A Water Quality Model for Shallow River-Lake Systems and Its Application in River Basin Management

A Water Quality Model for Shallow River-Lake Systems and its Application in River Basin Management

David Kneis

A dissertation submitted to the Faculty of Mathematics and Natural Sciences at the University of Potsdam, Germany for the degree of Doctor of Natural Sciences (Dr. rer. nat.) in Geoecology

Submitted 23 April 2007 Defended 19 June 2007 Published July 2007

Referees

Prof. Axel Bronstert University of Potsdam, Institute of Geoecology PD Dr. Dietrich Borchardt Kassel University, Center for Environmental Systems Research Prof. Jesús Carrera Institute of Earth Sciences ’Jaume Almera’, Barcelona This work is licensed under the Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/2.0/de/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.

Elektronisch veröffentlicht auf dem Publikationsserver der Universität Potsdam: http://opus.kobv.de/ubp/volltexte/2007/1464/ urn:nbn:de:kobv:517-opus-14647 [http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14647] Contents

Abstract 9

German abstract 11

1 Introduction 13 1.1 Context of the study ...... 13 1.2 Focus of this thesis ...... 14 1.3 Outline of the chapters ...... 15

2 State of the art and intentions for model development 17 2.1 The scope of water quality modeling ...... 17 2.2 A short overview on water quality modeling approaches ...... 18 2.2.1 Type of water body ...... 18 2.2.2 Implementation of turnover processes ...... 19 2.2.3 Spatial dimensionality ...... 20 2.2.4 Simulation time scale ...... 20 2.3 Objectives of model development ...... 21 2.3.1 Expert recommendations ...... 21 2.3.2 Speciﬁc targets ...... 21

3 The water quality simulation tool TRAM 25 3.1 Representation of the river network ...... 25 3.1.1 The reactor concept ...... 25 3.1.2 Discretization rules ...... 26 3.1.3 Network topology ...... 27 3.2 Hydrodynamics ...... 27 3.3 Turnover processes ...... 28 3.3.1 Terminology ...... 28 3.3.2 Use of the process matrix for model presentation ...... 30 3.3.3 User-deﬁned turnover processes in TRAM ...... 31 3.4 Handling of time series ...... 35 3.5 Computation algorithms ...... 35 3.5.1 Objectives of computations ...... 35 3.5.2 Reactive transport in stirred tank reactors ...... 35

5 6 Contents

3.5.3 Reactive transport in plug-ﬂow reactors ...... 38 3.5.4 Routing algorithm ...... 39 3.6 TRAM’s input data ...... 40 3.6.1 General remarks on data input ...... 40 3.6.2 Model units ...... 40 3.6.3 Network description ...... 41 3.6.4 Preprocessing of hydrodynamic information ...... 41 3.6.5 Deﬁnition of the turnover model ...... 45 3.6.6 Values of constants and initial concentrations ...... 49 3.6.7 Time series data ...... 49 3.6.8 Simulation control parameters ...... 50 3.7 TRAM’s output ...... 51 3.8 Possible enhancements of the current model version ...... 52 3.9 The source code ...... 52

4 Model application to the Lower Havel River 55 4.1 Objectives of modeling ...... 55 4.2 Introduction to the study site ...... 56 4.2.1 Hydrology ...... 56 4.2.2 The eutrophication problem ...... 57 4.2.3 Water quality overview ...... 61 4.3 Hydrodynamic submodel ...... 67 4.3.1 Sources and preprocessing of geometry data ...... 67 4.3.2 Boundary conditions ...... 68 4.3.3 Model calibration ...... 71 4.3.4 Exchange of data between the hydrodynamic model and TRAM ...... 71 4.4 Water quality submodel ...... 71 4.4.1 Simplified outline of the aquatic nutrient cycle ...... 73 4.4.2 Phosphorus storage in sediments and remobilization ...... 76 4.4.3 Representation of nutrient retention and remobilization in TRAM ...... 83 4.4.4 Parameter estimation and model performance ...... 91 4.5 Use of the calibrated model for system analysis ...... 99 4.5.1 Significance of phosphorus release ...... 99 4.5.2 Significance of nitrogen retention ...... 100 4.5.3 Visualization of patterns ...... 102 4.6 Management scenarios ...... 102 4.6.1 Integration of catchment models and TRAM ...... 102 4.6.2 Runoff and nutrient emissions from non-modeled subcatchments ...... 105 4.6.3 Description of the scenarios ...... 106 4.6.4 Simulation results ...... 108 4.6.5 Uncertainty of predictions ...... 112 Contents 7

5 Summary and conclusions 123 5.1 Model design ...... 123 5.2 Implications for water quality management of the Havel River ...... 124 5.2.1 Signiﬁcance of internal nutrient turnover ...... 124 5.2.2 Implications of simulated medium-term scenarios ...... 124 5.3 Challenges for future research ...... 126 5.3.1 Catchment modeling ...... 126 5.3.2 Classiﬁcation of the ecological status ...... 126 5.3.3 Simulation of nutrient turnover in rivers and lakes ...... 126 5.4 Final remarks ...... 128

List of ﬁgures 130

List of tables 135

Bibliography 136

Acknowledgments 147 8 Contents 9

Abstract port the implementation of the WFD in the lowland catchment of the Havel River located in North-East This work documents the development and appli- Germany. cation of a new model for simulating mass trans- Within the above-mentioned study TRAM was port and turnover in rivers and shallow lakes. applied with two goals in mind. In a first step, The simulation tool called ’TRAM’ is intended the model was used for identifying the magnitude to complement mesoscale eco-hydrological catch- as well as spatial and temporal patterns of nitro- ment models in studies on river basin management. gen retention and sediment phosphorus release in TRAM aims at describing the water quality of in- a 100 km stretch of the highly eutrophic Lower dividual water bodies, using problem- and scale- Havel River. From the system analysis, strongly adequate approaches for representing their hydro- simplified conceptual approaches for modeling N- logical and ecological characteristics. The need for retention and P-remobilization in the studied river- such flexible water quality analysis and prediction lake system were obtained. tools is expected to further increase during the im- In a second step, the impact of reduced external plementation of the European Water Framework nutrient loading on the nitrogen and phosphorus Directive (WFD) as well as in the context of cli- concentrations of the Havel River was simulated mate change research. (scenario analysis) taking into account internal re- The developed simulation tool consists of a tention/release. The boundary conditions for the transport and a reaction module with the latter be- scenario analysis such as runoff and nutrient emis- ing highly flexible with respect to the description sions from river basins were computed by project of turnover processes in the aquatic environment. partners using the catchment models SWIM and Therefore, simulation approaches of different com- ArcEGMO-Urban. Based on the output of TRAM, plexity can easily be tested and model formulations the considered options of emission control could fi- can be chosen in consideration of the problem at nally be evaluated using a site-specific assessment hand, knowledge of process functioning, and data scale which is compatible with the requirements of availability. Consequently, TRAM is suitable for the WFD. Uncertainties in the model predictions both heavily simplified engineering applications as were also examined. well as scientific ecosystem studies involving a According to simulation results, the target of the large number of state variables, interactions, and WFD – with respect to total phosphorus concentra- boundary conditions. tions in the Lower Havel River – could be achieved TRAM can easily be linked to catchment mod- in the medium-term, if the full potential for re- els off-line and it requires the use of external hy- ducing point and non-point emissions was tapped. drodynamic simulation software. Parametrization Furthermore, model results suggest that internal of the model and visualization of simulation re- phosphorus loading will ease off noticeably until sults are facilitated by the use of geographical in- 2015 due to a declining pool of sedimentary mo- formation systems as well as specific pre- and post- bile phosphate. Mass balance calculations revealed processors. that the lakes of the Lower Havel River are an important nitrogen sink. This natural retention effect TRAM has been developed within the research contributes significantly to the efforts aimed at re- project ’Management Options for the Havel River ducing the river’s nitrogen load. Basin’ funded by the German Ministry of Edu- If a sustainable improvement of the river sys- cation and Research. The project focused on the tem’s water quality is to be achieved, enhanced analysis of different options for reducing the nutri- measures to further reduce the immissions of both ent load of surface waters. It was intended to sup- phosphorus and nitrogen are required. 10 11

