xsu,i ) appears on the abscissa and prob(c > C|xst ) appears on the ordinate. The probability space (Ssu , Ssu , psu ) leads to distributions of CCDFs resulting from stochastic uncertainty as indicated in Fig. 5. When viewed formally, such distributions are defined by double integrals involving (Sst , Sst , pst ), f , and (Ssu , Ssu , psu ). The formal notation for such integrals is messy (Helton, 1996b) and will not be presented here. As previously indicated, integrals involving (Sst , Sst , pst ) and f are often evaluated with numerical procedures based on random sampling or importance sampling. Integrals involving (Ssu , Ssu , psu ) are often evaluated with procedures based on Latin hypercube sampling (Iman, 1992; McKay et al., 1979) because of its efficient stratification properties. In particular, Latin hypercube sampling shows less sampling variability than simple random sampling (McKay et al., 1979; Owen, 1992; Stein, 1987) and has been observed to produce stable results in several large analyses (Helton et al., 1995; Iman & Helton, 1988; Iman & Helton, 1991). Latin hypercube sampling operates in the following manner to generate a sample of size nLHS from nV variables. The range of each variable (i.e., the xj in equation (25)) is divided into nLHS intervals of equal probability and one value is selected at random from each interval. The nLHS values thus obtained for x1 are paired at random and without replacement with the nLHS values obtained for x2 . These nLHS pairs are paired at random and without replacement with the nLHS values of x3 to form nLHS triples. This process is continued until a set of nLHS nV-tuples is formed. These nV-tuples are of the form xsu,i = [x1i , x2i , . . . , xnV ,i ],

i = 1, . . . , nLHS,

(31)

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Fig. 6. Example of an LHS of size nLHS = 10 from variables U and V with U normal on [−1, 1] (mean = 0, 0.01 quantile = −1, 0.99 quantile = 1) and V triangular on [0, 4] (mode = 1) (Fig. 4, Helton, 1999).

and constitute the Latin hypercube sample (LHS). The individual xj must be independent for the preceding construction procedure to work; however, a method for generating LHSs from correlated variables has been developed by Iman and Conover (Iman & Conover, 1982; Iman & Davenport, 1982). As illustrated in Fig. 6 for an LHS from two variables, Latin hypercube sampling results in a dense stratification across the range of each variable (i.e., there is one value from each equal probability interval). In the 1996 WIPP PA, three replicated (i.e., independently generated) LHSs of size nLHS = 100 from the variables indicated in equation (28) and Table 3 were used (Iman, 1982). Further, the Iman/Conover restricted pairing technique (Iman & Conover, 1982; Iman & Davenport, 1982) was used to induce specified rank correlations between certain pairs of variables and also to ensure that all other pairs of variables had rank correlations close to zero. Each replicated sample was of the form xsu,i = [x1i , x2i , . . . , x57,i ],

i = 1, 2, . . . , nLHS = 100,

(32)

and pooling all three replicates (designated R1, R2, R3) resulted in a single sample of size 300. Replicated sampling provided a way to observe the stability of results obtained with Latin hypercube sampling. Specifically, it was possible to determine if similar results were obtained with the individual samples, which did indeed turn out to be the case. As a reminder, Latin hypercube sampling is being used to implement a numerical integration, and so it is the stability of this numerical integration that is being investigated with the replicated samples.

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Table 4 Mechanistic calculations performed as part of the 1996 WIPP PA (adapted from Table 4, Helton, 1999) BRAGFLO Individual calculations (6 cases; i.e., 6 different values for xst ): E0 (i.e., undisturbed conditions); E1 at 350, 1000 yr (i.e., drilling intrusion through repository that penetrates pressurized brine in the Castile Fm); E2 at 350, 1000 yr (i.e., drilling intrusion through repository that does not penetrate pressurized brine in the Castile Fm); E2E1 with E2 intrusion at 800 yr and E1 intrusion at 1000 yr. Total calculations: 6 nR nLHS = 6 · 3 · 100 = 1800. CUTTINGS_S Individual calculations (52 cases; i.e., 52 different values for xst ): Intrusion into lower waste panel in previously unintruded (i.e., E0 conditions) repository at 100, 350, 1000, 3000, 5000, 10,000 yr; Intrusion into upper waste panel in previously unintruded repository at 100, 350, 1000, 3000, 5000, 10,000 yr; Initial E1 intrusion at 350 yr followed by a second intrusion into the same waste panel at 550, 750, 2000, 4000 or 10,000 yr; Initial E1 intrusion at 350 yr followed by a second intrusion into a different waste panel at 550, 750, 2000, 4000 or 10,000 yr; Initial E1 intrusion at 1000 yr followed by a second intrusion into the same waste panel at 1200, 1400, 3000, 5000 or 10,000 yr; Initial E1 intrusion at 1000 yr followed by a second intrusion into a different waste panel at 1200, 1400, 3000, 5000 or 10,000 yr; same 23 cases for initial E2 intrusions as for initial E1 intrusions. Total calculations: 52 nR nLHS = 52 · 3 · 100 = 15, 600. BRAGFLO_DBR Same computational cases as for CUTTINGS_S. NUTS Individual calculations (15 cases; i.e., 15 different values for xst ): E0; E1 at 100, 350, 1000, 3000, 5000, 7000, 9000 yr; E2 at 100, 350, 1000, 3000, 5000, 7000, 9000 yr. Screening calculations: 5 nR nLHS = 1500. Total NUTS calculations: 594. Note: Screening calculations were initially performed for each LHS element (i.e., E0, E1 at 350 and 1000 yr, E2 at 350 and 1000 yr, which produces the multiplier of 5 in the calculation of the number of screening calculations) to determine if the potential for a radionuclide release existed, with a full NUTS calculation only being performed when such a potential existed. PANEL Individual calculations (7 cases; i.e., 7 different values for xst ): E2E1 at 100, 350, 1000, 2000, 4000, 6000, 9000 yr. Total calculations: 7 nR nLHS = 7 · 3 · 100 = 2100. Note: Additional PANEL calculations were also performed at 100, 125, 175, 350, 1000, 3000, 5000, 7500 and 10,000 yr for Salado-dominated brines and also for Castile-dominated brines to determine dissolved radionuclide concentrations for use in the determination of direct brine releases. SECOFL2D Individual calculations (2 cases; i.e., 2 different values for xst ): Partially mined conditions in vicinity of repository; Fully mined conditions in vicinity of repository. Total calculations: 2 nR nLHS = 2 · 3 · 100 = 600. SECOTP2D Individual calculations (2 cases; i.e., 2 different values for xst ): Partially mined conditions in vicinity of repository; Fully mined conditions in vicinity of repository. Total calculations: 2 nR nLHS = 2 · 3 · 100 = 600. Note: Each calculation is for four radionuclides: Am-241, Pu-239, Th-230, U-234. Further, calculations are done for unit releases at time 0 yr, which can then be used to construct transport results for the Culebra for arbitrary time-dependent release rates into the Culebra (Section 9, Ramsey et al., 2000).

Once the LHS is generated, PA results are calculated for each sample element. For EN2 and individual futures xst , this creates a mapping

 xsu,i , f(xst , xsu,i ) ,

i = 1, 2, . . . , nLHS,

(33)

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from uncertain inputs to results for specific futures. Typically, a relatively small number of futures is selected for detailed analysis of this type (e.g., 5–10). As a reminder, f is usually quite complex and in practice is composed of many models (e.g., see equation (10) and Table 2). Further, all models represented by f may not be used in the analysis of all futures selected for consideration. In particular, different models or suites of models might be used to analyze different futures at the facility under consideration or to calculate different outcomes of interest for a given future. As an example, the individual model calculations performed in the 1996 WIPP PA are indicated in Table 4. For CCDFs, a mapping of the form 

xsu,i , prob(c > C|xsu,i ) ,

i = 1, 2, . . . , nLHS,

(34)

is created, where prob(c > C|xsu,i ) defines the CCDF for some consequence of interest as indicated in equations (18) and (26) (i.e., a CCDF results when C is allowed to run across all possible values of the consequence). In most PAs, CCDFs are generated for a large number of consequences. The mappings in equations (33) and (34) provide the basis for both uncertainty analysis and sensitivity analysis. Uncertainty analysis designates the determination of the uncertainty in analysis outcomes that results from uncertainty in analysis inputs. Specifically, uncertainty analysis involves the determination of the effects of the uncertainty characterized by EN3 and thus is providing an answer to question Q4 as indicated in Section 1. Once the calculations that lead to the mappings in equations (33) and (34) are completed, the presentation of uncertainty analysis results simply involves the plotting of the predicted results with an equal weight (i.e., the reciprocal of the sample size) given to the individual results when either random or Latin hypercube sampling is used in the generation of the sample in equation (31). Sensitivity analysis designates the determination of the effects of individual variables contained in xsu on the observed uncertainty in predictions of interest. For sampling-based propagations of uncertainty, sensitivity analysis involves an exploration of the mappings in equations (33) and (34) with techniques based on examination of scatterplots, correlation analysis, regression analysis, and the identification of nonrandom patterns (Hamby, 1994; Helton, 1993b; Kleijnen & Helton, 1999). Additional information related to the ideas discussed in this section in the context of the 1996 WIPP PA is available in Helton et al. (2000c).

8. Example results from PAs for the WIPP 8.1. Two-phase flow in vicinity of repository The representation of two-phase flow in the vicinity of the repository in analyses for the WIPP is typical of the modeling problems encountered in a PA for a waste disposal site or some other facility. As discussed in conjunction with equations (11)–(17), the model for two-phase flow consists of three components: (i) a formal mathematical description of the model as a system of nonlinear partial differential equations (Section 2, Vaughn et al., 2000), (ii) a numerical solution procedure based on finite difference techniques (Section 9, Vaughn et al., 2000),

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Fig. 7. Uncertainty and sensitivity analysis results for pressure (Pa) in waste panel under undisturbed (i.e., E0) conditions: (7a) time-dependent pressures for 100 LHS elements in replicate R1 (left frame), and (7b) PRCCs obtained from analysis of all 300 LHS elements associated with replicates R1, R2 and R3 (right frame) (Fig. 5, Helton, 1999).

and (iii) a computer program, BRAGFLO, that actually implements the numerical procedure and carries out various input and output processing activities (WIPP PA (Performance Assessment) Department, 1996). For convenience, this model will be referred to as BRAGFLO, although technically BRAGFLO is just the computer program that implements a particular numerical solution procedure for the model. As is typical of many programs, BRAGFLO can implement a variety of models through appropriate specification of input. The use of this model must be supported by the development of appropriate data (Bean et al., 1996; Larson, 2000; Vaughn et al., 2000). Further, there exists considerable uncertainty in many of the inputs used by the model (e.g., see Table 3). In the context of the entities EN1, EN2 and EN3 introduced in Section 1, the two-phase flow model is part of EN2, and the characterization of the uncertainty in the inputs used by this model is part of EN3. Repository pressure under undisturbed conditions provides an example of the type of results calculated by BRAGFLO (Fig. 7a), with each curve resulting from a single calculation for one of the LHS elements associated with replicate R1 (see equation (32)). The spread of the curves is resulting from subjective uncertainty and is thus providing a representation of state of knowledge uncertainty with respect to predicted pressure in the repository. As Fig. 7a contains results for 100 LHS elements, its generation required solution of the underlying system of nonlinear partial differential equations 100 times. Thus, the numerical algorithm in use must be sufficiently efficient to allow this number of calculations. Further, the algorithm must also be reasonably robust in the sense of not requiring large amounts of analyst intervention to produce a suitably converged solution. Such intervention is acceptable for a few solutions but would result in unacceptable requirements for analyst time if it was required for a large number of solutions. Sensitivity analysis provides a way to determine which of the uncertain components of xsu are actually giving rise to the uncertainty in pressure shown in Fig. 7a. One way to make this determination is by calculating partial rank correlation coefficients (PRCCs) between