Kurzfassung wässer des Havel-Einzugsgebiets als Beitrag zur Erreichung der Ziele der WRRL. Die vorliegende Arbeit dokumentiert Konzept und Mit dem Einsatz von TRAM wurden zwei Anwendung eines Modells zur Simulation von Zielstellungen verfolgt: In einem ersten Schritt Stofftransport und -umsatz in Flüssen und Flach- wurden Bedeutung und Muster der gewässerin- seen. Das Simulationswerkzeug TRAM wurde als ternen Stickstoff-Retention sowie der Phosphor- Ergänzung zu mesoskaligen Wasser- und Stoff- Freisetzung aus See-Sedimenten quantifiziert haushaltsmodellen konzipiert, um die Beschaffen- (Systemanalyse). Auf dieser Basis konnten verein- heit einzelner Wasserkörper auf dieser räumlichen fachte, konzeptionelle Ansätze zur Beschreibung Skala in adequater Weise abbilden zu können. Die- von N-Retention und P-Remobilisierung abge- ser Aufgabenstellung kommt im Zuge der Umset- leitet werden. In einem zweiten Schritt wurden, zung der EU-Wasserrahmenrichtlinie (WRRL) be- unter Nutzung dieser Ansätze, die Auswirkungen sondere Bedeutung zu. verringerter externer Nährstoffeinträge auf gewäs- serinterne N- und P-Konzentrationen simuliert Das entwickelte Simulationsmodell TRAM (Szenario-Analysen) und die Unsicherheiten setzt sich aus einem Transport- und einem Re- der Modellrechnungen untersucht. Als Randbe- aktionsmodell zusammen. Letzteres zeichnet sich dingungen für die Szenario-Analysen dienten durch eine hohe Flexibilität hinsichtlich der Be- Simulationsergebnisse der öko-hydrologischen schreibung gewässerinterner Stoffumsatzprozesse Einzugsgebietsmodelle SWIM und ArcEGMO, aus. Es können mit geringem Aufwand unter- welche durch Projektpartner zur Verfügung ge- schiedlich komplexe Ansätze der Prozessbeschrei- stellt wurden. Die mittels TRAM berechneten bung getestet und die – je nach Problemstellung, Nährstoffkonzentrationen bildeten schließlich die Systemverständnis und Datenverfügbarkeit – an- Grundlage für eine Evaluierung der Handlungs- gemessene Modellformulierung gewählt werden. optionen anhand einer gewässertypspezifischen, TRAM eignet sich somit gleichermaßen für stark WRRL-konformen Bewertungsskala. vereinfachende Ingenieur-Anwendungen wie für Die Simulationsergebnisse zeigen, dass die Ziel- wissenschaftliche Analysen, die komplexe aquati- vorgabe der WRRL bezüglich Gesamt-Phosphor sche Ökosystemmodelle mit einer Vielzahl an Zu- im Falle der Unteren Havel mittelfristig erreicht standsvariablen, Interaktionen und Randbedingun- werden kann, wenn das Potential zur Senkung der gen erfordern. Einträge aus punktförmigen und diffusen Quel- Weitere Charakteristika von TRAM sind die len voll ausgeschöpft wird. Weiterhin kann im Koppelbarkeit mit öko-hydrologischen Einzugs- Zeitraum bis 2015 bereits mit einem merklichen gebietsmodellen sowie einem hydrodynamischen Nachlassen der internen Phosphat-Freisetzung auf- Modell, die Unterstützung von Modellparametri- grund einer Aushagerung der Sedimente gerechnet sierung und Visualisierungen durch Geografische werden. Mit Hilfe von Massenbilanzierungen ließ Informationssysteme (GIS) und ein klar struktu- sich zeigen, dass die Havelseen eine bedeutende riertes Daten-Management. Stickstoff-Senke darstellen. Dieser natürliche Re- tentionseffekt unterstützt wesentlich die Bemühun- TRAM wurde im Rahmen des BMBF- gen zur Verminderung der Stickstoff-Belastung. geförderten Forschungsprojekts ’Bewirtschaf- Im Sinne einer nachhaltigen Verbesserung der tungsmöglichkeiten im Einzugsgebiet der Havel’ Wassergüte der Unteren Havel erscheinen ver- entwickelt. Gegenstand dieses Projektes war die stärkte Anstrengungen zur weiteren Reduzierung Analyse von Handlungsoptionen zur Verminde- sowohl der Phosphor- als auch der Stickstoff- rung von Nährstoffeinträgen in die Oberflächenge- Emissionen geboten. 12 Chapter 1

Introduction

1.1 Context of the study phosphorus (LUA, 2005b). The project ’Manage- ment options for the Havel River Basin’ addressed The study presented here was undertaken in 2003– both problems but the clear emphasis was on the 2006 as a contribution to the joint research project eutrophication issue. ’Management options for the Havel River Basin’. The focal points of this research project are sum- As indicated by the project’s name, the spatial fo- marized below: cus was on the catchment of the Havel River which is located in the lowlands of North-East Germany • Target nutrient levels for rivers and lakes of and comprises a total area of about 24000 km2. different morphological and ecological types were derived (LUA, 2005a). The project was funded by the German Federal Ministry for Education and Research (BMBF) in • Management scenarios were designed which order to support the implementation of the Eu- include alternative options for reducing the ropean Water Framework Directive (WFD). The input of nitrogen (N) and phosphorus (P) into WFD became the legal basis of water management surface waters (Jacobs & Jessel, 2003). in the European Union in 2000 and its fundamen- • The impacts of the scenarios on water bud- tal aim is to achieve a good ecological status in gets, nutrient export rates, and in-river/lake all major rivers, lakes, aquifers, and coastal waters concentrations of N and P were analyzed by 2015 (European Commission, 2000) with lower quantitatively using eco-hydrological catch- targets being set for heavily modified water bodies. ment and water quality models (Biegel et al., When putting the WFD into practice, there are 2005; Habeck, 2006; Kneis et al., 2006; a number of aspects where scientific research is Krause, 2005). required, starting with the definition of the ’good • Finally, the scenarios and the included man- status’ for specific types of surface waters and pa- agement options were assessed with respect rameters. Scientific approaches are also neces- to effectiveness and efficiency in the light of sary for localizing and analyzing current deficits the previously identified water quality targets. and threats. Finally, the search for possible management strategies and their assessment involves The project’s results were intended to assist in the methods of quantitative system analysis. selection of management options by decision mak- With respect to the Havel River Basin, it is likely ers. A full outline of the project’s aims, research that the quality targets of the WFD will not be met guidelines, and major results can be found in the by 2015 in many rivers and lakes due to structural publications by Bronstert et al. (2005) and Bron- deficits and pollution by the nutrients nitrogen and stert & Itzerott (2006).

13 14 Chapter 1 Introduction

1.2 Focus of this thesis models being used in studies on river basin management. TRAM was designed to allow for a re- In the quantitative analysis and comparison of fined and scale-adequate representation of trans- management strategies, eco-hydrological catch- port and water quality processes in lowland rivers ment models are frequently used for simulating a and shallow lakes. river basin’s water budget and processes of mass transport (e.g. Kronvang et al., 1999; Schreiber The second part of this thesis covers the applica- et al., 2005). Since these models aim at simulat- tion of TRAM within the project ’Management op- ing meso- to large-scale catchments, their spatial tions for the Havel River Basin’. In this study, the discretization is usually coarse, typically with a focus of modeling was on the simulation of total grid size of 1×1 km. Such a coarse spatial reso- phosphorus and nitrogen concentrations in a large lution is chosen for the sake of computational effi- river-lake system under different management sce- ciency. Furthermore, the acquisition of detailed ge- narios. ographical information for whole catchments is of- From the analysis of field data and studies on ten impractical and strongly generalized data must similar systems it was known that internal turnover be used. processes such as nitrogen retention and phospho- As a consequence, the representation of individ- rus release affect the ambient nutrient concentra- ual water bodies in meso- or large-scale catchment tions in the studied river and river-lake systems models is mostly poor. The Water Framework Di- in general (Behrendt & Opitz, 2000; Kneis, 2002; rective, however, focuses just on the ecological sta- Schettler, 1995). Consequently, adequate model- tus of individual lakes and river sections. ing approaches had to be implemented in TRAM From this point of view, water quality simula- which take into account the turnover effects re- tion tools are required which can easily be linked ferred to earlier. Considering the complexity of the to (or integrated into) existing catchment models natural system on the one hand and the amount of (e.g. Van Griensven & Bauwens, 2003). The scope available data on the other hand, the development of such ’extensions’ must be a better representation and calibration of these approaches was a major of individual water bodies with respect to their hy- crux. draulic features and their source or sink functions For analyzing the impact of altered river basin in mass transport. In order to facilitate the coupling management, TRAM was coupled to the catch- of models, adjustable interfaces are an important ment models SWIM (Krysanova et al., 2000) and requisite. Furthermore, a flexible spatial and tem- ArcEGMO-Urban (Biegel et al., 2005). Thus, poral discretization of the water quality simulation TRAM’s boundary conditions, i.e. flow rates and tools is important for making coupled models ap- nutrient loads in the system’s inflows, were simu- plicable on different scales. lated by project partners for different management scenarios. The output from the coupled models fi- The development of a simple and flexible trans- nally provided the basis for assessing the effective- port and water quality model is addressed in the ness of possible management options with respect first part of this thesis. The model aims at bridg- to the requirements of the WFD. ing the identified gap between the requirements of the WFD on the one hand and the capabilities In summary, the two major aspects covered by this of present hydrological catchment models on the thesis are the design concepts of the newly devel- other hand. The model called TRAM1 is intended oped transport and water quality simulation tool to serve as an extension to hydrological catchment TRAM as well as the model’s first practical appli- 1TRAM simply stands for transport model cation in the context of river basin management. 1.3 Outline of the chapters 15

1.3 Outline of the chapters A major part of Chapter 4 is dedicated to the presentation of the hydrodynamic submodel for The subsequent text is sectioned into four major the Havel River and the approaches for simulating chapters: nutrient turnover in the studied river-lake system (Sect. 4.3–Sect. 4.4). An outline of aquatic nu- The introductory Chapter 2 provides an overview trient cycles is included in the description of the of the basic approaches to water quality modeling turnover submodel, giving an idea of the system’s with respect to the simulated type of water body, complexity. This basic knowledge is believed to be the description of turnover processes as well as the essential for understanding both the need for strong representation of space and time. simplifications in the model as well as their conse- Sect. 2.3 presents the intentions behind the de- quences. velopment of a new water quality model TRAM. After dealing with parameter estimation, The major objectives of model design are derived Sect. 4.5 illustrates which quantitative information from deficits of existing models and the require- on nutrient dynamics can be attained from the ments of river basin management. calibrated model. Chapter 3 provides a comprehensive outline of the The last part of the chapter (Sect. 4.6) deals basic concepts of TRAM. It is described how river with the simulation of management scenarios fo- networks can be approximated by a set of coupled cusing on a reduction of nutrient emissions. In reactors and the dependence on a hydrodynamic this context, the coupling of TRAM to other mod- model is addressed (Sect. 3.1 & Sect. 3.2). Further- els and the nature of the scenarios are addressed. more, the chapter describes how TRAM allows for In Sect. 4.6.4, the impact of altered nitrogen and the implementation of user-defined turnover sub- phosphorus loading on in-river nutrient levels as models and which algorithms are used for solv- predicted by TRAM is presented. The subsequent ing the equations of reactive transport (Sect. 3.3– Sect. 4.6.5 attempts to uncover the uncertainty as- Sect. 3.5). A substantial part of Chapter 3 covers sociated with the simulation results. the description of TRAM’s input data and methods Chapter 5 of preprocessing (Sect. 3.6). It is pointed out which In the final , major results from the mod- kinds of information are required for a successful eling study are summarized and conclusions are application of the model. However, the chapter is drawn. Suggestions for future research are made, not intended to replace a user’s manual. In the final based on the experiences described in Chapter 4. sections, some aspects of the model’s source code and possible objectives for further development are discussed.

Chapter 4 describes the use of TRAM for simulating the dynamics of nitrogen and phosphorus in the Lower Havel River. In Sect. 4.1, the objectives of water quality simulation within the project ’Management options for the Havel River Basin’ are explained in detail. Subsequently, an introduction to the study site is given which provides basic information on hydrological conditions as well as the river’s major water quality problem which is eutrophication. 16 Chapter 1 Introduction Chapter 2

State of the art and intentions for model development

2.1 The scope of water quality of the WFD four basic steps can be distinguished modeling (adapted from Von Keitz & Schmalholz, 2002): 1. the identification of significant threats to wa- In general, a water quality model is a simplified ters, mathematical representation of an aquatic system. 2. the definition of appropriate and site-specific Water quality models aim at describing the dynam- quality targets, ics or steady state of the system’s major physical, chemical, and biological state variables. In order to 3. the evaluation of potential management ac- do so, the models take into account transport phe- tions, nomena as well as biogeochemical turnover pro- 4. the implementation of measures which are cesses. proven to be most efficient. Water quality models are primarily used in re- In step 2–4, quantitative methods of system anal- search on complex aquatic ecosystems. Simula- ysis can be helpful. By using appropriate water tion tools provide a means to integrate the present quality simulation tools it becomes possible: understanding of the system’s functioning and to compare model predictions with observations. The • to evaluate the contribution of different deviation between simulated and observed data in- sources of pollution (pressures and impacts dicates how incomplete our understanding of the analysis), system actually is. From analyses of the simula- • to identify reasonable quality targets, e.g. tion errors, the impact of those processes which are concentrations for nutrients, toxins, etc., not explicitly considered in the model can be fig- to assess the impact of management actions ured out. Furthermore, the interpretation of resid- • on water quality before actual measures are uals stimulates the formulation of new hypotheses taken (scenario analyses). By identifying ef- on the system’s functioning. ficient and inefficient measures costs can be Models which have proven their value for a cer- saved and management can be optimized. tain problem are also used in water resources management. For example, water quality control is one The demands for water quality modeling rise con- of the main issues of the European Water Frame- stantly. Although new simulation tools are devel- work Directive. In the implementation process oped and existing ones are improved, the currently

17 18 Chapter 2 State of the art and intentions for model development available simulation models often cannot yet fully essarily focus on the same state variables and in- satisfy the requirements of the practice. teractions. The classical objective of RWQM was to simulate the effect of organic pollution on a river’s 2.2 A short overview on water oxygen level because of its significance for aquatic quality modeling approaches life and its variability in space and time. The pio- neering Streeter-Phelps model, describing the bal- Numerous water quality models have been devel- ance between O2 consumption and reaeration, is oped in recent decades and new approaches for presented in all relevant textbooks (e.g. Chapra, special applications are frequently published. A 1997). Another important field of RWQM is the tabular overview of established, full-fledged river simulation of the transport and decay of toxic sub- water quality modeling systems can be found in stances such as organic chemicals or heavy metals Reichert et al. (2001) and Borchardt (1998). which are harmful to river biota and humans. For Instead of discussing selected models in de- example, the Rhine Alarm Model or the ’TOXI’ tail, the following sections give a more general module of the WASP model package (Ambrose overview on the different approaches to water qual- et al., 2001) are specialized on problems like these. ity simulation. The focus is placed on (1) the type In recent decades, the focus of RWQM has shifted of water body, (2) the implementation of turnover towards a more complete description of the river processes, (3) the spatial dimensionality, and (4) ecosystem, often considering nutrient cycling and the time scale. the growth of both heterotrophic and autotrophic organisms. QUAL2K (Chapra et al., 2006) which 2.2.1 Type of water body is an enhanced version of the famous QUAL-2E (Brown & Barnwell, 1987), the ’EUTRO’ module First of all, a distinction between river water qual- of WASP (Ambrose et al., 2001), QSIM (Kirch- ity models (RWQM) and lake models (LWQM) is esch & Schöl, 1999), or the ATV model (Müller, necessary. 2002) are examples of more complex RWQM. Rivers are characterized by a continuous, mostly The typical scope of LWQM is to analyze how unidirectional, movement of water. The depth is in-lake nutrient concentrations are related to nutri- usually small (in the order of several cm to a few ent loading from the catchment. For a long time, meters) and turbulences resulting from shear stress statistical-empirical models were used (e.g. Vol- at the river bed prevent vertical stratification. In a lenweider, 1979; Vollenweider & Kerekes, 1982). typical lake, the through-flow rate Q is small com- As in RWQM, state of the art dynamic simulation pared to the total volume of the water body V . models include a detailed description of nutrient Thus, the theoretical residence time of the water in cycling and primary production. Often, organisms a lake V/Q is large compared to a channel section of different trophic levels (phytoplankton, bacte- of equal length. Mixing of the lake water is usually ria, zooplankton, sometimes even fish) are con- controlled by wind action and – in deep lakes of the sidered as state variables. Examples of sophisti- temperate zone – thermal stratification occurs. cated process-oriented lake ecosystem models are Obviously, different approaches are required PCLake (Van Puijenbroek et al., 2004) or SALMO for describing the hydrodynamics in running and (Petzoldt et al., 2005). standing waters. Due to the great differences in Although a separate development of RWQM water residence time, turbulence, and depth, the and LWQM is reasonable on the one hand, both dominating turnover processes may be different as types of models need to be consistently linked for well. Therefore, RWQM and LWQM do not nec- many applications, because lakes usually receive 2.2 A short overview on water quality modeling approaches 19 their water through tributaries and some rivers ex- acidic waters such as mining ponds. In this light, hibit lake-like expansion zones. existing water quality models can be distinguished A rather new simulation software which is ca- by their flexibility with respect to the description pable of handling stratified as well as vertically of turnover processes. mixed systems while solving the complete hy- Models which are specially designed for a well drodynamic equations is CE-QUAL-W2 (Cole & defined problem usually have a fixed set of state Wells, 2006). variables. Also fixed is the description of their Another approach is taken by multi- interaction and only parameter values may be ad- compartment water quality models (MWQM) justed to fit the model to the river or lake of interest. like AQUASIM (Reichert, 1998). This model While the development time for such models may allows multiple types of ’reactors’ to be simulated be (relatively) short, they suffer from a lack of flex- that differ with respect to hydrodynamics and the ibility. If, for example, a new process or variable dominant transport processes. By linking advec- turns out to be relevant but has not been consid- tive, dispersive, or advective-dispersive reactors, ered in the original code, changes to the model’s it becomes possible to approximate river-lake core are required. The same is true if simplifica- systems as well. tions to the process description become necessary For a proper simulation of river and lake wa- because data for a certain variable of parameter are ter quality, the inclusion of sediment-water inter- unavailable – which is often the case. actions is often essential. In many water quality models, benthic fluxes can be taken into account Different approaches have been taken in the past as boundary conditions only. However, advanced to tackle this problem. For example the WASP MWQM such as AQUASIM also allow for the sim- package (Ambrose et al., 2001) was split into the ulation of transport phenomena in porous bed sed- modules EUTRO and TOXI. While the first han- iments. dles oxygen dynamics and nutrient cycles, the sec- While a number of water quality models in- ond module is designed for simulating the trans- clude routines to calculate the hydrodynamics, oth- port of organic chemicals, heavy metals, and sed- ers only (or optionally) provide an interface for im- iment. Furthermore, the EUTRO module may be porting hydrographs of stage, flow, and other hy- run in different levels of complexity. draulic variables computed by external software. Flexibility is taken one step further in open- structure water quality models. In these models 2.2.2 Implementation of turnover pro- it is the user, not the developer, who decides on cesses the number and kind of state variables, boundary conditions, and the nature of their interactions. 1 Although the turnover processes of interest are of- The AQUASIM tool (Reichert, 1998) and the Bio- ten similar for a certain type of water body, differ- geochemical Reactions Network Simulator BRNS ent environmental conditions may require adapta- (Regnier et al., 2002; Thullner et al., 2005) are ex- tions. For example, primary production is of minor amples for such open-structure models. The sim- importance in small creeks but it may significantly ulation tools ’ECO Lab’ (DHI, 2006) and DU- influence the water quality of large rivers. And, − FLOW follow a similar approach. The open- whereas bicarbonate (HCO3 ) can be assumed to structure model WEST (World Wide Engine for be always available in most lakes for assimilation Simulation, Training and Automation), which is by algae, it may be a limiting factor in heavily mainly applied to simulations of wastewater treat- 1The terms turnover and conversion process/model are ment plants, was also used for river water quality used synonymously throughout this text. modeling (Dekissa et al., 2004). 20 Chapter 2 State of the art and intentions for model development

Open-structure water quality models have the A one-dimensional representation is often suit- great advantage of being applicable to systems of able for rivers where turbulence guarantees fast lat- any complexity with respect to the number of sim- eral and vertical mixing. Then, concentration gra- ulated variables and interactions. They support dients are mainly oriented parallel to the stream empirical, conceptual as well as process-oriented lines. However, a 1D approximation is insufficient system descriptions. They also allow for determin- if one is interested in near-field phenomena (e.g. istic as well as stochastic simulations. The open the concentrations just downstream of an effluent structure provides the basis for adapting the model or tributary) where lateral and vertical homogene- to the present knowledge of process functioning ity is not yet achieved. 1D vertical models are and the available amount of information (initial employed in the simulation of deep lakes because values, boundary conditions, parameters). stratification often causes sharp vertical concentra- While a model with a fixed set of processes and tion gradients. variables ’only’ needs to be supplied with data, A two-dimensional representation of the model open-structure models require the model itself to domain may be suitable for simulating wide but be formulated first. If no template is available for shallow channels, channel junctions and estuaries the specific problem this may cost additional time. (2D horizontal) or deep, narrow reservoirs with a However, the investment pays out: Once a tem- single major inflow (2D vertical). plate is created, it can easily be changed and ex- The application of three-dimensional models is tended for new applications. Secondly, the mod- only necessary if spatial gradients of the state vari- eler is forced to develop a deeper understanding of ables occur with respect to all three spatial axes. the simulated processes and their simplified repre- This might be the case if hydrodynamics are re- sentation. A modeler with this insight is probably ally complex as in deep reservoirs with multiple able to assess the simulation results and their relia- tributaries or in the ocean. In many surface water bility more realistically. In the author’s opinion, an applications a restriction in the dimensionality is open model structure is, in most situations, prefer- possible and, if so, should be adopted. able. 2.2.4 Simulation time scale 2.2.3 Spatial dimensionality Water quality models are applied on all time scales. Water quality models may be zero-, one-, two-, or For example, the simulation of an accidental spill three-dimensional with respect to the spatial repre- of toxic substances into a river may be limited to sentation of the model domain. The proper choice a time period of several hours or days, depending depends on the occurrence of spatial gradients in on the initial concentration, decay characteristics, the state variables. For example, a volume of wa- dilution rates, and the river length. In contrast, ter which is laterally and vertically fully mixed can the simulation of eutrophication or restoration of be handled with a zero dimensional model as there a large lake with a long residence time may re- are no spatial concentration gradients at all. Each quire a simulation over many decades. In order of the simulated state variables is a time depen- to limit computation times to a reasonable level, dent scalar. Shallow ponds and lakes can often the degree of simplification and generalization usu- be treated in this way. Also river sections have ally increases with the length of the simulation pe- been approximated by a series of interlinked, fully riod. A long-term water quality model focusing on mixed water bodies which is known as the tanks- the estimation of monthly average concentrations in-series approach (Jokiel, 1995; Dekissa et al., rather than on instantaneous values was developed 2004). by Schlaeger (2003). 2.3 Objectives of model development 21