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Fig. 8. Example of estimated CDF and CCDF for repository pressure at 10 000 yr under undisturbed conditions (i.e., y = WAS_ PRES) obtained from the 300 LHS elements that result from pooling replicates R1, R2 and R3 (see equation (32)).

the elements of xsu and the pressures above individual times on the abscissa of Fig. 7a. The resultant PRCCs can then be plotted to produce a time-dependent display of variable importance (Fig. 7b). As indicated in Fig. 7b, the uncertainty in pressure is dominated by the inputs WMICDFLG, WGRCOR, WASTWICK and HALPOR, with pressure tending to increase as each of these variables increases. Many additional techniques for sensitivity analysis are also available (Hamby, 1994; Helton, 1993b; Kleijnen & Helton, 1999). As described in conjunction with equations (29) and (30), the uncertainty associated with xsu leads to a distribution of possible analysis outcomes. For time-dependent repository pressure, this distribution is in essence shown by a plot of the resultant pressure curves (Fig. 7a). The more formal representation associated with equations (29) and (30) can be calculated for individual times on the abscissa of Fig. 7a (e.g., at 10 000 yr as illustrated in Fig. 8). The curve labeled CCDF in Fig. 8 corresponds to a plot of the approximation to prob(c > C|xst) defined in equation (30), with the consequence associated with c and C corresponding to repository pressure at 10 000 yr, the future xst corresponding to undisturbed conditions (i.e., no human disruption of the repository), and the summation taken over the nS = 300 LHS elements that result from pooling replicates R1, R2 and R3 (see equation (32)). The results summarized in Fig. 8 are for pressure at a single time (i.e., 10 000 yr). For results of the form in Fig. 7a, selected quantiles (e.g., 0.10, 0.50, 0.90) and also the expected value can be calculated at selected times on the abscissa and then connected to form curves that

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Fig. 9. Mean and quantile curves for three replicated LHSs for pressure in lower waste panel under undisturbed conditions (Fig. 6, Helton, 1999).

present quantiles and means as a function of time (Fig. 9). The indicated calculations involve evaluating the results in Fig. 8 at a sequence of times. The results in Fig. 9 actually involve three sets of curves, with each set having been generated from one of the three replicated LHSs of size nLHS = 100 each (see equation (32)). The purpose of replicating the LHSs was to assess the stability of the associated integration over the probability space (Ssu , Ssu , psu ) in the 1996 WIPP PA. As the closeness of the individual curves in Fig. 9 shows, the pressure results obtained with an LHS of size 100 were quite stable. Indeed, there were no cases in the 1996 WIPP PA when the individual replicates would have lead to different assessments of the potential behavior of the site. 8.2. Construction of CCDFs As discussed in conjunction with equations (18), (19) and (21), construction of CCDFs resulting from stochastic (i.e., aleatory) uncertainty can be viewed as a numerical integration problem involving the probability space (Sst , Sst , pst ) and models that predict the analysis result under consideration. Thus, construction of a CCDF involves both EN1 and EN2. Further, as discussed in conjunction with equation (26), different values for the uncertain input xsu lead to different CCDFs, with a distribution of CCDFs resulting from the probability space (Ssu , Ssu , psu ) that corresponds to the entity EN3. Although a number of different release pathways to the accessible environment were considered in the 1996 WIPP PA, only the direct release pathways (i.e., cuttings and cavings,

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Fig. 10. Distributions of CCDFs resulting from subjective uncertainty and obtained with replicate R1 for the following release modes: (10a) cuttings and cavings, (10b) spallings, (10c) direct brine release, and (10d) total (i.e., cuttings and cavings, spallings, direct brine) (Figs 6, 9, Helton et al., 1998b)). The statements in Frames 10b and 10c that observations are offscale means that the largest normalized release for each of these observations is less than 10−5 EPA units, with the largest release for each of these observations typically being zero.

spallings, and direct brine release) produced nonzero releases. As a result, the function f in equation (10) reduces to



   (35) f (xst ) = fC (xst ) + fSP xst , fB (xst ) + fDBR xst , fSP xst , fB (xst ) , fB (xst ) , and CCDFs were constructed for fC , fSP , fDBR and the total release f (Fig. 10). As discussed in Section 6, the individual CCDFs resulted from a numerical integration over (Sst , Sst , pst ) based on Monte Carlo procedures. Further, the multiple CCDFs in each frame resulted from a numerical integration over (Ssu , Ssu , psu ) based on Latin hypercube sampling (see equation (32)).

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Fig. 11. Mean and quantile curves for distribution of total release CCDFs in Fig. 10d (Fig. 7, Helton et al., 1998b).

The primary regulatory requirement placed on the WIPP is that the CCDF for total release to the accessible environment (Fig. 10d) fall below the boundary line designated “EPA Limit” in Fig. 10 (p. 38086, U.S. EPA (U.S. Environmental Protection Agency), 1985). Further, an additional regulation specifies that the uncertainty in the value of this CCDF is to be determined and presented (pp. 5242–5243, U.S. EPA (U.S. Environmental Protection Agency), 1996a). As examination of Fig. 10 shows, the CCDFs fall below the EPA limit when the effects of subjective uncertainty are assessed and propagated through the analysis. The EPA limit appearing in Fig. 10 is an example of what is sometimes called the Farmer limit line approach to the definition of acceptable risk (Cox & Baybutt, 1982; Farmer, 1967; Helton, 1993a; Munera & Yadigaroglu, 1986). Mean and quantile curves were used to summarize the distribution of pressure curves (Figs 7a, 9). A similar summary can be used for a distribution of CCDFs and provides a more quantitative summary of where the CCDF under consideration is believed to be located than is obtained from a visual examination of the corresponding distribution of CCDFs (Fig. 11). Formally, the results in Fig. 11 can be represented as a double integral involving (Sst , Sst , pst ) and (Ssu , Ssu , psu ), with these results being approximated with procedures based on random sampling and Latin hypercube sampling as previously described. The replicated LHSs discussed in conjunction with equation (32) can also be used to examine the stability of the estimated CCDF distributions. The least stable estimates were obtained for the direct brine releases (Fig. 12). However, even in this case, the numerical variability in the estimated distribution is unimportant given the wide separation of the distribution from the EPA limit. The estimates for the other CCDF distributions were very stable, with the individual replicates producing mean and quantile curves that were almost indistinguishable.

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Fig. 12. Outcome of replicated sampling for distribution of CCDFs for normalized release to the accessible environment over 10 000 yr due to direct brine release (Fig. 12, Helton, 1999).

9. Summary A PA for a radioactive waste disposal site or some other facility is a complex mathematical calculation. In such a calculation, it is important to have a clear view of both the formal mathematical components on which the PA is based and the numerical procedures used in the approximation of these components. Appropriate numerical procedures and software that implements these procedures are essential parts of a PA. However, it is also important to maintain the conceptual distinction between a mathematical model and the numerical procedures used to implement that model. Significant uncertainties exist in most PAs. Failure to acknowledge and represent this uncertainty can result in serious criticism of a PA. Indeed, a PA that does not provide a representation of the uncertainty in its outcomes is incomplete. As discussed, most PAs involve both stochastic (i.e., aleatory) and subjective (i.e., epistemic) uncertainty. The incorporation of uncertainty into a PA is essentially a multidimensional integration problem and can place significant computational burdens on the analysis. Again, good numerical procedures and appropriate computational strategies are essential. The resolution required in both models and numerical procedures is influenced by the uncertainty associated with the data that underlie a PA. Ultimately, the quality of model predictions is constrained by the quality of both the models in use and the available data to support these models. Although erroneous or nonconverged numerical procedures produce results of no value, acceptable errors in appropriately operating numerical procedures are, to some extent, determined by the uncertainty in model predictions that results from lack of resolution

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in the model or uncertainty in its inputs. Uncertainty and sensitivity analyses are important components of a PA and help provide guidance on both the uncertainty in outcomes of interest and where to best invest additional effort to reduce this uncertainty. The modeling of physical processes is often done on two levels in a PA. First, very detailed models are used to develop and demonstrate an understanding of particular processes. Such models are often conceptually complex and numerically demanding and help identify the essential physical processes that must be considered in the final, integrative calculations of the PA. Second, less detailed models that embody the essential processes identified in prior analyses are often used in the final, integrative calculation. These models are less computationally intensive and serve to incorporate information gained from the more detailed models and other sources into the PA. Reducing computational cost through the use of good numerical procedures and appropriate computational strategies is an important part of any large PA. Designing the PA in a way that will facilitate its implementation, documentation and review is also an important consideration. In most PAs, the cost of the human time to implement, document and review the analysis is much greater than the cost of the computation itself. Quality documentation is an essential part of a successful PA. Such documentation requires a clear conceptual understanding of the PA and the role that various numerical procedures play in the computational implementation of the PA. Most PAs underestimate the time and resources required to produce adequate documentation. Yet, without such documentation, much of the value of a PA is lost. Performance assessment should be an iterative process and should be initiated early in the analysis of a particular site or facility. Initially, PA can be primarily exploratory and relatively simple. Early PA efforts provide opportunities for (i) interactions between experimentalists and modelers, (ii) uncertainty and sensitivity analysis, (iii) development of insights with respect to the systems under study, (iv) outside review, (v) regulator and stakeholder involvement, (vi) initiation of a Quality Assurance (QA) program, and (vii) education of analysis participants on what to expect in later analyses. From such efforts, guidance emerges on (i) experimental programs and data development, (ii) model development, including the appropriateness of more or less complex models, and (iii) computational structure for a fully integrated analysis. Intermediate PAs are typically more complex than initial PAs. At times, intermediate PAs are pushed towards unnecessary complexity or complexity that exceeds available data by model developers or outside reviewers. However, PAs tend to improve through multiple iterations as (i) more data becomes available or increased understanding improves the match between available data and model requirements, (ii) the appropriateness and implementation of individual models improves, (iii) the design and assembly of the entire PA improves due to increased understanding of both the overall system and results obtained from individual models, and (iv) the benefits of an appropriately implemented QA program are realized. In addition, continued iterations provide opportunities for (i) review of alternative models, (ii) uncertainty and sensitivity analysis, (iii) identification of errors in analysis or model implementation, and (iv) continued education of analysis participants, regulators, and stakeholders. The final PA (e.g., a PA supporting a regulatory decision) benefits from the experiences of the prior analyses, including (i) a well-defined and well-tested analysis system that will produce no surprises in the final analysis, (ii) an analysis focused on recognized needs and

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requirements, (iii) use of models appropriate for the requirements of the analysis, including use of simplified models when prior analyses have indicated that this is acceptable due to either analysis requirements or resolution in available data, (iv) QA procedures emplaced and employed with models and an analysis system that have been previously used and tested, and (v) analysts, reviewers, regulators and stakeholders who are familiar with the analysis due to prior involvement.

Acknowledgements Work performed for Sandia National Laboratories (SNL), which is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000. Many individuals at SNL and its contractors contributed to the results in this presentation, including D. R. Anderson, K. Aragon, J. E. Bean, D. G. Bennet, J. W. Berglund, R. Blaine, M. B. Crawford, F. J. Davis, K. Economy, D. A. Galson, J. W. Garner, F. D. Hansen, T. W. Hicks, J. D. Johnson, H.-N. Jow, M. K. Knowles, K. W. Larson, M. E. Lord, R. J. MacKinnon, M. G. Marietta, J. Miller, D. G. O’Brien, J. L. Ramsey, L. C. Sanchez, J. D. Schreiber, J. Schatz, A. Shinta, L. N. Smith, D. M. Stoelzel, C. Stockman, P. N. Swift, T. W. Thompson, M. S. Tierney, P. Vaughn, M. Wallace, M. Williamson and R. D. Wilmot. Reivew provided at SNL by K. Byle, C. Crawford and M. J. Shortencarier. Editorial support provided by L. Harrison, T. Allen and H. Radke of Tech Reps, Inc.

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Chapter 13

The Chernobyl accident and aquatic biota Ivan I. Kryshev, Tatiana G. Sazykina∗ , Alexander Kryshev Scientific & Production Association “Typhoon”, 82 Lenin Ave., Obninsk, Kaluga Region, 249020 Russia

1. Introduction This chapter emphasizes the modelling of radioecological consequences of the Chernobyl accident for aquatic biota in the cooling pond of the Chernobyl Nuclear Power Plant. The “Cooling Pond” scenario was developed by the Post Chernobyl Data Working Group of the International BIOMOVS II Programme (Biospheric Model Validation Study, Phase II) (BIOMOVS II, 1996; Kryshev et al., 1996; Hoffman et al., 1996). The objective of the “Cooling Pond” Scenario (Scenario CP) is to test models for radioactive contamination of aquatic ecosystems. The Scenario is based on data from the Chernobyl Nuclear Power Plant (NPP) cooling pond, which was contaminated in 1986 as a result of damage to Unit 4 of the NPP. One of the most heavily contaminated water bodies in the world, the cooling pond is a suitable model for studying the dynamic processes of radionuclide migration and accumulation in closed aquatic systems (Kryshev, 1992). Dynamic processes worthy of detailed investigation include 137 Cs accumulation in different trophic levels, the accumulation and removal of 137 Cs from fish flesh, and the dependence on size of the accumulation of 137 Cs by various species of fish. The calculation endpoints of the Scenario are in two stages. In Stage I of the Scenario, the endpoints are model predictions of the dynamics of ecosystem contamination with 137 Cs. This long-lived, biologically active radionuclide determines the radioecological consequences of the Chernobyl accident for the cooling pond over a long time period. For this stage of the Scenario, model predictions can be tested using the existing data base of 137 Cs concentrations in water, sediments and biota. In Stage II, the endpoints are dose and risk assessment calculations, both for aquatic biota and for humans. This stage provides modelers with the opportunity to compare dose and risk estimates made with different dosimetric models, using the same input information. Of particular interest is the assessment of doses received in the first two years after the accident. During this period a wide set of radionuclides contributed to the cooling pond contamination; taken together, their contribution was greater than that from cesium isotopes. * E-mail: [email protected]

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2. The “Chernobyl cooling pond” scenario description The input data presented in Scenario CP include characteristics of the cooling pond ecosystem and estimates of the amounts of different radionuclides in the Chernobyl cooling pond as of 30 May 1986. The input data for dose assessment include measured concentrations of the most important radionuclides in water, sediments, and biota for several years after the accident. 2.1. Ecological conditions The Chernobyl NPP, consisting of 4 units of 1 GW each, is located in the eastern part of the Polesye, Ukraine, on the bank of the Pripyat River, which flows down to the Kiev reservoir. Water is supplied to the Chernobyl NPP from a cooling pond which is located southeast of the plant site and which was made by cutting off part of the river flood plain with a dam (Kaftannikova et al., 1987; Protasov et al., 1991). The cooling pond resembles an oval in shape. There is a water intake canal in the western part of the pond and a water discharge canal in the southern part. The morphometric characteristics of the cooling pond are summarized in Table 1. The cooling pond has no proper outflow. Its water reaches the Pripyat River by filtration through a dam, with a discharge flow of 1.2 × 105 m3 /y. The thermal regime (Table 2) of the pond is strongly influenced by the release of heated waters; their discharge flow was 206 m3 s−1 in 1985. The strip of land which runs through the middle of the cooling pond acts as a dam, separating the heated waters from the cooler waters. The transfer of water from the discharge canal to the water intake canal takes about 8 days. Water losses from the cooling pond are compensated for using a water-pumping station located on the bank of the Pripyat River. The water of the cooling pond is moderately mineralized (260–430 mg l−1 ) and hard (2.50– 4.35 mg equivalent per liter) (Table 3). The average concentration of potassium in the cooling pond water is 4 mg l−1 . The naturally occuring concentration of suspended material in the water is about 10 g m−3 (for no-wind conditions). When all four units at Chernobyl were running, operation of circulation pumps would lead to an increase in the concentration of suspended material in the cooling pond water by 15–20 g m−3 , with maximum values up to 25–30 g m−3 .

Table 1 Morphometric characteristics of the Chernobyl NPP cooling pond Characteristic

Value

Volume (m3 ) Area (km2 ) Length (km) Width, average (km) Depth, maximum (m) Depth, average (m)

1.5 × 108 22 11.4 2.0 20 6.6

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Table 2 Temperature of water in the cooling pond (natural conditions) Month

Temperature (◦ C)

Month

Temperature (◦ C)

January February March April May June

0.0 0.1 0.4 ± 0.2 8.1 ± 1.3 14.0 ± 1.6 19.6 ± 1.3

July August September October November December

21.5 ± 0.7 24.5 ± 0.4 15.6 ± 0.8 9.0 ± 2.8 4.3 ± 2.6 0.7 ± 0.6

Note. The temperature of water in the heated zone was higher than the natural temperature by 9.4 ± 1.6◦ C. Table 3 Hydrochemical parameters of the cooling pond Parameter

Range

Average

Total ions (mg/l) Total hardness (mg/l) Sulphates (mg/l) Chlorides (mg/l) pH Iron (mg/l) Ca (mg/l) Mg (mg/l) K (mg/l) Oxygen (mg/l) Mineral phosphorus (mg/l) Organic phosphorus (mg/l) Nitrate (mg/l) Ammonia (mg/l) Organic nitrogen (mg/l)

250–350 2.90–4.16 10–53 14–28 6.5–8.4 0.1–1.3 40–53 5–9 3–5 1.6–16.5 0.01–0.51 0.01–0.68 0.05–2.34 0.15–3.46 0.01–3.28

300 3.5 30 20 7.5 0.5 43 7 4 8.0 0.06 0.06 0.54 0.9 1.0

The Chernobyl NPP did not operate from May to September 1986, following the accident. Operation of Units 1–3 of the Chernobyl NPP was later resumed: Unit 1, in October 1986; Unit 2, in November 1986; and Unit 3, in December 1987. The pond serves as a habitat for more than 500 species and taxa of algae, over 200 species of invertebrates, and over 20 fish species; these include both nonpredators (carp, silver carp, silver bream, bream, roach, goldfish, goby) and predators (pike-perch, pike, perch, asp). The average biomass of different types of aquatic organisms in the cooling pond are presented in Table 4. The most typical and widespread species of aquatic organisms with a sufficiently representative number of samples (as a rule, not fewer than 20) were used for analysis. These include the following: • Aquatic plants: fennel-leaved pondweed (Potamogeton pectinatus); clasping-leaved pondweed (Potamogeton perfoliatus); spiked water milfoil (Myriophyllum spicatum); dark-

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Ivan I. Kryshev et al. Table 4 Annual averaged biomass of aquatic organisms in the cooling pond Aquatic organisms

Biomass (tonnes)

Bacterial plankton Phytoplankton Zooplankton Mollusks Zoobenthos Phytoperiphyton Fish (total) Nonpredatory fish Predatory fish

300 ± 160 1000 ± 600 270 ± 150 500 ± 200 10 ± 5 7±2 60 ± 40 50 ± 32 10 ± 6

green hornwort (Ceratophyllum spicatum); green filamentous alga (Cladophora glomerata Kuetz). • Mollusc: Dreissena bugensis; Anadonta piscinalis; Viviparus viviparus; bivalves of the Unionidae family (Unio pictorum, Unio timidus Phill, and Unio crassum Phill). • Fish: common bream (Abramis brama); silver bream (Blicca bjoerkna); roach (Rutilus rutilus); carp (Cyprinus carpio); goldfish (Carassius auratus gibelio); silver carp (Hypophthalmichthys molitrix); goby (Neogobius); perch (Perca fluviatilus); pike-perch (Stizostedion lucioperca); pike (Esox lucius). When calculating internal exposure to aquatic organisms, radionuclides are assumed to be distributed uniformly in the organisms; in each case the tissue density is assumed to be unity (Woodhead, 1976, 1979; Kryshev & Sazykina, 1986, 1990). The shape and size of an organism (generally in terms of a mean geometrical factor) must be considered for calculation of the absorbed dose rate from gamma radiation (Woodhead, 1976, 1979). The following default values for the geometric sizes of organisms were provided for dose calculation: (1) fish: cylinder 50 cm in length and 10 cm in diameter, g = 41 cm; (2) molluscs: flat cylinder 1 cm in height and 4 cm in diameter, g = 10 cm; (3) algae Cladophora: flat layer of 2 cm thickness. Specific geometric factors for cooling pond biota are as follows: bream 1 y of age, g = 23 cm; bream 5 y of age, g = 51 cm; pike-perch 1 y of age, g = 31 cm; pike-perch 5 y of age, g = 75 cm; adult silver carp, g = 94 cm. The age structure of the fish population in the cooling pond prior to the accident was given for pike-perch (percentage in the population): age 0–1 y, 13%; 1–2 y, 17%; 2–3 y, 36%; 3–4 y, 30%; older than 4 y, 4%. For the calculation of external exposure, two cases were considered: (1) the fish spends most of the time (> 70%) in water away from the bottom (for example, silver carp); (2) the fish resides mainly close to the bottom (for example, bream). 2.2. Radioactive contamination The reactor of Unit 4 at the Chernobyl NPP, a RBMK-1000 reactor, was put into operation in December 1983. At the time of the accident (26 April 1986), it had been in service for 865 d (IAEA, 1986; Borovoy et al., 1990).

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Table 5 Estimates of amounts of radionuclides in the Chernobyl cooling pond (decay-corrected to 30 May 1986) Radionuclide

Sediments (1012 Bq)

Water (1012 Bq)

137 Cs

110 ± 50 60 ± 40 860 ± 400 640 ± 280 220 ± 100 700 ± 360 1200 ± 500 1100 ± 400 400 ± 140 280 ± 120 30 ± 10 50 ± 20

60 ± 30 30 ± 15 30 ± 20 50 ± 30 20 ± 10 40 ± 15 50 ± 30 70 ± 40 120 ± 70 80 ± 40 250 ± 60 6±4

134 Cs 144 Ce 141 Ce 106 Ru 103 Ru 95 Zr 95 Nb 140 Ba 140 La 131 I 90 Sr

Primary contamination of the cooling pond as a result of the Chernobyl accident was due mainly to atmospheric fallout from 26 April through May of 1986. Estimates of the amounts of different radionuclides in the Chernobyl cooling pond (decay-corrected to 30 May 1986) are given in Table 5. These estimates were made from sampling data of the radioactive contamination of water and bottom sediments (Kryshev, 1995). In the early period of radioactive contamination (April–May 1986), 90–98% of the total radioactivity of the soils and bottom sediments of the close-in zone of the Chernobyl NPP consisted of depositions of fuel components. Hot aerosol particles ranged in size from a few to hundreds of microns. The early period of radioactive contamination of the cooling pond with 137 Cs included wash-off from the catchment area during snow melting and rainfall, washout from flooded parts of the flood plain, wind resuspension and deposition of radioactive aerosols, and direct discharge into the cooling pond of liquid effluents containing 137 Cs. The total contribution of these secondary sources of contamination does not exceed 20% of the total amount of 137 Cs that entered the cooling pond from all sources.

3. Model testing using the Chernobyl data 3.1. Ecological model ECOMOD The dynamic model ECOMOD was developed to simulate radioecological processes in the aquatic ecosystems contaminated with radionuclides (Sazykina, 1994, 2000; Sazykina & Kryshev, 1996; Kryshev & Sazykina, 1986, 1990, 1995; Kryshev et al., 1999). The modification of this model, ECOMOD-W was used for calculations on Scenario CP. The model consists of three interrelated modules, which are operated as a set of interrelated programmes: • “ECOSYSTEM” is a dynamic model of the aquatic ecosystem;

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Fig. 1. Schematic structure of the radioecological model “ECOMOD”.