In some applications one is less interested in the relevant features of river regulation and water dynamics of a system but in the steady state result- resources management. ing from certain boundary conditions (e.g. long- term lake nutrient balances). Then, a dynamic sim- • As water quality simulation models produce ulation may be completely unnecessary. However, large amounts of data, the development of im- for highly non-linear systems, steady state con- proved methods of pre- and postprocessing as ditions can often be obtained most conveniently well as data visualization is proposed. The as the result of a very long-term simulation with fact that many programs do not support batch steady input. imports of large data sets (e.g. boundary condition time series or geometry data) is identified as a major hindrance for efficient model 2.3 Objectives of model develop- application. ment • It is suggested that newly developed models should be platform independent and well doc- 2.3.1 Expert recommendations umented to make them useful for others. It is In 2002, the German Federal Water Institute pub- considered a serious waste of resources that lished a list of recommendations for future re- many existing models of general interest lack search and development in the field of water qual- sufficient documentation or cannot be ported ity modeling (BfG, 2002). Those recommenda- from one platform to another. tions which were considered in the development of TRAM include the following: 2.3.2 Specific targets • First of all, water quality modeling should be The subsequent paragraphs summarize the funda- regarded as an optimization problem which mental objectives which guided the development aims at balancing the level of model complex- of the TRAM model presented in Chapter 3. These ity with the reliability of model predictions specific targets were derived from the requirements and the effort for data acquisition. Against of river basin management in the context of the this background, the development and use of Water Framework Directive (see Sect. 1.2), the modular simulation tools which can be run in above recommendations by BfG (2002), as well as different levels of complexity is proposed. personal modeling experience.

• According to BfG (2002), a better represen- Open-structure turnover model tation of sediment-water interactions in water The required complexity of a turnover model de- quality simulations should be given special at- pends on the nature of the problem, the knowledge tention. In spite of many decades of research about the relevant processes, and the availability of on this topic, a need for combined field stud- data for model calibration and verification. Fur- ies and model development is diagnosed. thermore, in different applications, the focus of modeling may be on different state variables and • The use of state-of-the-art hydrodynamic ap- interactions. Consequently, TRAM’s conversion proaches is recommended in order to make model was intended to be of the open-structure water quality models applicable to river net- type (see Sect. 2.2.2) so as to offer maximum flex- works of complex geometry, including looped ibility. In particular, TRAM should support the systems. It is pointed out that the hydrody- use of simple empirical as well as more advanced, namic submodels must be able to consider all process-based model formulations. Also, the pos- 22 Chapter 2 State of the art and intentions for model development sible number of state variables or boundary condi- postprocessing software for visualization and the tions should not be restricted in any way. creation of maps2. For many biogeochemical processes, model In order to guarantee maximum flexibility, a concepts were already published decades ago. tight coupling of TRAM to any specific software With its open structure, TRAM was intended to al- was to be avoided. Instead, the model should sup- low for a convenient re-implementation of such export the import and export of data in simple stan- isting concepts in an up-to-date simulation frame- dard formats which can easily be created and read work. by any GIS with little conversion effort. In most of the existing water quality models, a GIS interface Hydrodynamic submodel is still lacking. The poor representation of the hydrodynamics in Handling of data hydrological catchment models was identified as a As pointed out in BfG (2002) the usefulness of a severe deficit in Sect. 1.2. In order to overcome model strongly depends on how efficient data input this, TRAM was intended to make use of detailed and output is. Some guidelines for efficient and cross-section data for representing the geometry of safe data transfer which have been considered in rivers and lakes. Furthermore, the simulation of the development of TRAM are listed below. control structures such as weirs, gates, and flow diversions was considered to be essential because of • The model’s input files conform to exist- its influence on turnover processes. ing standards and they can easily be created Since well established hydrodynamic simulation by preprocessing software or by data base models exist, it was planned to link TRAM with queries. such external software. However, depending on the • The model’s output files are directly loadable nature of the river system, the availability of geom- by software for plotting, statistical evaluation, etry data, and the required accuracy, different ap- or data storage. proaches for simulating the hydrodynamics may be preferred. Therefore, instead of being tightly cou- • Different types of information are stored in pled to a specific software, TRAM was designed to separate files. All data items are identified by read stage and flow hydrographs from input files. keywords or described by comments. Such files can easily be created from the output of • The model can be run in batch mode without whatever hydrodynamic or hydrological model. any user interaction.

Linkage to geographic information systems Execution speed The linkage of TRAM to a geographical informa- If a water quality model is to be used for long-term tion system (GIS) was considered to be essential simulations over several years or decades, execu- for several reasons. First of all, model parametrization speed matters. Speed becomes even more rel- tion can be sped up significantly if all data on evant if automatic calibration procedures are ap- model geometry (river network plan, cross-section plied and if sensitivity or uncertainty analyses are data, lake bathymetry, etc.) are preprocessed us- to be carried out. Thus, TRAM had to be imple- ing GIS facilities. Furthermore, a full spatial refer- mented in a computer language which, in the first ence of all simulated water bodies is helpful when place, allows for the creation of fast and platform TRAM is coupled to a hydrodynamic or catchment independent code. model because the locations of boundary conditions can easily be identified. Finally, simulation 2Water quality maps are an essential part of status reports results may be transfered back into GIS or other and management plans according to the WFD. 2.3 Objectives of model development 23

Availability of software and source code Among the existing water quality models some are free software but others require the payment of license fees and the source code is unavailable. While proprietary water quality models seem appropriate for commercial use, free modeling software is needed in research and education. This is not only a matter of cost. The availability of the source code guarantees that the model can be adapted to a certain scientiﬁc application and last but not least it may promote further development. Consequently, it was planned to make TRAM available to the public at an adequate state of development and documentation.

When the work on this thesis began in 2003, none of the freely available water quality models fulﬁlled the requirements addressed above. The development of the TRAM model introduced in Chapter 3 was a logical consequence. 24 Chapter 2 State of the art and intentions for model development Chapter 3

The water quality simulation tool TRAM

3.1 Representation of the river whether due to molecular diffusion or turbulence, network is assumed to be negligible. However, within a control volume itself, spatial homogeneity of the 3.1.1 The reactor concept concentrations is assumed. Any CV entering a PFR at its upstream end is released at its down- In TRAM, the natural river system is represented stream boundary with a time lag controlled by the by a network of interlinked reactors. Two basic flow velocity. The PFR concept is used for repre- types of reactors are currently available: stirred senting river and channel sections where advective tanks (STR) and plug-flow reactors (PFR). Both transport dominates over dispersive processes. concepts are widely used in water quality modeling The disregard of dispersion effects in PFR is (e.g. Chapra, 1997; James, 1993; Reichert, 1998). quite a significant simplification compared to natural conditions. If longitudinal mixing in river or Stirred tank reactors channel sections cannot be neglected1, one must The fundamental property of a STR is complete use an array of coupled PFR and STR in order to mixing of the contained water which implies an approximate advective-dispersive transport. Sev- instantaneous spreading of incoming dissolved or eral authors (e.g. Jokiel, 1995; Dekissa et al., 2004) suspended matter all over the water body. Thus, a used tanks-in-series models instead for represent- STR is characterized by spatial homogeneity and ing advective-dispersive transport in river sections. any mass transport through a STR is due to dis- In this approach, the magnitude of dispersion at persive processes only. Advection-bound transport a given flow rate is controlled by the spatial dis- is assumed to be negligible. The stirred tank con- cretization of the model, i.e. the number of STR cept allows for a reasonable approximation of mass per reach length (Jokiel, 1995). transport in shallow, polymictic (i.e. non-stratified) There are cases where it is justifiable to neglect lakes. mixing in river or channel sections, for instance if Plug-flow reactors the model is driven by boundary conditions with a In contrast to STR, the transport of dissolved or coarse time resolution. Since input loads and flow suspended matter in plug-flow reactors is due to rates are averages over the length of a model time advection only. The flow is assumed to be unidi- step the input is ’already mixed’. Another exam- rectional and mass transport can be imagined as 1 a sequence of separate control volumes (CV) en- Dispersion cannot be neglected if large spatial concentration gradients occur, e.g. due to highly time-variable bound- tering and leaving the reactor in tight succession. ary conditions. The magnitude of dispersion is largely con- The exchange of matter between neighboring CV, trolled by the occurrence and relative size of dead zones.

25 26 Chapter 3 The water quality simulation tool TRAM

ple is river-lake systems where mixing primarily Reach a occurs in the lakes. Concentration hydrographs at a lake’s outlet are always smooth, i.e. dc/dt La(t) is low. Consequently, spatial concentration gradi- Qb1 (t) Qb2 (t) ents (dc/dx) in river sections downstream of lakes Reach b1 are low too, and thus dispersion is of minor impor- Reach b2 tance.