• “RADIONUCLIDE DISTRIBUTION” calculates the dynamics of individual radionuclide concentrations in the ecosystem components (water, sediments, plankton, algae, fish); • “DOSE ASSESSMENT” calculates external and internal exposure doses for aquatic organisms and the human population. A schematic structure of “ECOMOD” model is given in Fig. 1 (Sazykina, 1994, 2000; Kryshev, 2000). The model complex is implemented in the form of FORTRAN code. If necessary, each module can be used separately from other model blocks; also new blocks can be added to the model complex. 3.1.1. ECOSYSTEM program The “ECOSYSTEM” program includes as variables essential components of the ecosystem: phytoplankton, macroalgae, zooplankton, fish of different feeding and living habits (plankton-

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eating, predators, omnivorous). The “ECOSYSTEM” program calculates seasonal dynamics of biomass density for ecosystem components (per unit volume), mortality, growth rate and food consumption rate, intensity of predation, etc. The equations describing the dynamics of ecosystem components were derived using the theory of organism growth in the environment with a limiting nutrient (Alekseev et al., 1992). The physical factors such as light and temperature are considered as external factors causing ecological processes to change. The model equations are detailed elsewhere (Sazykina, 1994, 2000; Kryshev & Sazykina, 1986, 1990). A simplified modification of ecological equations may be used, if necessary (for instance, only the fish model). The output information of “ECOSYSTEM” program (biomass density, growth rate and mortality during the growing period) provides input information for the “RADIONUCLIDE DISTRIBUTION” program. 3.1.2. RADIONUCLIDE DISTRIBUTION program The “RADIONUCLIDE DISTRIBUTION” module is intended to calculate the dynamics of radionuclide transfer between the aquatic medium and the food chains of aquatic organisms. This module consists of two subprogrammes. The first subprogramme calculates the dynamics of the radionuclide distribution in abiotic components of an aquatic system, in particular in water and in sediments. Any appropriate standard hydrological model can be used for this purpose. We used model GIDRO which is based on the standard procedure recommended by Goskomgidromet (Russia) for forecasting the transport of radionuclides in water bodies in the case of an accident at the Nuclear Power Plant (Goskomgidromet, 1988). The following processes were considered in the model GIDRO: the homogeneous distribution of radionuclide in water column; sorption/desorption processes governed by distribution coefficient Kd ; particle scavenging; interactions between the water column and the upper (effective) layer of bottom sediments, including sedimentation/resuspension, sorption, diffusion, bioturbation, burial to deeper sediment layers. The main subprogramme calculates the dynamics of radionuclides transfer in the biomass of aquatic biota. The programme can be run for several radionuclides simultaneously. The modeling of the radionuclide migration in an aquatic ecosystem is based on the general methodology of integrating radioecological and ecological models, desribed in the chapter “Radioactivity and Aquatic biota (Chapter 7)”, see also publications (Kryshev & Sazykina, 1986, 1990; Sazykina, 1994, 2000). Radioactive isotopes are considered as tracers, circulating in the ecosystem in parallel with their stable analogous elements. The outputs of the program “RADIONUCLIDE DISTRIBUTION” are concentrations of radionuclides in water, sediments, and the biomass of aquatic organisms (algae, zooplankton, several species of fish, etc.). This information serves as input for the next program “DOSE ASSESSMENT”. 3.1.3. DOSE ASSESSMENT program Internal and external doses to aquatic organisms living in a radioactively contaminated water body are calculated. The doses to aquatic organisms are estimated as consisting of the following components: external irradiation from water and sediments, external irradiation of organisms living in the contaminated thick aquatic vegetation, internal irradiation of organisms from incorporated radionuclides. For estimation of doses with experimental data about

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radionuclides concentrations in components of aquatic ecosystems, methods described in the literature are used (IAEA, 1976; Woodhead, 1979; Kryshev & Sazykina, 1986, 1990, 1995). In the calculations, consideration was given to the geometry of a radiation source, geometric characteristics of aquatic organisms, composite structure of radiation spectra, differences in behavior and dosimetric parameters of parent and daugther nuclides, accumulation factors for scattered gamma radiation, feeding and living habits of aquatic organisms. Doses to people consuming the contaminated fish or other aquatic organisms are also calculated in the “DOSE ASSESSMENT” program. The “DOSE ASSESSMENT” program can be used independently from the other parts of the program complex as in the case of employing the experimental data as input information. 3.2. Application of the ECOMOD model to the Chernobyl cooling pond case The first calculations of radioactive contamination of the Chernobyl cooling pond ecosystem were made in May–August 1986, just after the Chernobyl accident. In that period the information on contamination of the biota was scarce and was unavailable to modellers. In the first calculations the estimated values of the Chernobyl radionuclide fallout to the cooling pond were used as input information. The available biological data on the cooling pond ecosystem were also taken into account. The radionuclide distribution in the abiotic components of the pond was calculated, using a simple “water-sediment” compartmental model. The dynamics of the radioactive contamination of aquatic biota were calculated for a number of radionuclides; for some of them, adsorption processes were considered. The block “ECOSYSTEM” included the following components: vegetation (phytoplankton and macroalgae), zooplankton, non-predatory and predatory fish. The following set of differential equations was used to describe the dynamics of i-th radionuclide activity (Bq/kg w.w.) in the biomass of ecosystem components: Vegetation (i)



  dyk = βk T 0 , LC(t) · S(t) · −yk(i) + CF(i,stab) · X(i) (t) − λ(i) yk(i) ; dt

(1)

Consumers (zooplankton and fish) (i)

dyj

dt

(i) = −yj ·

 (i,stab)   Qj 1 dMj (i) (i) (i) λ + εj + γm · FOODm · ym ; + (i,stab) · Mj dt Qj,food

(2)

Adsorption on the surface of organisms   (i) dysorp X(i) (i) (i) (i) = qsorp ; · (i) − qdes · ysorp dt Qstab

(3)

where: indices k = 1, 2 refer to phytoplankton and macroalgae; indices j = 1, 3 refer to zooplankton, non-predatory fish, and predatory fish;

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(i)

yk , yj are activity concentrations of i-th radionuclide in the biomass of ecosystem components; Mj is the biomass of the j -th ecosystem component; βk (T 0 , LC(t)) is the coefficient of biomass production by photosynthesis as a function of water temperature T 0 and light condition LC(t), which in turn is a specific function of the latitude and season; S(t) is the concentration of dissolved limiting nutrient in the aquatic medium; the values of water temperature, light conditions and nutrient concentration in the aquatic medium, were used as input data; CF(i,stab) is the coefficient of accumulation of a stable element, which is analogous to i-th radionuclide by vegetation; X(i) (t) is the activity of the i-th radionuclide in the aquatic medium at the time t; the values of X(i) (t) were calculated in the hydrological block of the model; λ(i) is radioactive decay constant; εj(i) is the coefficient of metabolic losses of the i-th element; (i,stab)

(i,stab)

, Qj,food are the concentrations of stable analogous element in the biomass of Qj aquatic organism and in its diet;  (i) γm · FOODm · ym is intake of i-th radionuclide by an organism from its diet; (i) ysorp is the activity adsorbed on the unit of surface of organism (Bq/cm2 ); (i) (i) qsorp , qdes are parameters of sorption and desorption of the i-th radionuclide and its stable “twin”; (i) Qstab is the concentration of stable analogue of i-th radionuclide in the aquatic medium. Table 6 presents the model predictions compared with measured (averaged) concentrations of some radionuclides, taken from (Kryshev, 1992). As seen from Table 4, the ECOMOD model predictions are in relatively good agreement with the observations even for an extreme non-equilibrium radiological situation, such as was observed in the Chernobyl cooling pond in the very early days after the accident. 3.3. Comparison of model calculations with test data of the Scenario CP 3.3.1. Predictions for Stage I The requested endpoints for Stage I were the 137 Cs concentrations in the following components of the Chernobyl cooling pond: water (monthly averages for May–December 1986, annual means for 1987–1990); sediments (annual means for 1986, 1988, 1990); nonpredatory, predatory and unsorted fish (annual means for 1986–1990). The observed dynamics of 137 Cs content in the cooling pond water, sediments, and biota are characterized by the following features: • Water: The highest levels of 137 Cs concentrations in water were in May, 1986; a small secondary maximum for water contamination was observed in October–November, 1986, after start up of units 1 and 2 of the Chernobyl NPP. This secondary peak was probably the result of resuspension of contaminated sediments due to the action of water pumps. In 1987–1990, further decrease of 137 Cs concentrations in water was observed, with an average environmental half-time of 1 year (BIOMOVS II, 1996).

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Table 6 Model predictions and measured concentrations of radionuclides in the components of the Chernobyl NPP cooling pond, in the early period after the accident (May–August 1986), kBq/kg w.w. Component 30 May 1986 Water: measurements model predictions Fish: measurements model predict. Algae: measurements model predict. 15 June 1986 Fish: measurements model predict. Algae: measurements model predict. 18 July 1986 Algae: measurements model predict. 7 August 1986 Algae: measurements model predict. Fish: measurements model predict.

131 I

137 Cs

1.7 ± 1.4 3.0

0.4 ± 0.2 0.2

90 Sr

0.04 ± 0.026 0.018

0.2 ± 0.13 0.12

30 ± 20 50

50 ± 30 85

300 ± 200 400

150 ± 80 120

10 ± 6 26

80 ± 50 115

100 ± 60 150

140 ± 60 100

15 ± 7 9

400 ± 200 500

–

110 ± 60 80

15 ± 8 9

150 ± 70 100

– –

100 ± 60 50

12 ± 6 8

110 ± 70 50

– –

110 ± 60 160

9

0.3 ± 0.2 0.2

144 Ce

16 ± 10 9

0.5 ± 0.3 0.3

0.9 ± 0.4 0.6

90 ± 50 115 800 ± 500 850

110 ± 50 140

– –

• Sediments: In 1986, radionuclides were deposited primarily in the upper layer of the bottom sediments (0–4 cm). In 1987–1990, a slow decrease in 137 Cs content in the upper layers of sediments took place as a result of the continuous deposition of relatively uncontaminated new material and the subdominant influence of vertical migration. • Fish: The 137 Cs content in nonpredatory fish was characterized by a maximum value in 1986 and subsequent decrease, with an environmental half-time of about 1 year. The maximum content of 137 Cs in predatory fish was observed in 1987–1988. The calculation results were compared with the test data (Figs 2–5). In general, all model predictions of 137 Cs concentrations in cooling pond components were in rather good agreement with the observed data, indicating the satisfactory description of the processes of radionuclide distribution in the aquatic ecosystems. For predatory fish, ECOMOD-W model accurately simulated the peak fish contamination that occurred in 1987–1988.

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Fig. 2. Comparison of model predictions by ECOMOD with observations for 137 Cs concentrations in Chernobyl cooling pond water. Here and in Figs 3–5, dark circles represent the test data, with the 95% confidence intervals represented by vertical lines. The solid lines represent the model predictions, and the dashed lines (when present) represent 95% subjective confidence intervals on the predictions.

Fig. 3. Comparison of model predictions by ECOMOD with observations for 137 Cs concentrations in the bottom sediments of the Chernobyl cooling pond.

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Fig. 4. Comparison of model predictions by ECOMOD with observations for 137 Cs concentrations in nonpredatory fish (silver carp) in the Chernobyl cooling pond.

Fig. 5. Comparison of model predictions by ECOMOD with observations for 137 Cs concentrations in predatory fish (pike-perch) in the Chernobyl cooling pond.

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Table 7 Assessment of total dose to fish in 1986–1990 (Gy) Assessment ECOMOD Benchmark estimates

Type of fish Bream

Pike-perch

Silver carp

4.2 ± 2.5 4.5 ± 3.0

3.3 ± 2.0 3.3 ± 2.2

2.3 ± 1.2 1.8 ± 1.3

Table 8 Assessment of potential dose to humans from consumption of 1 kg of fish from the cooling pond (mSv) Assessment

1986

1987

ECOMOD Benchmark estimates

7.4 ± 4.3 5.6 ± 3.1

5.5 ± 3.2 3.5 ± 2.2

3.3.2. Stage II model predictions The endpoints of Stage II (doses to fish; doses and health risk to humans) cannot be tested with measurements, as none were made. Instead, benchmark estimates were made by the authors of the Scenario on the basis of experimental data on contamination of the components (90 Sr, 95 Zr, 95 Nb, 106 Ru, 134 Cs, 137 Cs, 144 Ce) of the cooling pond ecosystem. Predictions of doses to fish and ‘dose-from-fish-consumption’ demonstrated satisfactory agreement with the benchmark estimates (Tables 7, 8). 3.4. Reasons for misprediction The main reasons for model misprediction fall into three categories: the description of processes by the model; selection of the values of model parameters; deficiency of the input information given in the Scenario. 3.4.1. Mispredictions in ‘137 Cs-in-water–sediments’ calculations The reconstruction of the non-equilibrium processes of 137 Cs distribution in the water– sediments system of the Chernobyl cooling pond was a complicated task for modelling. For the first year after the accident (1986), the accuracy of model predictions of 137 Cs concentrations in water and sediments were strongly dependent on the correctness in the description of the sedimentation processes. Better results were produced by the calculations, in which the different rates of sedimentation were used for accidental fuel particles and natural suspended matter, whereas the assumption of a constant gross sedimentation rate for the whole year led to overestimation of water contamination (Kryshev et al., 1999). For the period 1987–1990, the mispredictions in ‘137 Cs in water–sediments’ calculations were mainly associated with the deficiency of information about the destruction of fuel particles, and resuspension of 137 Cs from bottom sediments.

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3.4.2. Mispredictions in ‘137 Cs-in-fish’ calculations First of all, the mispredictions in ‘137 Cs-in-fish’ calculations were directly determined by the mispredictions in ‘137 Cs-in-water’ calculations, because the radioecological model used data on water contamination as the input information for radioecological calculations. The second source of mispredictions in ‘137 Cs in fish’ calculations was associated with the non-correct selection of the values of sensitive radioecological parameters (rates of 137 Cs bioassimilation and elimination by fish; concentration factors for aquatic organisms). The use of the parameter values, taken from literature, resulted in unacceptable uncertainty ranges in the predictions. 3.4.3. Mispredictions in dose calculations The mispredictions in the ‘internal-dose-to-fish’ calculations were associated with the mispredictions in ‘137 Cs-in-fish’ calculations. Mispredictions in estimates of external doses from sediments were dependent on the assumptions about the time that fish spend near the bottom, and also on the mispredictions in ‘137 Cs-in-sediments’ calculations. Mispredictions in ‘dose-from-fish-consumption’ totally resulted from mispredictions in fish contamination. 4. Modelling the size-effect in bioaccumulation of 137 Cs by fish populations in the Chernobyl cooling pond Radiocaesium accumulation by fish is a complex dynamic process, which is determined by both environmental and physiological factors such as contamination of feedstuffs, feeding intensity, position in the food chain, etc. In turn, the physiological characteristics of fish growth and metabolism depend on fish age (size). As a consequence, a specific phenomenon, called the ‘size effect’ is observed in many water bodies contaminated by the Chernobyl depositions (Evans, 1988; Lindner et al., 1990; Hakanson, 1991, 1999; Elliott et al., 1992; Kryshev, 1992; Kazakov, 1995; Grimas et al., 1996; Hadderingh et al., 1996). More often, a ‘positive size effect’ has been revealed – older fish are more contaminated with 137 Cs than younger members of the same species. More rarely, mostly during the first year following the accident, a ‘negative size effect’ was observed – the specific activity of 137 Cs in older fish was lower than that in young fish of the same species. A number of authors observed a maximum in the distribution of radiocaesium activity in fish by age during the second year following the accidental contamination. A mathematical model which enables a description of all types of size effect and other features of radiocaesium accumulation by fish is presented in the paper (Kryshev & Ryabov, 2000). 4.1. Description of the model 4.1.1. Basic model equations In a natural aquatic ecosystem, a population of each fish species is represented by a set of individual fish of different age classes. Each age class is characterized by a specific growth rate, the amount and assortment of the food components, and the activity of the metabolic processes. Therefore, the rates of accumulation and elimination of the radiocaesium in fish is specific for each age class. In the case of accidental contamination of a water body, it seems

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convenient to divide the population of each species of fish into discrete age classes and to express the fish age as τ = τ0 + t, where τ0 is the age of the fish at the time of the accident, and t is the elapsed time since the occurrence of the accident. It is assumed in the model that the major part of the radiocaesium enters the fish via contaminated food. The basic equation of the model can be written as (Kryshev & Ryabov, 2000)     QA 1 dM dy 1 dM W W = − λr + εA + · y + 1A · + εA · X, dt M M dt M dt M Q0

(4)

where y is the activity concentration of the radionuclide in the muscle of the fish, in Bq/kg; λr is the radioactive decay constant; εA is a coefficient of proportionality between the elimination rate of the bioelement and the general rate of metabolism W ; M = M(τ ) is the fish weight averaged over a particular age class, in kg; QA 1 is the concentration of the bioelement in the fish, QA is the concentration of the bioelement in the food of the fish, X is the concentration 0 of radionuclide in the food of the fish. Equation (4) can be written for each discrete generation of fish and can be solved numerically. 4.1.2. Relationships between the fish growth rate and the metabolic rate The specific increase of fish biomass and the value for specific metabolic rate are related to each other by the ‘energy balance equality’ (Winberg, 1956): γ ·F =

1 dM W + , M dt M

(5)

where γ characterizes the degree of food assimilation in the gastrointestinal tract (GIT) of the fish and does not depend on the radionuclide; F is equal to total amount of food consumed by the fish during the time period divided by the average biomass of fish during this period. The value of the metabolic rate W , is proportional to the power function of the fish weight (Winberg, 1956; Ivlev, 1962): W = α1 M α2 · 0.0048 · Kmov · β0 · exp(β · TEMP).

(6)

In this formula, W is expressed in grams (wet weight) per hour; Kmov is a correction factor allowing for the fact that the fish spend energy not only on internal processes but also on movement. According to Ivlev (1962), Kmov = 2. The numerical values of α1 and α2 for some species of freshwater fish were determined by Winberg (1956) and Melnichuk (1975). β is a parameter for the temperature dependence of the fish metabolism. β0 is a normalizing factor selected in such a way that at 20◦ C, β0 × exp(β × TEMP) = 1, where TEMP is the temperature in ◦ C. The coefficient 0.0048 converts W expressed in ml O2 /h to units of wet weight (g/h). Using equation (3) and knowing the specific biomass growth rate of the fish, it is possible to calculate the assimilated diet of the fish. The rate of biological elimination of the radiocaesium from the fish (ε) depends on the fish weight and on temperature, as well as on the organs and tissues in which the radionuclide is accumulated. In this model, the rate of radiocaesium elimination is assumed to be proportional to the rate of the overall metabolism W/M: ε = εA · (W/M), where εA = 0.3 ± 0.1 (Kryshev & Ryabov, 2000).

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4.1.3. Contamination of the fish food The assortment of feedstuffs, consumed by fish is strongly dependent on fish size (age), as well as on the position of a given fish species in the food chain. The contamination of fish food is estimated separately for each fish species and each age class. If the fish feeds on small organisms, for example on plankton, yfood can be estimated using the constant equilibrium concentration factor CFfood (Kryshev & Sazykina, 1994). This simplification is justified since an equilibrium in the exchange of radionuclides between such hydrobionts and the water is established rapidly because of a large specific surface area for interaction with the environment. Then X = CFfood · Cw (t), where Cw (t) is the activity of the radionuclide in water, and equation (4) takes the form dy = −ay + G · Cw (t), dt

(7)

A where a = λr + εA (W/M) + (dM/M dt), G = (QA 1 /Q0 ) · (εA (W/M) + (dM/M dt)) · CFfood . The specific activity of radiocaesium in predatory species of fish is determined from the system of equations of types (4) and (7), where the activity in the prey fish is calculated by the first equation:

      dyprey dM W = − λr + εA + yprey dt M prey M dt prey  A      Q1 dM W + · ε + CFfood · Cw (t), A M prey M dt prey QA 0 prey       dypred W dM = − λr + εA + ypred dt M pred M dt pred  A      Q1 dM W + · ε + yprey(t). A M pred M dt pred QA 0 pred

(8)

(9)

In equations (8)–(9) the index prey is related to the prey fish, index pred is related to the predatory fish, CFfood , is the average concentration factor for organisms which are consumed by the prey fish. 4.2. Modelling of fish contamination in the Chernobyl cooling pond Two fish species from the Chernobyl cooling pond were selected for model testing: carp (nonpredatory fish) and pike-perch (predatory fish). Carp from the cooling pond of the Chernobyl NPP feed chiefly on chironomid larvae, oligochaetes, planktonic crustaceans, mollusks and detritus, whereas pike-perch are piscivorous predators from the second year of their life onwards. Data for the average observed annual biomass growth rates of fish from the Chernobyl NPP cooling pond was taken from Kazakov (1995). The dynamics of the 137 Cs activity in water of the Chernobyl NPP cooling pond Cw (t) were estimated earlier (Sazykina & Kryshev, 1996; Kryshev et al., 1999).

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Table 9 The numerical values of coefficients used in the model calculations (Kryshev & Ryabov, 2000) Parameter

Symbol

Carp

Pike-perch

Food assimilation in GIT

γ

0.6

0.8

Ratio between K concentration in fish and fish food

Q1 /Q0

1.0

1.2

Averaged concentration factor of 137 Cs in feedstuff of carp

CFfood

1986: τ <1 1<τ <2 2<τ <4 τ >4 since 1987: for all τ

Parameters of metabolic rate

α1 α2

0.467 0.8

– 600 500 400 320 500 0.446 0.83

The numerical values of the coefficients used in the model calculations are shown in Table 7. In the early period after the radioactive contamination of the water body, the concentration factors in the food of fish-benthophages may depend on fish size. For example, large individual carp feed on benthic organisms predominantly from deeper layers of the bottom sediments (5–8 cm) (Nikolsky, 1974), into which the radionuclide has not as yet managed to migrate. Therefore different numerical values were selected for the concentration factors in the food of carp in the first year following the accident, depending on fish age (Table 9). The calculations were performed for a 4 year period (1986–1990) after the accidental contamination. The model predictions were made for fish from the generation of 1978 up to the generation of 1988. The model predictions were compared with the experimental data (Kryshev & Ryabov, 2000). The calculated dynamics of activity concentration in the muscles of carp of different generations are presented in Fig. 6. Carp specimen of younger age classes are seen to accumulate radiocaesium more actively and to reach a maximum level of contamination earlier than the older carps. However, following the decrease of 137 Cs activity in the water, young carp lost radiocaesium more intensively than the older of the species. Due to their more intense metabolism, the activity of 137 Cs in the young fish follows rather rapidly the change in the contamination of their food. Radiocaesium accumulation by fish of the older age groups is slower because of the low relative diet, but its removal from the organism is also slow. This is due to the fact that the rate of biological removal ε is proportional to the rate of overall metabolism in the organism divided by the weight (W/M), which in turn decreases with increasing weight of fish (for example, for carp ε ∼ (W/M) ∼ M −0.2 ). Different types of size effect are observed in different periods of time after an accidental contamination. During the first year after an accident, carp specimens of the younger age groups are contaminated more than the older carps, i.e. a negative size effect applies. An additional contribution to this phenomenon is due to the fact that carp of younger age feed on the more contaminated food. Beginning from the second year after the accident, older carp