Further compartments Figure 3.1: Three reaches forming a split flow junction. Many turnover processes in surface waters are La(t): Export load hydrographs of reach a, Qb1(t) & bound to interfaces between the liquid and solid Qb2(t): Inflows of the downstream reaches. phases due to sorption effects and the role of attached biofilms. Often, the interstitial of the bot- is true when TRAM is applied to impounded chan- tom sediments and the surfaces of aquatic vege- nels or free-flowing rivers with significant changes tation or rock are of special relevance and must in bed slope. This is because TRAM assumes that therefore be considered as additional model com- the slope of the water surface (dW/dx) within a partments (e.g. Borchardt & Reichert, 2001; Re- plug-flow reactor is constant, i.e. dW/dx 6= f(x). ichert et al., 2001). Although TRAM does not pro- In the application to the Lower Havel River (Chap- vide a special type of reactor for representing such ter 4), accuracy requirements allowed for the use compartments, turnover processes at non-mobile of plug-flow reactors with a length of several hun- surfaces can already be simulated for STR (see dred meters to about 2 km because the slope of the Sect. 3.3). water surface does not exceed a few cm/km. A special case emerges if river networks with 3.1.2 Discretization rules split flow junctions (looped networks) as shown in Fig. 3.1 are simulated. In the example, the import The application of the above concept to a natural load hydrographs for the reaches b1 and b2 down- river network requires the latter to be subdivided stream of the junction are computed from their in- into reactors. Clearly, a subdivision is always nec- flow hydrographs Qb1(t) and Qb2(t) and the export essary where advection-dominated river sections load hydrograph of the upstream reactor La(t) ac- (represented as PFR) interface with mixed water cording to the following equations: bodies (modeled as STR). Apart from this, a further discretization of river Qb1(t) reaches into a sequence of separate PFR may be Lb1(t)= La(t) · (3.1) Qb1(t)+ Qb2(t) advisable if hydraulic features change significantly Q (t) along the reach. This is necessary because TRAM L (t)= L (t) · b2 (3.2) b2 a Q (t)+ Q (t) makes use of reach-averaged values of basic hy- b1 b2 draulic variables in turnover computations instead For Eq. 3.1 and 3.2 to be correct, the inflow rates of accounting for the situation at every single Qb1(t) and Qb2(t) must not contain contributions cross-section (see Table 3.3 in 3.3.3 and Eq. 3.17– of any additional lateral inflow to the reactors b1 3.20 in Sect. 3.6.4 for details). Consequently, river and b2. Thus, a subdivision of plug-flow reactors or channel sections with significant morphologi- might be required if lateral inflow just downstream cal irregularities (flow depth, wet perimeter, etc.) of a split flow junction is to be simulated. should not be modeled as a single PFR but by a Finally, plug-flow reactors might need to be sub- sequence of shorter plug-flow reactors. The same divided into multiple PFR in order to simulate dif- 3.2 Hydrodynamics 27 fuse pollution. Since the current version of TRAM 1 does not allow for a uniform distribution of load 5 4 8 boundary conditions over the length of the reach, 3 7 the use of a couple of short PFR, each receiving a 9 6 2 fraction of the total diffuse load, is necessary. 4 3 1 5

3.1.3 Network topology Figure 3.2: A sample network of plug-flow and stirred The routing of matter through a system of inter- tank reactors with stream orders increasing in flow direction. connected PFR and STR requires information on the network topology or hierarchy. Each reactor must ’know’ its upstream neighbor(s) which sup- Fig. 3.2 shows an example of a simple network of ply the flow and load boundary conditions. In the reactors with assigned stream orders. case of flow diversions, a reactor must also know its ’parallel’ reactors which share the same boundary conditions (see Eq. 3.1 & Eq. 3.2). 3.2 Hydrodynamics Essentially, stream orders must be assigned to all reactors in order to reflect the upstream- If the transport of dissolved or suspended matter downstream relations. The stream order takes a in surface waters is to be modeled, a simulation of value of 1 for the most upstream section(s) of the the hydrodynamics is a prerequisite. For example, network and the numbers increase in flow direc- information on the flow and the storage volume of tion. The highest stream order is always attached a reach is required in order to determine the trav- to the reactor located furthest downstream, i.e. the eltime of control volumes in a plug-flow reactor. system’s outlet. Similarly, information on the stage-dependent re- A utility program was developed for automatic actor volume and flow rates are required for com- stream order assignment which is also capable puting the residence time of water and mass in a of handling looped river networks. Input to this STR. If the simulated components undergo reac- program are two files: The first file defines the tions (e.g. degradation or settling), it is the travel- shape and spatial position of the plug-flow reac- or residence time which regulates the concentra- tors’ stream lines, the second optional file provides tions in the reactor’s outflow. the corresponding information for STR (as poly- Because the simulation of hydrodynamics is not gon outlines). included in the TRAM program external software The preprocessing yields a table listing the fol- must be used. In a study on the Havel River (Chap- lowing information for each reactor: ter 4), the unsteady 1D hydrodynamic model HEC- RAS (USACE, 2002) was applied. The use of an 1. the user-defined reactor name, external program has a number of pros and cons. On the one hand, the separation of hydrodynamics 2. the assigned stream order, and water quality simulations is advantageous as 3. a list of the reactor(s) that supply the upper the user can choose a flow computation approach boundary conditions, of the desired complexity, e.g. a hydrodynamic model or a simple hydrologic routing procedure. 4. a list of reactors that share the same upstream On the other hand, the use of separate models load boundary condition (only non-empty for is a disadvantage because feedbacks between wa- reactors downstream of split flow junctions). ter quality and hydrodynamic processes cannot be 28 Chapter 3 The water quality simulation tool TRAM taken into account without costly iterative simu- for a linear combination of species that is not lations2. An example where the consideration of affected by aqueous equilibrium reactions. such feedbacks might be necessary is the growth Species: If an element of interest occurs in a num- of macrophytes in streams. Due to increased flow ber of different compounds, the latter are of- resistance (higher Manning values), flow velocities ten called species with respect to this element. can drop significantly and the flow depth increases For example, the component dissolved inor- accordingly. This again, may have an influence on ganic nitrogen incorporates the three species − − further water quality parameters, such as the reten- NO , NO , and NH+. tion of phosphorus (Schulz, 2004). 3 2 4 For the sake of computational efficiency, TRAM In most instances, the universal term component does not read raw stage3 and flow hydrographs as will be used throughout this text. Often, a basic output by a hydrodynamic model. Instead, the time distinction into organic and inorganic as well as series of hydrological variables need to be prepro- dissolved and suspended (i.e. particulate) compo- cessed as described in Sect. 3.6.4. nents is useful. Only a few components which can be found 3.3 Turnover processes in natural waters are conservative or inert. In- stead, most components in the aquatic environment 3.3.1 Terminology are not only subject to transport but they also undergo conversion or turnover. The two synony- mous terms turnover and conversion are quite un- Basic terms specific and may cover processes like: Prior to describing how turnover processes can be simulated with TRAM, it is worth defining some • the sorption of ions basic terms. • settling of suspended matter Element: A class of atoms listed in the Periodic • precipitation of insoluble compounds Table. Elements are easily referenced by their • decay of organic compounds unique symbols (C: Carbon, O: Oxygen, ...). • takeup by organisms Compound: A compound is a substance with a •··· fixed ratio of elements determining its composition. E.g. all organic substances and wa- Obviously, some of these turnover or conversion ter itself are compounds. processes are true chemical reactions (e.g. precip- Component: A component can be either a com- itation). In other cases, some fraction of a compound or even a lumped group of compounds. ponent may be converted into another one without A component’s concentration is often ex- reactions taking place. Settling for example is ac- pressed with reference to an element of in- tually a transport phenomenon, not a reaction, but terest (e.g. dissolved inorganic nitrogen or it converts the suspended fraction of a component organic carbon). According to Molins et al. into the corresponding sediment bound fraction. In (2004), the term component should be used TRAM, all turnover processes are represented in the same way, no matter whether these are true 2 Alternatively, the hydrodynamic and water quality model chemical reactions or not. could be coupled on-line with data exchange in every time In the study of mass balances, the term retention step. 3The terms stage and water surface elevation are used syn- is often used for those turnover processes which onymously throughout this text. cause a decrease in a component’s concentration 3.3 Turnover processes 29 during its passage through a lake or river section. For example, the sorption of ions to mineral sur- Retention usually means that some fraction of the faces or the dissociation of acids may take place load is either retained in the sediment, or it is trans- in a period of milliseconds. In such cases, Tp is formed (such as degradable organic compounds) or much shorter than the residence time of the water it is released to the atmosphere. in the considered control volume Tr, unless flow velocities are extraordinarily large or the control Mobile and immobile components volume is very small. When reactions take place Often, sediment-water interactions such as settling, on a very short time scale compared to the time storage, and remobilization significantly affect the scale of transport processes, i.e. T ≪ T , lo- quality of the overlying water column or pelagic p r cal equilibrium can be assumed (Saaltink et al., zone. In many water quality models, these inter- 2004). Consequently, the term equilibrium reac- actions are considered by source and sink terms tions or equilibrium processes is used (Stumm & only. If mass balances are to be closed, elemen- Morgan, 1996). tal concentrations in the sediment or attached (ses- sile) biomass must be considered as variables in the In contrast to that, conversion processes which model. Consequently, TRAM distinguishes mobile are controlled by the activity of organisms are of- and immobile components. ten rather slow. For example, the half life of Mobile components are generally transported most organic matter compounds in sewage treat- with the flow, i.e. they are either dissolved or sus- ment may be in the order of some minutes to days pended in the water column. Their concentration is while other organic compounds may be degradated typically specified in g m−3 which is equivalent to within decades only. In those cases where Tp is mg l−1 or ppm. Only loads of mobile components greater than Tr equilibrium cannot be assumed and may be exchanged between adjacent reactors. one speaks of kinetic processes. Immobile components are somehow attached to The distinction between kinetic and equilibrium the bottom of a river or lake. Typically, immo- processes has special importance: While the simu- bile components exist in the sediment and the cor- lation of kinetic processes generally requires ordi- responding unit is either g m−3, if the thickness nary differential equations (ODE) to be integrated of the considered sediment layer is specified, or over time, the concentrations of components which g m−2 (areal concentration). In the first case, to- are in equilibrium can often be computed more tal concentrations (mass per volume of sediment) easily using algebraic equations (Reichert, 1998; and pore water concentrations (mass per volume Thullner et al., 2005). of pore water) need to be distinguished. At present, immobile components can be sim- Process order ulated in STR only, because the implementa- Kinetic processes are often characterized by their tion for plug-flow reactors is still incomplete (see reaction order. The order of a reaction reflects the Sect. 3.8). dependence of the reaction rate on the concentration of the reactants. Most elementary reactions are Kinetic vs. equilibrium processes first- or second-order as they involve either one or In general, the conversion processes taking place in two reactants (Stumm & Morgan, 1996). For ex- a control volume of water will tend to an equilib- ample, the elementary reaction A → X is of first rium. After external perturbations, e.g. due to the order as the corresponding rate expression is input of mass, equilibrium conditions will reestab- lish within a characteristic time Tp. For some conversion processes, equilibrium dCA conditions may be reached almost instantaneously. = −k1 · CA (3.3) dt 30 Chapter 3 The water quality simulation tool TRAM