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Fig. 6. The dynamics of 137 Cs activity concentration in the muscle of carp of different generations, Chernobyl cooling pond (model calculations).

specimens are contaminated to a greater extent than younger carp. Along with differences in the rates of accumulation and removal in the different age groups of fish, a sound reason for a positive size effect in radiocaesium accumulation is its decreased activity in the water and hence in the food of the fish. The fish born within a year after the accident or later are less contaminated. Figure 7 shows the calculated dynamics of the activity concentration in the muscles of the pike-perch of different generations. Although the observed effects, on the whole, are similar to those for carp, note that for predatory fish the size effect is more pronounced at all stages after

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Fig. 7. The dynamics of 137 Cs activity concentration in the muscle of pike-perch of different generations, Chernobyl cooling pond (model calculations).

the accident. In the first year following the accident, a negative size effect is enhanced by the fact that pike-perch of the first year of life feed as non-predatory fish. After their change to a predatory type of feeding the secondary maximum of contamination takes place. The maxima in activity concentration in pike-perch come later than in carp due to the delay caused by the passage of 137 Cs through the trophic chain (the “effect of trophic levels”). The change from a negative size effect to a positive one occurs, both in pike-perch and carp, beginning in the second year after the accident but in pike-perch it occurs more smoothly. Figure 8 shows the calculated activity concentrations of 137 Cs in the muscles of pike-perch as function of their weights together with the measurement data. It is seen that the model describes the size-effect quantitatively as well as qualitatively. Figure 9 shows the age distribution of activity concentration of 137 Cs in pike-perch for the cooling pond of the Chernobyl NPP on the 180th, 270th, 540th and 900th day after the accident. It can be seen that the size effect changes sign progressively, as the contamination maximum for pike-perch moves from younger to older ages.

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Fig. 8. The specific activities of 137 Cs in muscles of pike-perch from the Chernobyl cooling pond as a function of their weights – model calculations in comparison with the measurement data.

5. Reconstruction of radioecological processess in the Chernobyl cooling pond in the early period after the accident of 1986 The reconstruction of radioecological impact on aquatic biota in the early period is an important and complicated task. During the first months after the accident many short-lived radionuclides with various chemical behavior were present in aquatic media of the Chernobyl cooling pond along with long-lived radionuclides. Due to the high initial activity, the contribution of short-lived radionuclides to radiation doses received by aquatic biota in the early post-accidental period exceeded the doses from long-lived radionuclides. The following groups of radionuclides were considered in calculations (Kryshev, 1992): • Long-lived radionuclides with half-life of about a year and more (137 Cs, 134 Cs, 90 Sr, 106 Ru, 144 Ce). • Radionuclides with half-life from a week to two months (131 I, 140 Ba, 136 Cs, 141 Ce, 103 Ru, 95 Zr, 95 Nb, 89 Sr).

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Fig. 9. The distribution of 137 Cs activity in pike-perch from the Chernobyl cooling pond by age at four different time points: 180, 270, 540 and 900 days after the accident.

• Short-lived radionuclides with half-life of several days and less (132 I, 239 Np, 99 Mo etc.).

133 I, 135 I, 132 Te,

Modelling of radionuclide distribution between ecosystem components in the cooling pond shows the complex non-equilibrium character of the processes (Fig. 10) (Kryshev, 2000).

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Fig. 10. Dynamic of radionuclide concentration in fish of the Chernobyl cooling pond in the initial period after the accident (model calculation).

The results of dose reconstruction to aquatic organisms permit the following conclusions: • In the initial period after the accident the major contribution to exposure of plankton, algae, fish and benthos (60–80%) was made by radionuclides with half-life from a week to two months, primarily 131 I (Table 10). • Long-lived radionuclides (half-life over a year) were responsible for 20–40% of exposure for plankton, algue and fish and for 10% of external dose of bottom-dwelling organisms.

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Table 10 Relative contributions (in %) of radionuclides in exposure of aquatic organisms in the Chernobyl cooling pond (26 April–10 May 1986) Group of radionuclides

Phyto-plankton

Zoo-plankton

Macro-algae

Fish

From bottom

From water

Long-lived radionuclides of half-life of about a year and more (137 Cs, 134 Cs, 90 Sr, 106 Ru, 144 Ce)

16/23

16/20

30/38

24/37

7/11

4/7

Radionuclides of half-life from 8 to 64 days (131 I, 140 Ba, 136 Cs, 141 Ce, 103 Ru, 95 Zr, 95 Nb, 89 Sr)

77/74

76/78

61/58

59/55

74/82

70/83

Short-lived radionuclides of half-life less than 3 days (132 I, 133 I, 135 I, 132 Te, 239 Np, 99 Mo)

7/3

8/2

9/4

17/8

19/7

26/10

Note. Numerator is averaged estimate for the first week, denominator – for the second week after the accident.

• Short-lived radionuclides (half-life less than 2–3 days) accounted for less than 10% of exposure dose of plankton and algae in the first week after the accident and 20% for bottomdwelling organisms and fish. During the second week after the accident, the input of shortlived radionuclides to exposure of aquatic organisms was reduced by more than a factor of 2. • It is of interest that in terms of exposure of bottom-dwelling organisms and fish in the first week after the accident, the significance of short-lived radionuclides was comparable to that of long-lived. • The most significant contribution of short-lived radionuclides to external dose was that from water (about a quarter of the dose in the first week after the accident), the dose from water itself, however, being much lower than the dose from internal irradiation of aquatic organisms and external irradiation from the bottom. In June of 1986 the most important radionuclides for radioactive contamination of aquatic organisms were 95 Zr, 95 Nb, 144 Ce, 106 Ru, 141 Ce, 103 Ru etc. Further, with the reduction in radioactivity as a result of radioactive decay, of major significance in terms of contamination of aquatic biota were the long-lived radionuclides such as 137 Cs, 134 Cs, 90 Sr. In this period, increased doses over a long time period in fish, macroalgae and benthic organisms were observed (Kryshev, 1992; Kryshev & Sazykina, 1995).

6. Conclusions The “Chernobyl cooling pond” scenario provides modellers with an opportunity to test their models using the unique data set on the consequences of heavy accidental contamination for

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natural freshwater ecosystems. The results of modeling, obtained by the use of the ecologoradioecological model ECOMOD demonstrates the capacity of the model to reconstruct the dynamics of the radioecological situation in the aquatic ecosystem in the early period as well as in the later period after the Chernobyl accident. The dynamic model, which takes into account the characteristic features of the metabolism, feeding and behaviour of the fish, enables the modeller to describe and explain all known types of size effect in the radiocaesium accumulation by fish. The discrepancy between the calculated and observed data is within the uncertainty of the experimental data. There is a need to develop radioecological models not only for 137 Cs, but also for other ecologically significant radionuclides, such as 131 I, 90 Sr, 95 Zr, 95 Nb, 106 Ru, 144 Ce, 239 Pu, etc. The consideration of a set of radionuclides is particularly important for evaluating the radioecological consequences of radiation accidents in the early “acute” period of accidental contamination. Finally, in addition to the utility of the exercise for evaluation of various models, the use of Post-Chernobyl data in Scenario CP provided a valuable opportunity for training students and specialists in the field of radioecology and environmental protection.

References Alekseev, V. V., Kryshev, I. I. & Sazykina, T. G. (1992). Physical and Mathematical Modelling of Ecosystems (367 pp.). St.-Petersburg, Russia: Hydrometeoizdat (in Russian). BIOMOVS II (1996). Assessment of the Consequences of the Radioactive Contamination of Aquatic Media and Biota. Model testing Using Chernobyl Data. BIOMOVS II Technical Report No. 10. Stockholm, Sweden: Swedish Radiation Protection Institute. Borovoy, A. A., Ibragimov, G. D. & Ogorodnik, S. S. (1990). The state of the Chernobyl NPP unit and nuclear fuel in it. Chernobyl: The Expedition of Kurchatov’s Institute of Atomic Energy (in Russian). Chernobyl Symposium (1989). Vols 1, 5, 6. Chernobyl (in Russian). Goskomgidromet (1988). The procedure for forecasting the transport of radionuclides in water reservoirs in the case of an accident at the Nuclear Power Plant. RD 52.26 174–188. Goskomgidromet (in Russian). Elliott, J. M., Hilton, J., Rigg, E., Tullett, P. A., Swift, D. J. & Leonard, D. R. P. (1992). Sources of variation in post-Chernobyl radiocaesium in fish from two Cumbrian lakes (north-west England). Journal of Applied Ecology, 29, 98–119. Evans, S. (1988). Accumulation of Chernobyl-related 137 Cs by fish populations in the biotest basin, northern Baltic Sea. Studsvik Report. STUDSVIK/NP-88/113 (pp. 1–30). Grimas, U., Karas, P. & Neumann, G. (1996). Observations of effects on individual and population levels of perch (Perca fluviatilis L.) with high concentrations of fallout cesium. Proc. of International Symposium on Ionizing Radiation, Stockholm, 20–24 May, 1996 (pp. 230–239). Hadderingh, R. (1989). Distribution of 137 Cs in the aquatic foodchain of the Ijsselmeer after the Chernobyl accident. The Radioecology of Natural and Artificial Radionuclides. Proc. of XV Regional Congress of IRPA, Visby, 10–14 September, 1989 (pp. 325–330). Hadderingh, R., Nasvit, O., Ryabov, I., Van Aerssen, G., Prostantinov, V. & Belova, N. (1996). Field studies of fish size effect in 137 Cs accumulation. Int. Sci. Collaboration of the Consequences of the Chernobyl Accident. ECP-3 Final Report (pp. 84–94). Hakanson, L. (1991). Ecometric and dynamic modelling exemplified by cesium in lakes after Chernobyl. Lecture Notes in Earth Sciences. Springer-Verlag. Vol. 35 (pp. 1–158). Hakanson, L. (1999). A compilation of empirical data and variations in data concerning radiocesium in water, sediments and fish in European lakes after Chernobyl. Journal of Environmental Radioactivity, 44 (1), 21–42. Hoffman, F. O., Thiessen, K. M. & Watkins B. (1996). Opportunities for the testing of environmental transport models using data obtained following the Chernobyl accident. Health Physics, 70, 5–7.

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IAEA (1976). Effects of Ionizing Radiations on Aquatic Organisms and Ecosystems. Technical Reports Series, No.172, IAEA, Vienna. IAEA (1979). Methodology for Assessing Impacts of Radioactivity on Aquatic Ecosystems. Vienna: IAEA. Technical Report, Ser, No. 190: p. 43–96. IAEA (1986). The accident at the Chernobyl NPP and its consequences. Information compiled for the IAEA Expert’s Meeting (25–29 August 1986, Vienna). Ivlev, V. S. (1962). The method for calculation of food amount consumed by growing fish. Biologia vnutrennih vodojomov Pribaltiki. pp. 132–138 (in Russian). Kaftannikova, O. G., Protasov, A. A., Sergeeva, O. A., Kalinichenko, R. A., Vinogradskaya, T. A., Lenchina, L. G., Kosheleva, S. I., Novikov, B. I., Afanasiev, S. A., Sinitsina, O. O., Movchan, N. B. & Pankov, N. G. (1987). The ecology of an NPP’s cooling pond. Kiev: Ukraine Academy of Sciences (in Russian). Kazakov, S. V. (1995). Management of Radiation Conditions of Water Cooling Reservoirs of Nuclear Power Plants. Kiev, Tehnika. pp. 1–192 (in Russian). Kryshev, A. I. (2000). Dynamic Model of Radionuclides Migration in Freshwater Ecosystems. PhD Dissertation. Moscow State Unuversity. Biophysical Department (in Russian). Kryshev, A. I. & Ryabov, I. N. (2000). Dynamic Model of 137 Cs Accumulation by Fish of Different Age Classess. Journal of Environmental Radioactivity, 50 (3), 221–233. Kryshev, A. I., Abramenkov A. A., Avramova D. A., et al. (1996). Validation Experience of Models of 137 Cs Radioactive Contamination of the Chernobyl cooling pond Ecosystem. Izvestiya VUZov (Transactions of Institutions of Higher Education). Nuclear Power Engineering (3), 16–20 (in Russian). Kryshev, I. I. (Ed.) (1992). Radioecological Consequences of the Chernobyl Accident (pp. 1–142). Moscow: Nuclear Society International. Kryshev, I. I. (1995). Radioactive contamination of aquatic ecosystems following the Chernobyl accident. Journal of Environmental Radioactivity, 27, 207–219. Kryshev, I. I. & Sazykina, T. G. (1986). Mathematical Modeling of Radionuclide Migration in Aquatic Ecosystems. Moscow: Energoatomizdat (in Russian). Kryshev, I. I. & Sazykina, T. G. (1990). Simulation Models for Ecosystem Dynamics Under Anthropogenic Effects of Heat Electric Generation and Nuclear Power Plants (190 pp.). Moscow: Energoatomizdat (in Russian). Kryshev, I. I. & Sazykina, T. G. (1994). Accumulation factors and biogeochemical aspects of migration of radionuclides in aquatic ecosystems in the areas impacted by the Chernobyl accident. Radiochimica Acta, 66/67, 381–384. Kryshev, I. I. & Sazykina, T. G. (1995). Assessment of Radiation Doses to Aquatic Organism’s in the Chernobyl Contaminated Area. Journal of Environmental Radioactivity, 28 (1), 91–103. Kryshev, I. I., Sazykina, T. G., Ryabov, I. N., Chumak, V. K. & Zarubin, O. L. (1996). Model testing using Chernobyl data: II. Assessment of the consequences of the radioactive contamination of the Chernobyl Nuclear Power Plant cooling pond. Health Physics, 70, 13–17. Kryshev, I. I., Sazykina, T. G., Hoffman, F. O., Thiessen, K. M., Blaylock, B. G., Feng, Y., Galeriu, D., Heling, R., Kryshev, A. I., Kononovich, A. L. & Watkins, B. (1999). Assessment of the consequences of the radioactive contamination of aquatic media and biota for the Chernobyl NPP cooling pond: model testing using Chernobyl data. Journal of Environmental Radioactivity, 42, 143–156. Lindner, G., Becker, M., Eckmann, R., Frenzel, P., Kleiner, J., Petermann-Seyboldt, D., Pfeiffer, W., Wahl, U. & Recknagel, E. (1990). Biological transfer and sedimentation of Chernobyl radionuclides in Lake Constance. Large Lakes. Ecological Structure and Function (pp. 265–287). Springer-Verlag. Melnichuk, G. L. (1975). Ecology of nutrition, food requirements and energy balance of young fishes from the Dnieper water reservoirs. Izvestija GOSNIORH, 101, 1–290 (in Russian). Nikolsky, G. V. (1974). Fish ecology (pp. 1–357). Moscow: Vysshaja shkola (in Russian). Protasov, A. A., Sergeeva, O. A., Kosheleva, S. I., Kaftannikova, O. G., Lenchina, L. G., Kalinichenko, R. A., Vinogradskaya, T. A., Novikov, B. I., Afanasiev, S. A. & Sinitsina, O. O. (1991). Hydrobiology of cooling ponds of thermal and nuclear power plants in Ukraine. Kiev: Naukova Dumka (in Russian). Sazykina, T. G. (1994). Modelling the transfer of fission products in aquatic ecosystems. 4th Int. Conf. on the Chemistry and Migration Behaviour of Actinides and Fission Products in the Geosphere. Charleston, SC, USA, 12–17 December (pp. 727–731). Sazykina, T. G. (2000). ECOMOD – An ecological approach to radioecological modelling. Journal of Environmental Radioactivity, 50 (3), 207–220.

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Sazykina, T. G. & Kryshev, A. I. (1996). Radioecological model for transport of radionuclides and exposure in aquatic ecosystems. Proc. of International Symposium on Ionizing Radiation, Stockholm, 20–24 May, 1996 (pp. 503– 509). Winberg, G. G. (1956). Rate of metabolism and food requirenments of fishes. Fish. Res. Board. Can. Transl. Ser. (194), 1–239. Woodhead, D. S. (1976). Concentrations of radionuclides in aquatic environments and the resultant radiation dose rates received by aquatic organisms. In Effects of ionizing radiation dose rates received by aquatic organisms and ecosystems. Tech. Rep. No. 172 (pp. 5-54). Vienna: IAEA. Woodhead, D. S. (1979). Methods of dosimetry for aquatic organisms. In Methodology for assessing impacts of radioactivity on aquatic ecosystems. Tech. Rep. No. 190 (pp. 43–96). Vienna: IAEA.

Concluding remarks E. Marian Scott

Are there limits to what can be achieved with a mathematical model and how can we define the conditions which determine those limits? How can we demonstrate the validity, applicability and the worth of a model? Directly and indirectly within the chapters here, the authors have attempted to define the current achievements of the models for environmental radioactivity used in the atmosphere, biosphere, marine and freshwater systems; to demonstrate the validity, applicability and value of such models through studying specific case-studies drawn from relevant and current radioactivity issues and to identify the limits of what so far can be achieved. It is clear that in some environmental compartments, and for some applications, the diversity of models used is very limited, and that particular models have become linked to regulations and are considered the ‘norm’. It is also clear that, in many systems, the physical basis of the models is relatively simple (linked advection/diffusion equations) with ‘add-ons’ as required to reflect some of the peculiarities of radionuclides and perhaps even as ‘fixes’ to deal with model-observation mis-matches in a pragmatic and empirical manner. Many of the developments, particularly in atmospheric and ocean modelling, stem not from an interest in environmental radioactivity but in global climate change so these developments may not be all that is needed for environmental radioactivity. Improvements in terrestrial and freshwater system models have been driven forward by the need to model, to predict and to understand the behaviour of caesium (predominantly) released from Chernobyl and globally distributed. Model development has often been opportunistic and not as a result of a designed experiment but it is clear that radioactivity modelling no longer is stifled by computational limitations. With few exceptions, no chapter in this book devotes more than a few paragraphs to computational issues relating to solutions to model equations. Many of the radionuclides dealt with in the chapters are present in the environment as pollutants, released from waste processing, or as a result of accident, others are naturally occurring, but their activities and distributions are technologically enhanced. Regardless of source, with their presence, we have the opportunity to learn more about the environment in which we live and the models provide one tool for synthesis of current knowledge and observation. 417

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A frequent debating point concerns putting more science into models, particularly when the science introduces further complexity, which may not be supportable given the observation base. There are, as we know, leads, lags and feedback loops in the model–science–observation system. As modellers, we always want to make ‘the model performance better’ but are limited by the experimental data; as experimentalists, we might often claim that all possibilities can be demonstrated by a model, so there is no discriminatory power to choose between alternative explanations. There is some truth contained in each of these statements, and the decisionmaker must assess when model development is sufficient and when the ‘latest science’ needs to be included in the model. A relevant comment to bear in mind is that a model needs only to be fit for its purpose. The chapters in Section 1 are intended to provide the reader with sufficient information to form an educated opinion about the position which modelling science has reached, to identify any limitations and extensions still required, all with the objective of informing the reader as to whether the model he/she intends to use is fit for its purpose. The chapters are quite different in terms of the modelling approaches used and the system complexities modelled but they have similarities in respect of good modelling practice; they also exemplify a common concern with model validation although they contain only a short debate about whether we can actually validate a model. Importantly and reflecting growing awareness and trends in modelling, the authors have all indicated an awareness of the issues in quantifying and handling uncertainties, although the degree of sophistication of methodology used does vary considerably. Clearly the handling of uncertainties has moved upwards in the modellers’ priorities. There is a clear statement of realism and practicality with the underlying message being that ‘complexity of model does not always bring the anticipated rewards’. What lessons are learnt from studying applications? For the most part, as in many areas of science, few of the authors have presented case studies where the model did not perform reasonably well and, as an educational tool, often it is in the situations where ‘things’ did not work that we learn the most. Situations where the model performs poorly provide the driver for much new science and new model development and thus the translation to a logical progression in improvements unfortunately is not clearly defined either in the general radioecology literature or in the application chapters in this book. Interestingly for me, for some of the applications, the driving principle of matching our perceived (through a cloudy glass of experiment and observation) reality and model predictions is not always appropriate. This is especially true in the case where the model presents a stochastic view (e.g. probability of a breach in containment of a waste repository) in a hypothetical future (several thousand years of climate change, and how can we include the unknowable of societal change?). This identifies that there remains a pressing need for objective and quantifiable procedures for model testing and evaluation. It is also clear to me that links between design of experiments for hypothesis and model generation and testing need further development and formalisation.

In summary On the whole, the chapters illustrate the health of the modelling activity in radioecology, but we need to be aware that much of the impetus for the current healthy state stems from the

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burst of activity following Chernobyl and from the research investment in global environmental issues. For how much longer will radioecology in general and radioecological modelling remain a research and development priority? The signs are probably already there that funding for this work is declining and, to maintain the health of the discipline, there needs to be a steady input of funds. The perception and use of models within the physical sciences is changing. The developments in computing power have freed environmental models but our scientific knowledge of the whole might be best described as a lack of knowledge of how to fit together complex environmental parts. Our data assimilation tools are limited by our ability (usually funding and time) to collect appropriate data. The challenge still remains to design better models, which are systems based and stochastic, but always driven by the underlying scientific objective. Environmental radioactivity provides an invaluable tool for gaining understanding of the natural and anthropogenic perturbations of the whole earth system and will continue to do so in new and exciting ways. The ultimate challenge for the new millennium is therefore to increase our understanding of the dynamics of the system by better sampling and more meaningful measurement feeding into more integrated modelling activities and then to be able to convey this improved understanding to regulators, managers and the general public.

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Index of Authors Kryshev, A., 391 Mayall, A., 13 McKinley, I. G., 109 Monte, L., 147 Periáñez, R., 223 Pröhl, G., 87 Sanchez, A., 261 Sazykina, T. G., 201, 391 Scott, E. M., 1 Shaw, G., 315 Smith, P. A., 109 Zhuchenko, T., 315

Abril, J. M., 223 Alexander, W. R., 109 Avila, R., 315 Beresford, N., 261 Burchard, H., 55 Crout, N., 261 Dvornik, A., 315 Fesenko, S., 315 Håkanson, L., 147 Harms, I. H., 55, 287 Helton, J. C., 353 Karcher, M. J., 55, 287 Kryshev, I. I., 391