where CA is the concentration of A and k1 is a rate The process matrix also reflects mass balances constant (t−1). Examples of complex first order if proper units are chosen for the simulated com- processes are the decay of a degradable substance ponents (i.e. all components with a common base by microbes but also the growth of a population in element X are expressed in g X m−3). If the sum its initial phase. of the stoichiometry coefficients in one row is zero, The reaction A + A → X is second order be- the mass balance with respect to the process listed cause the corresponding rate expression is in the row header is closed. If different units are used for the components (e.g. mass/volume and mass/area) unit conversion factors must be taken dCA 2 into account in mass balance checks. = −k2 · CA (3.4) dt The change in the concentration of component − i due to process j is calculated by multiplying the where the unit of the constant k is ([C] · t) 1. 2 stoichiometry coefficients in column i and row j The rate law of a third reaction A + B → X is (Q ) with the corresponding process rate R . R shown in Eq. 3.5 with C being the concentration i,j j j B expresses the velocity of the conversion process. of reactant B. As in Eq. 3.4, the unit of k is ([C] · 2 The full differential equation reflecting the impact t)−1. of all m conversion processes is obtained by sum- ming up the products Qi,j · Rj for all rows of the dC matrix (Eq. 3.6). A = −k · C · C (3.5) dt 2 A B m In this case, the reaction is said to be first order in dC i = Q · R (3.6) A and first order in B, but the total reaction order is dt i,j j j=1 two. Thus, the total order of a reaction/process is X equal to the sum of the exponents associated with The mass of a component within a control volume each species appearing in the rate expression. is not exclusively affected by conversion processes but also by the exchange of mass with neighbor- 3.3.2 Use of the process matrix for model ing control volumes, i.e. by transport phenomena. presentation The corresponding transport terms are usually not presented in the process matrix. For comprehensively presenting a multi- Fig. 3.3 shows a simplified detail of a water component water quality model the use of a quality model for algal growth in a stirred tank so called ’process matrix’ has been proposed which is suitable for demonstrating the use of the (Reichert, 1998; Reichert et al., 2001). The basic process matrix. In this example, phytoplankton layout of such a matrix is shown in Table 3.1. growth is only limited by the availability of a dis- If the process matrix is read by columns, one solved nutrient Xdis and temperature whereas the can see which processes (row headers) have an in- effect of light intensity is neglected. A simple fluence on the concentration of a specific compo- Michalis-Menten type formulation is used for de- nent (column header). If the stoichiometry coef- scribing nutrient limitation. Furthermore, settling ficient Qi,j is zero, the process corresponding to is the only loss process considered. It is repre- row j does not affect the component in column i. sented by a first order approach as suggested by If Qi,j > 0, the concentration of component i rises, Scheffer (1998). Due to settling, algal carbon if Qi,j < 0, the concentration of component i de- (Calg) is transferred to the sediment together with clines due to the action of the j-th process. the organic bound nutrient X. At the sediment’s 3.3 Turnover processes 31

Table 3.1: Basic layout of a process matrix for presenting a multi-component water quality model. The process names (1. . .m) appear as row headers while the names of the simulated components (1. . .n) form the column headers. The stoichiometry coefﬁcients Qi,j ﬁll the center of the matrix (so-called stoichiometry matrix). The process rates which describe the velocity of the conversion processes are listed in the rightmost column. Usually these are lengthy mathematical expressions presented as a separate list.

Processes Component 1 Component 2 ... Component n Process rates

Process 1 Q1,1 Q2,1 ... Qn,1 Rate expr. 1 Process 2 Q1,2 Q2,2 ... Qn,2 Rate expr. 2 ...... Process m Q1,m Q2,m ... Qn,m Rate expr. m

3.3.3 User-defined turnover processes in Water TRAM column Xdis Calg 3.3.3.1 Potential and limitations Sediment C surface set When TRAM was designed, a main objective was Figure 3.3: Simplified detail of a water quality model to guarantee maximum flexibility with respect to for demonstrating the use of the process matrix. See the description of turnover processes. At the cur- Table 3.2 for the declaration of symbols. rent state of development the number of components whose fate can be simulated simultaneously with TRAM is virtually unlimited; the number of surface, the settled organic material (expressed as turnover processes considered is also chosen by the carbon; Cset) is recycled by microbial activity and user. the released nutrient X is returned to the water col- With respect to the kinds of reactions which can umn. be simulated there is one important restriction: All The process matrix corresponding to Fig. 3.3 is conversion processes are treated by TRAM as ki- presented in Table 3.2. Note that only the mass bal- netic processes, i.e. the evolution of concentra- ance for the nutrient X is closed in this example, tions is exclusively described by differential equa- while the carbon balance is not. In order to close tions. Thus, if fast equilibrium processes are to the balance for C too, dissolved inorganic carbon be modeled, these must also be described by ODE (DIC) had to be introduced as an additional vari- using large values for the rate constants. This able. In practice this is unnecessary since DIC is is possible in principle (Reichert, 1998; Reichert always present in excess unless the pH is very low. et al., 2001) if appropriate methods of integration It should be noted that the process matrix shown are used. However, it is pointed out that there are in Table 3.2 could be simplified. The simplification more elegant strategies for coping with equilibrium could be achieved by expressing the concentration reactions involving solvers for mixed algebraic- of the phytoplankton and the settled organic matter differential equation systems (e.g. Reichert, 1998; as nutrient equivalent, not as carbon. Then, the nu- Thullner et al., 2005) and strategies for decoupling trient to carbon ratio QXC would be canceled from components (Molins et al., 2004). Advanced so- the matrix of stoichiometry coefficients. In the rate lution algorithms for equilibrium processes are ap- expressions, Calg and Cset would be substituted by plied in reactive transport models for porous media Xalg and Xset, respectively. (Saaltink et al., 1998). 32 Chapter 3 The water quality simulation tool TRAM

Table 3.2: Example of a process matrix. The three components are (1) a dissolved nutrient Xdis, (2) phytoplankton carbon Calg, and (3) settled phytoplankton material Cset. The three considered processes are (1) nutrient-limited phytoplankton growth, (2) settling of phytoplankton, and (3) recycling of the nutrient from settled phytoplankton material due to microbial decay. This example refers to a stirred tank reactor.

Xdis Calg Cset Processes (g X m−3) (g C m−3) (g C m−2) Process rates

Xdis Growth −QXC 1 0 g(T ) · · Calg Xdis + HX u Settling 0 −1 V/A · C D alg

Recycling QXC · A/V 0 −1 k · Cset

−3 Xdis: dissolved nutrient in the water column (g X m ) −3 Calg : phytoplankton carbon in the water column (g C m ) −2 Cset : settled phytoplankton carbon at the sediment surface (g C m ) g : potential phytoplankton growth rate (s−1) −3 HX : half-saturation constant for growth limitation due to shortage in X (g X m ) −1 QXC : ﬁxed stoichiometric ratio of X and carbon in phytoplankton (g X (g C) ) u : settling velocity of phytoplankton in turbulent environment (m s−1) k : rate constant of mineralization (s−1) T : water temperature (◦C) V : volume of the stirred tank reactor (m3) A : sediment surface area (m2) D : mean water depth (m) 3.3 Turnover processes 33

It should be acknowledged that biologically is entirely hidden from the user and no compila- driven processes are often the focus of surface tion is necessary. In fact, the idea is quite attractive water quality simulation, e.g. in eutrophication and it was used by an earlier version of TRAM. or BOD models. Here, kinetic formulations are But its main deficiency is execution speed. Al- clearly appropriate. though, the parsing of expressions is done only once, the interpreted code turned out to run sig- TRAM’s flexibility with respect to turnover mod- nificantly slower than plain compiled FORTRAN eling is due to the fact that the declaration of the code. This was expected in advance, but the dif- i = 1 ...n simulated components, the process rate ference in execution speed by a factor of 20, even expressions Rj (j = 1 ...m) for all m processes, for simple turnover models, demanded alternative and the values of the stoichiometry coefficients solutions (see below). Qi,j are not static parts of the source code. In- stead, the components’ names, the expressions Rj, Dynamic code generation and the values of Qi,j, are user-defined. Hereby, all This method primarily aims at transferring the information contained in the process matrix can be user-supplied expressions for the process rates Rj adapted to the specific problem at hand. into native FORTRAN code. Obviously, this results in a multi-step procedure: 3.3.3.2 Implementation of the open structure In a first step, the user supplied expressions are parsed and translated into standard FOR- While the components’ names are just strings and TRAN code by a preprocessor program TRAMP- the stoichiometry coefficients Qi,j are numbers, CODEGEN. In this procedure, the names of vari- the process rates Rj are mathematical expressions ables and constants are substituted by references which may be rather complex. In principle there to the elements of dynamically allocated arrays are two technical alternatives for evaluating user- which hold the respective values at runtime. Ba- supplied expressions: the use of an interpreter or sically, two source files are generated in this first the generation of dynamic code. step. The first file is a module declaring the names Interpreter method of constants and variables. The second generated Using an interpreter means that the mathematical source file contains a function to return the deriva- expressions Rj (j = 1 ...m) which are supplied tives of the components’ concentrations using in- by the user as a string are disassembled into tokens stantaneous values of all involved variables and (numbers, variable names, function names, opera- constants as arguments. tors and parenthesis) at the very start of a model In a second step, the generated dynamic parts run. The order of tokens is then converted from of the code must be compiled and linked with the native infix- to so-called postfix-order which TRAM’s core sources. The built executable is then allows for the evaluation of the expressions by a ’tuned’ for the specific application. Because free series of simple stack operations. When the ex- compilers for modern FORTRAN became avail- pressions are evaluated in each time step, only the able recently (e.g. www.g95.org, 2006), the nec- values of variables need to be updated and the stack essary rebuild of the code no longer conflicts with operations must be performed. The actual parsing the objective of making TRAM freely available at of the expressions needs to be done only once. a later stage of development. If the interpreter method is used, different The major advantage of the dynamic code ap- turnover models can be implemented without the proach is execution speed. Apart from that, it has need to make any changes to TRAM’s source code. the advantage that a large number of mathematical This is very convenient as the abstract source code functions and language constructs is available in 34 Chapter 3 The water quality simulation tool TRAM

FORTRAN as a standard while each of these items Table 3.3: Hydrologic variables in stirred tanks (STR) would need specific support by the interpreter. and plug-flow reactors (PFR) which can be referenced Automatically generated FORTRAN code is in the formulation of process rates. Variables with a + sign are available for the corresponding type of reactor also used by the Biogeochemical Reaction Net- while those marked with – are unimplemented. Note work Simulator (BRNS) developed at Utrecht Uni- that only reach-averaged values are available for plug- versity, The Netherlands (Regnier et al., 2002). flow reactors (see Sect. 3.6.4). The BRNS uses a MAPLE preprocessor for code generation from user-specified information. Variable Short name STR PFR average depth ADEPTH + + 3.3.3.3 Items in user-supplied expressions storage volume VOLUME + – surface area SFAREA + – The format used for writing user-supplied expres- avg. flow velocity AVELOC – + sions is discussed in Sect. 3.6.5 along with the de- wet perimeter WETPRM – + scription of the model input files. At this point top width TWIDTH – + however, the nature of the process rate expressions Rj needs to be examined in more detail. Different types of variables and constants can be used in the Boundary condition variables formulation of process rates: Boundary condition variables are user-declared Component concentrations variables whose dynamics are prescribed rather In many cases, the change in a components’ con- than being simulated by TRAM. For example, time centration is somehow related to its current value. series of water temperature may be supplied by the For example, in first order decay of an organic pol- user and the values can then be referenced to adjust lutant X, the change in the concentration is given temperature-dependent rate constants in each time step. by dCX /dt = k · CX with k being a temperature dependent rate constant. Thus, the concentration Constants CX appears in the process rate. Along with component concentrations, hydrologic Hydrologic variables variables and boundary condition variables, the process rates probably always contain fixed param- Often, changes in the concentrations are affected eter values (constants). Typical examples are po- by hydrological variables. For instance, the rate tential growth and decay constants at a certain ref- of settling in the turbulent environment of a STR − erence temperature. (s 1) was expressed by the quotient of the settling − velocity u (m s 1) and the average water depth Time D (m) in Table 3.2. D can be derived from the Diurnal and seasonal variations in boundary con- reactor volume V and the water surface area A ditions (e.g. light intensity) may conveniently be (D = V/A). Another example is the exchange of modeled as functions of the Julian day or the day- oxygen between river water and the atmosphere. In time. In order to support this, the simulation time many models, the essential reaeration coefficient is may be referenced in user-defined expressions. estimated by empirical relationships from the hydrological parameters stream depth and flow ve- In order to facilitate the writing of process rates, locity (McCutcheon, 1989). The number and types TRAM automatically considers the transport terms of available hydrologic variables are different for which account for the import and export of mass plug-flow and stirred tank reactors as shown in Ta- via the reactor’s in- and outflow. These terms, ble 3.3. which contain the time-variable import load of mo- 3.5 Computation algorithms 35 bile components as a fifth type of variable, must be used for calculating the desired information on not be included in the user-supplied process rate concentrations. Also, the output load time series expressions. The names of the auxiliary variables serve as boundary conditions in the simulation of are generated from the user-declared component downstream reactors. Within a complete model names by adding the prefix ’impload’. Since the run, the computation is carried out for each sin- transport terms for immobile components are zero, gle reactor and the procedure advances from the TRAM omits them automatically. upstream end(s) of the river network to the downstream model boundary. The algorithms for computing the transport of 3.4 Handling of time series mobile components in PFR and STR (Sect. 3.5.3, Sect. 3.5.2) were designed with two basic objec- Because the computations described in the subse- tives in mind: They should be efficient with respect quent sections include a lot of time series handling, to execution speed and memory consumption and some general remarks need to be made. In TRAM, they should make the implementation of turnover a time series is considered as a function where the submodels easy. time t is the depended variable (the argument) and x(t) is the independent variable. As the model deals with discretized time series, values for x(t) 3.5.2 Reactive transport in stirred tank are available only at intervals ∆t which are not reactors necessarily regular. TRAM generally relies on the assumption that the value x(t) is constant within the interval ∆t. Thus, time series are handled Mass balance equation like step functions with the values of x(t) chang- The computation of reactive transport in a stirred ing abruptly at the boundaries of time intervals. If tank reactor is based on a numerical solution of the for example a time series has the two arguments mass balance equation (Eq. 3.7). 01.01.2000 00:00:00 and 01.02.2000 00:00:00 and the corresponding values are 1.0 and 2.0, a constant value of 1.0 is assumed for all days in Jan- d(V C ) i = C Q − C Q + Z (3.7) uary. This representation significantly facilitates dt i,in in i out i time series processing but it sets severe restrictions on the resolution ∆t in case of highly variable data. In this equation, V (m3) is the reactor volume and A more accurate alternative may be implemented − C (g m 3) is the concentration of the i-th mobile in a future version of TRAM. i component in the reactor. Furthermore, C is the The terms hydrograph and time series are used i,in concentration in the inflow, and Qin and Qout are synonymously throughout this text. the rates of in- and outflow (m3 s−1), respectively −1 (see Fig. 3.4). The term Zi (g s ) accounts for 3.5 Computation algorithms all conversion processes in the reactor which affect the concentration of component i. 3.5.1 Objectives of computations By rewriting the left hand side of Eq. 3.7 as in Eq. 3.8 (chain rule), TRAM’s basic task is to compute the load hydrographs for all simulated components at the outlets of the simulated reactors. Together with the cor- d(V Ci) dV dCi responding flow rates, these load hydrographs can = Ci + V (3.8) dt dt dt 36 Chapter 3 The water quality simulation tool TRAM

Qin (t), C i,in (t) Qout (t), C i(t) The current version of TRAM also allows for the Ci(t) simulation of immobile components. In contrast V(t) to mobile components, no import into the STR via the inflow occurs for immobile components nor is there any export with the reactor’s outflow. Con- sequently, the governing differential equation for Figure 3.4: Stirred tank reactor with inflow and out- immobile components is simplified in that the term flow of water (Q) and mass of a component i. Input Qin/V · (Ci,in − Ci) disappears from Eq. 3.11. and output loads (L) are given by Li,in = Qin · Ci,in If the fate of m components is simulated, a and Li,out = Qout · Ci where Ci,in is the concentra- system of m ordinary differential equations like tion in the inflow and Ci is the spatially homogeneous Eq. 3.11 has to be solved. Usually, these are cou- concentration in the reactor of volume V . pled equations as the concentration of a component may appear in multiple process rate expres- and taking into account Eq. 3.9 which reflects the sions. This is illustrated by the example presented reactor’s water balance, in Table 3.2. None of the three components can be simulated individually. For instance, the differential equation for the concentration of the nutrient dV = Q − Q (3.9) X cannot be integrated without taking into ac- dt in out dis count the simultaneous change in the concentration Eq. 3.7 can be rearranged to yield Eq. 3.10. A com- of the phytoplankton Calg. parison with Eq. 3.7 reveals that the reactor’s out- Numerical solution flow rate Q was eliminated. out Several algorithms are available for solving systems of coupled ODE. The current version of d(C ) TRAM offers two alternatives, both adapted from V i = Q (C − C )+ Z (3.10) dt in i,in i i Press et al. (2002). As a standard solver, an adaptive stepsize 5-th order Runge-Kutta scheme From Sect. 3.3.2 (Eq. 3.6) it should be known was implemented which has the advantage of be- that the term Z is equal to product of the stoi- i ing robust and quite fast. In some applications, chiometry coefficients Q and the process rates i,j however, a stiff ODE system may be encountered R (j = 1 ...m) summed up for all m conversion j where Runge-Kutta methods fail4. This is the processes after multiplication with the reactor vol- case if some processes cause the simulated con- ume V . Thus, the complete derivative of the i-th centrations to change in the time scale of hours component’s concentration with respect to time is or days (e.g. due to a biological growth pro- given by Eq. 3.11. It should be realized that the cess) while other processes result in almost in- input load (C · Q ) can actually be a sum, if i,in in stantaneous changes (e.g. equilibrium reactions). the modeled STR has multiple inflows. A term for TRAM allows for the integration of stiff ODE sys- direct input of mass (e.g. by atmospheric deposi- tems using a Rosenbrock-Algorithm. By nature, tion) was not included in Eq. 3.11 since this can be taken into account by adding an appropriate row to 4A stiff system of ODE is one in which some components the process matrix. of the solution are slowly varying while other components are rapidly decaying. Unfortunately, it is necessary to follow the variation in the solution on the shortest length scale (time m scale) to maintain stability of the integration, even though ac- d(Ci) Qin = (Ci,in −Ci)+ Qi,j Rj (3.11) curacy requirements would allow a much larger step size to be dt V used (Press et al., 2002). Xj=1 3.5 Computation algorithms 37 this algorithm is somewhat slower than a simple relative change in a partial derivative falls below a Runge-Kutta method as it involves the evaluation termination threshold. of the Jacobian matrix by numerical differentiation and its subsequent inversion5. The Jacobian ma- Step size control trix holds the partial derivatives of the ODE right Apart from the automatic step size adaptation hand sides with respect to the components’ con- in the Runge-Kutta and the Rosenbrock integra- centrations. For example, in a problem with two tors, TRAM generally adjusts the computational components X1 and X2 and two processes P1 and time step in order to satisfy the Courant criterion. P2, the ODE system to be solved is The maximum possible step size ∆t is given by Eq. 3.12. dCX1/dt =QX1,P 1 RP 1 + QX1,P 2 RP 2

dCX2/dt =QX2,P 1 RP 1 + QX2,P 2 RP 2 with the stoichiometry coefﬁcients Q and the pro- V ∆t ≤ (3.12) cess rates R (transport terms were neglected). The Q corresponding Jacobian matrix is shown below.