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Subject Index Komsomolets, 300 Kursk, 301 “coupled” codes, 112 ‘hindcast’ scenario, 292 1D dispersion model, 227 1D-hydrodynamic, 228 2D depth averaged model, 227 accumulation curve, 209 ADMS, 32 adsorption, 203, 207, 210, 212, 218–220 advection, 58, 66, 113, 229, 235, 240 advection schemes, ‘up-stream’, 67 advection schemes, centred, 67 advection-diffusion equation, 58, 66 advective, 227, 235, 240, 243 advective transport, 160 AERMOD (USA), 36 age classes, 404 aggregated, 266 aggregated transfer factor, 331, 334, 335 aggregation, 183 aleatory uncertainty, 353, 361 alternative models, 354, 370, 383 anchor ice, 304 aquatic biota, 201, 207, 210, 218, 391, 397, 398, 410, 413 aquatic pathways, 148 Arctic Ocean, 56, 74, 291, 304 atmospheric sources, 287 atmospheric stability, 16 Barents Sea, 56, 74, 292, 301 Batemann equations, 329 Bayesian methods, 9 Beaufort Gyre, 304 beta radiation, 49 bioaccumulation, 203, 209, 404 bioassimilation, 203–205, 207, 209, 211, 212, 216, 218, 219 bioavailability, 264 bioconcentration factor, 153

biofuel, 318 biological compartments, 153 BIOMASS, 319, 336, 343, 346, 349 biomass, 90, 92, 93, 95, 153, 211–220, 316, 320, 322– 325, 327, 329, 333, 335, 336, 344–346, 348, 393, 394, 397–399, 405, 406 biomass dynamics, 211 biotic processes, 167 bioturbation factor, 161, 167 biouptake, 151–154, 168–171, 176 Black Sea, 56 boundary condition, 68, 72, 186 boundary layer, 23 box model, 76 building shielding, 49 building wake effects, 30 calibration, 2, 4, 5 catchment basin, 188 catchment sub-model, 155 chemical retardation, 114 Chernobyl, 315–322, 327, 329, 332, 333, 340, 343, 344, 349 Chernobyl cooling pond, 392, 395, 398, 399, 401–404, 406, 408–413 chronosequence, 344 Chukchi Sea, 296 climatological atmospheric forcing, 296 collective parameters, 149 compartment model, 76 complementary cumulative distribution functions, 354, 367 complexity, 3 Compliance Certification Application, 357 computational implementation, 353, 354, 361, 383 concentration ratio, 261 conceptual structures, 3 coordinate, sigma, 59 coordinates, general vertical, 59 coordinates, geopotential, 59 coordinates, isopycnal, 61 countermeasures, 105

424

MODELLING RADIOACTIVITY IN THE ENVIRONMENT

coupled ice–ocean model, 302 coupled processes, 113 coupling mode, off-line, 69 coupling mode, on-line, 69 coupling modes, 291 deposition, 13, 27–30, 43, 45, 227, 235, 238, 240, 244, 258, 316–320, 322, 325, 329, 333, 335, 336 deposition velocity, 28 design and conceptualisation, 2 differential equations, 126, 354, 362–364, 367, 373, 376, 377, 398 diffusion, 15, 21, 23, 24, 34, 41–44, 58, 66, 113, 160, 225, 229, 235, 238–240 diffusion coefficient, 113 diffusion, ‘Monte Carlo’, 68 diffusion, numerical, 67, 68 dimensionless moderator, 165 discretisation, 59 discretisation, horizontal, 61 discretisation, temporal, 62 discretisation, vertical, 59 dispersion, 13, 14, 17, 18, 20–24, 27, 29–32, 34–37, 39– 47, 49–51 dissolved, 169, 171 dissolved phase, 162 distribution coefficient, 78, 153, 227, 233, 245–247, 251– 253, 264, 397 DOSE ASSESSMENT, 392, 396–398 dose reconstruction, 412 drifting sea ice, 290 dry deposition, 29 dynamic, 269–271 East Greenland Current, 293 East Siberian Sea, 292 Eastern Urals, 315 ecological models, 210, 211, 219 ecosystem contamination, 391 ecosystem modelling, 147, 153, 174 eddy diffusivity, 58, 66, 67 eddy viscosity, 58 edible fungi, 318 effective diffusion coefficient, 181 English Channel, 56, 69 epistemic uncertainty, 353 equilibrium, 178, 268 equilibrium distribution coefficients, 233 erosion, 227, 235, 238, 240, 244, 258 estimation, 2 estuary, 223 Eulerian, 43–45 expert judgement, 2

external dose, 397, 404, 412, 413 external irradiation, 397 external radiation, 48 Farmer limit line, 381 field experiments, 134 finite cloud, 48 finite differences, 62, 236, 240 finite elements, 62 fit for the purpose, 3 flaw leads, 305 flushing, 303 flux, 315–317, 319, 320, 327, 329, 331, 343–345 food assimilation, 405 food chain, 217, 261 forcing, atmospheric, 73 forcing, climatological, 74 forcing, external, 72 forcing, tidal, 73 forcing, variability, 74 forest, 315–325, 327, 329–333, 335–337, 340, 343–349 FORESTLAND, 319, 321–323, 336–338, 340, 343, 344, 347, 349 FORESTLIFE, 332, 333, 335–338, 340, 343, 344 FORESTPATH, 319 fractured rocks, 122 frayed edge site, 162 frazil ice, 304 gamma radiation, 48 gastrointestinal tract, 405 Gaussian, 21, 22, 24, 26, 27, 29, 31, 34–37, 39–43, 47, 49 Gaussian plume model, 21 geologic disposal, 355, 357 geosphere, 109 geosphere structures, 122 geosphere transport modelling, 109 global scale, 297 gravitational settling, 29 Greenland Sea, 80 groundwater flow rates, 113 half life decay, 66 heterogeneity, 122 horizontal standard deviation, 24 hydrodynamic equations, 235 hydrodynamic model, 57, 287 IAEA, 318, 319, 321, 336, 343, 346 ice–ocean model, 290 importance sampling, 368, 369, 373 ingrowth, 116

Subject Index interception, 92–96, 99, 105, 320, 324 intercomparison, 37 interface layer, 186 internal exposure, 394, 396 internal irradiation, 397 internal loading, 164 International Arctic Seas Assessment Program (IASAP), 288 interstitial water, 186 inventory, 190 ion-exchange, 114 Irish Sea, 56, 79, 292 isotropic, 126 Kara Sea, 292 Kara Sea, dump sites, 56, 296, 306 kinetic, 234, 235, 241, 246, 255 kinetics, 116 Lagrangian, 44, 45 Lagrangian trajectories, 306 LaHague discharge, 69 lakes, 147, 148, 151–154, 157, 158, 161–166, 168–170, 172–176, 187, 197 LANDSCAPE, 319 landscape, 316, 323 Laptev Sea, 292 Latin hypercube sampling, 373, 374, 376, 380, 381 local scale, 297 logistic equation, 329 long range dispersion modelling, 43 marine sources, 287 mass-balance calculations, 163 mathematical description, 354, 360, 376 matrix, 345 matrix diffusion, 114 mechanical dispersion, 113 metabolic, 268–271, 273, 275, 276, 283 metabolic factors, 268 metabolic rate, 405 Michaelis-Menten, 100 migration, 89, 105 model complexity, 346 model domain, 61 model evaluation, 1, 2, 4, 5 model grid, 61 model scenario, ‘hindcast’ scenario, 292 model scenario, ‘worst case’ scenario, 297 model scenario, examples, 288 model simplification, 135 model testing, 127, 395, 406 model uncertainties, 77, 310 model validation, 290

425

model-building, 3 Monin–Obukhov length, 18 Monte Carlo, 346, 347 Monte Carlo technique, 361, 368 multi-compartment models, 269 Muroroa Atoll, 56 mushrooms, 321, 323, 325, 337 natural radionuclides, 223 Navier–Stokes equations, 57 near field, 290 non-equilibrium processes, 403 Nordic Seas, 74, 293 North Atlantic, 56, 291 North Sea, 56, 69, 292 Norwegian Atlantic Current, 291 Norwegian Coastal Current, 291 Norwegian Sea, 292 Novaya Semlya, 56, 296, 306 nuclear submarines, 300 Ob, 306 OML (Denmark), 37 Pacific Ocean, 56 paper, 318, 337 parameter, 3, 6, 7 parameter estimation, 4 parameter uncertainty, 8 parameter values, 131 particle reactive, 304 particle tracking, 68, 297, 308 partition coefficient, 79, 162, 178 Pasquill stability category, 17 passive adsorption, 203 performance assessment, 353–357, 367, 377, 383 plume rise, 30 plutonium, 317 Poisson process, 359, 360, 362 precipitation, 115 predictive (or blind) modelling, 133 probability distribution, 331 probability spaces, 354 puff models, 21 RADFORET, 344 radioactive decay process, 110 radioactive waste disposal, 353, 357, 360–362, 382 radioactivity, anthropogenic, 287 radioactivity, dissolved, 66, 77 radioactivity, particle-bound, 77 radiocaesium, 262–266, 268–270, 275–278, 280–283, 315–319, 321, 325, 327, 328, 337, 340, 343, 344, 346, 349

426

MODELLING RADIOACTIVITY IN THE ENVIRONMENT

radioecological models, 201, 210, 214 radioiodine, 268, 269, 273–275 radionuclide concentration factors, 203 radionuclide mobility, 110 radiostrontium, 262, 263, 269–273 range of predictions, 6 rank correlations, 374 real time decision support, 45 reality checks, 138 regional scale, 297 release function, 289, 290 reliability, 2, 4, 7 repository performance assessment (PA), 109 resuspension, 103–105 resuspension velocity, 238 retardation, 126 retardation factor, 181 RIFE, 319, 327–332, 336–338, 340, 343, 344, 347, 349 rivers, 147, 148, 172, 176, 181, 188, 189, 197 RODOS, 45 root sorption, 100 root uptake, 98–100, 102, 105

spatial, 261, 263, 266, 267, 276–279, 282, 283 spatial scales, 117 speciation, 98, 100 stability, 16–19, 21, 23, 24, 26, 28, 31, 34–37, 40, 43 stability criteria, 63, 69 stable analogous elements, 203 stable analogue, 207, 212, 219, 269 steady state, 316, 331, 332 stochastic uncertainty, 354, 361, 373 stratification, 160 structural uncertainty, 8 subjective uncertainty, 354, 372, 377, 380, 381 substrate–prey–predator, 217 sufficient confidence, 134 surface roughness, 15 suspended load, 165 suspended matter, 79, 224, 225, 227, 233–235, 237, 239, 240, 243–249, 251, 252, 255, 258, 304 suspended particulate matter, 161 suspension freezing, 304

sampling based, 7 saturation, 116 saturation mechanism, 274 scavenging, 304 scavenging coefficient, 27 scenario uncertainty, 8 sea ice, 304 seasonal dynamics, 397 seasonal variation, 90, 91, 95 sediment, 78, 224, 225, 227, 233–235, 238, 240, 242, 244–247, 249–251, 255, 258, 304 sedimentation, 160 selective assimilation, 205 Sellafield discharge, 76, 80, 291 semi-infinite cloud, 48 semi-mechanistic, 263, 283 SEMINAT, 319, 327, 329–332 sensitivity, 6–8, 10 sensitivity analysis, 376–378, 383 serial resistances, 90 settling velocity, 166, 238 shielding, 49 simulations, 280 size-effect, 404 Skagerrak, 296 soil–plant–animal, 261 soil-to-plant transfer factor, 261 solid phases, 233 sorption, 114, 203, 207, 209, 210, 219 sorption/desorption, 397 source depletion model, 29

temperature inversion, 16 tide, 223–225, 227, 229–232, 235, 236, 252 trajectory, 306 transfer, 261–263, 265–277, 281–283 transfer function, 188 transfer rate, 88, 101 transfer ratio, 266 translocation, 96–98, 105 Transpolar Drift, 304 transport model, Eulerian, 66, 291, 297 transport model, Lagrangian, 68, 291 transport model, conservative, 66 transport model, non-conservative, 77 transport modelling, 119 transuranic (TRU) waste, 355 tritium, 269, 270, 275 trophic level, 391, 409 turbidity, 224 turbulence, 14–16, 18, 19, 21, 26, 29, 31, 32, 35, 36, 51, 178 uncertainty, 2, 4–10, 39, 331, 338, 345, 346 uncertainty analysis, 376 underground laboratories, 129 uniform deposition, 280 validation, 2, 4, 6, 9, 10, 39, 173, 278, 290 VAMP, 318, 319

Subject Index

427

ventilation, 302 vertical standard deviation, 24

wet deposition, 27, 88, 94, 95 wood, 315–317, 321, 322, 325, 326, 329, 333–338, 346

Waste Isolation Pilot Plant, 353, 355, 356 weathering, 95, 96, 105

Yenisei, 306

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