∂(dCX1/dt) ∂(dCX1/dt) The quotient of the stirred tank reactor’s volume V ∂CX1 ∂CX2 and the flow rate Q is equivalent to the residence ∂(dCX2/dt) ∂(dCX2/dt) time of the water. In TRAM, the actual size of a ∂CX1 ∂CX2 time step is chosen to be always smaller than half the maximum possible ∆t. The reason why the Jacobian matrix is evaluated numerically is this: As described in Sect. 3.3.3 the TRAM user must supply the right hand sides of Calculation of average loads the differential equations describing the evolution Except for the most upstream reactor(s) of the sim- of the simulated concentrations in time. If the Ja- ulated network, the input load of a reactor within a cobian matrix was to be calculated analytically, the time step ∆t is equivalent to the output load from user had to supply the partial derivatives of these its upstream neighbor(s) within the same ∆t. For expressions too. In principle this is possible but it conservation of mass, the load hydrographs passed is rather inconvenient in case of complex turnover from one reactor to another must comprise aver- models. age loads for each time step, not instantaneous val- In TRAM, numerical estimates of the partial ues. However, the numerical solution of the mass derivatives are calculated by the so-called ’central balance equations for a STR (Eq. 3.11) yields in- derivatives approach’. In this approach, derivatives stantaneous concentration values and some post- are determined as the slope of a quadratic spline lo- processing is necessary in order to obtain the de- cally fitted to the function to be derived. Since only sired ’average-load hydrographs’. the slope of the spline is of interest, not the spline In general, the average load L over the period coefficients, the method requires not three but only ∆t is defined by Eq. 3.13, where Q is the flow rate two evaluations of the ODE right hand sides: one and C is the concentration. to the left and another one to the right of the location of interest. The increment of the variable with respect to which derivatives are computed is suc- t0+∆t cessively reduced by a factor of 0.5 until the largest 1 L = C(t) · Q(t) dt (3.13) ∆t 5The matrix inverse is computed by LU decomposition. tZ0 38 Chapter 3 The water quality simulation tool TRAM

Due to the condition of constant flow within ∆t Q(t), L i(t) V (see Sect. 3.4) Eq. 3.13 simplifies to Eq. 3.14, with Mi...n Q being the average flow rate. flow

∆xT t0+∆t Q L = C(t) dt (3.14) ∆t Figure 3.5: Approximation of the movement of water tZ0 and mass through a plug-flow reactor of the total length ∆xT by a sequence of control volumes. Each of the The integral in Eq. 3.14 has to be evaluated us- control volumes is characterized by its water volume V ing concentrations at discrete times. In the cur- and the contained masses Mi of the n simulated mobile rent version of TRAM the trapezoid rule is used components. The concentration of the i-th component for the sake of runtime efficiency. Thus, the av- is defined as Mi/V . The upstream flow and load bound- erage load over a time step is computed as L = ary conditions are denoted Q(t) and Li(t), respectively. Q · 1/2 · (C(t0)+ C(t0 + ∆t)). Advanced integration methods, e.g. using cubic splines, could be implemented if critical mass balance errors need to water balance equation (Eq. 3.15). The informa- be cured. tion on Qin as well as V is provided by the hydrodynamic model (see Sect. 3.6.4).

3.5.3 Reactive transport in plug-ﬂow re- dV actors Q = Q − (3.15) out in dt

Transport algorithm As a plug-flow reactor is a purely advective system In reality, there is a continuous input of water and by definition, the first in – first out principle ap- mass into a river reach as described by the inflow plies. In other words: In each time step, the ’old- hydrograph and the corresponding concentration est’ water is exported from the reactor. time series (chemographs). In TRAM, these input In practice, the traveltime of a control volume time series are discretized into time steps of dura- is hardly an exact multiple of the simulation time tion ∆t. Within a single time step, a certain volume step because a plug-flow reactor can be of arbitrary of water (forming a control volume) and a corre- length and flow rates are variable. Therefore, mul- sponding mass of each component enter the plug- tiple control volumes may contribute to the amount flow reactor. The volume V is given by V = Q·∆t of water and mass leaving the reactor within a simulation time step. Averaging the loads from several and the contained mass Mi of component i is given control volumes must result in a kind of numerical by Mi = Ci ·V if the flow Q and the concentration dispersion. However, the dispersion effect is re- Ci is constant within ∆t (see Sect. 3.4). Alterna- stricted to the moment when the control volumes tively, Mi can be expressed as Li · ∆t with Li being the load. Thus, the input into a plug-flow reac- pass the reactor’s outlet. In some other numeri- tor resembles a sequence of control volumes, each cal solutions of the advection equation, which are containing a certain amount of water and mass of based on a static space and time discretization, nu- the n simulated components (Fig. 3.5). merical dispersion affects each control volume in every single time step. For each time step, the PFR’s outflow rate Qout can be computed from the inflow rate Qin and the Turnover processes change in the reactor’s storage volume V using the Because the exchange of water between adjacent 3.5 Computation algorithms 39 control volumes is neglected and spatial homo- step. It is therefore wise to subdivide river sec- geneity6 within each control volume is assumed, tions with highly irregular geometry into several the mathematical description of conversion pro- PFR (see Sect. 3.1.2). cesses is equal to that used for stirred tanks (see As mentioned earlier, TRAM currently does not Sect. 3.5.2). The only difference is that the terms support the simulation of immobile components in which describe the mass flux due to in- and outflow plug-flow reactors. are omitted. Also, the amount of water contained in the control volume remains constant during the 3.5.4 Routing algorithm passage of the plug-flow reactor. The time available for any turnover to take place In contrast to many spatially distributed models, in a control volume is given by its traveltime T . TRAM does not update the simulated variables af- However, T may be different for the front and the ter each simulation time step in the entire model back end of a control volume (Tfront, Tback) if the domain. Instead, transport and turnover in a single flow is unsteady. Because the time-dependence of reactor are always calculated over the full simula- concentrations is usually nonlinear (e.g. exponen- tion period before the computation proceeds with tial law for first-order reactions), the arithmetic av- another reactor of equal or higher stream order. erage 1/2 · (Tfront + Tback) is not a representative In terms of programing, this means that TRAM’s estimate of the control volume’s mean traveltime. time loop is nested within the loop over the reac- A representative mean traveltime can be derived tors (spatial loop). analytically for special cases only. For the sake of Because there is no need to store instantaneous general applicability, TRAM computes the change values of all simulated variables in all spatial units in the components’ concentrations twice, first us- of the model domain, this method is efficient with ing Tfront and another time using Tback. The mean respect to memory consumption. Only the vari- of the resulting concentrations is then taken as an ables which are involved in the simulation of a sin- estimate of the homogeneous average concentra- gle reactor need to be allocated. In fact, memory tion in the control volume after passage of the PFR. consumption would be a minor problem if only An even more exact solution could be obtained if STR were simulated as these do not exhibit spa- the component’s concentrations Ci were also eval- tial variability in concentrations. However, instan- uated at locations x in the middle of the control taneous concentrations in every single control vol- volume with length ∆x: Spline polynomials Ci(x) ume of all PFR would need to be kept in memory could then be fitted to describe the spatial concen- if TRAM was solving for the state variables in the tration gradient in the control volume and the aver- entire model domain in every time step. age concentrations could be calculated by solving Another advantage of the method is that it facil- Ci(x) dx/∆x. itates a stepwise model calibration. Once the pa- It should be noted that the values of all bound- rameters for all reactors with a stream order ≤ k R ary condition variables appearing in the descrip- have been determined, TRAM can be forced to tion of turnover processes are considered as reach- start the computation with reactors of stream order averages. This means that user-specified data (e.g. k + 1. The required output from the upstream re- water temperatures) but also the values of hydro- actors with stream orders ≤ k is available from the logic variables (such as flow velocities or the hy- previous model run(s). Thus, only those sections draulic depth; see Sect. 3.6.4) are assumed to be of the network which were not already calibrated valid for the entire plug-flow reactor within a time need to be simulated again. Of course, advantages have their price. It is an 6vertically and horizontally obvious drawback of this algorithm that flow re- 40 Chapter 3 The water quality simulation tool TRAM versal cannot be handled within a model run as TRAM’s input files can easily be generated by util- the upstream-downstream relations (i.e. the reac- ity software, such as scripts that query a SQL data tor’s stream orders) would change. Although flow base. Moreover, well structured text files provide reversal is not too common in natural rivers, it an automatic documentation of each model run. If may sometimes occur in lowland rivers due to sig- the need for a graphical user interface ever arises, nificant withdrawal of water or input of wastew- it could be easily implemented as a separate ap- ater7 or due to backwater effects during flood plication which prepares the simulation input files events8. Also, flow directions may be variable and then starts TRAM as a batch job or calls it as a in shortcut channels between two reaches which DLL routine. have their confluence downstream of the shortcut In order to make data input convenient and safe, (there are several examples in the Havel River; see TRAM uses a standard format for most of its in- Fig. 4.5). Fortunately, the amount of water and put files: the so-called ’.ini’-format. This stan- mass transported against the usual flow direction dard is used by many WINDOWS applications is mostly small. Thus, substituting negative flow that do not access the registry for storing config- rates by very small positive values is often a tol- uration data. The ’.ini’-format requires that each erable makeshift. Certainly, the current version of data value or array is preceded by a descriptive TRAM is not at all suitable for studying tidal rivers keyword (e.g. key= value). Keyword-value pairs where flow-reversal occurs regularly. again can be grouped by context into separate ’sections’ whose names appear in square brackets. The 3.6 TRAM’s input data use of this (or a XML-based) structuring convention is believed to be essential for serious model The following sections give an overview of the data applications. All input data are instantaneously and control parameters which are required for run- human-readable without additional comments and ning a TRAM simulation. In many cases, parts of the chance of data confusion is very low. Finally, input files will be presented in order to illustrate the use of unique keywords for data identification which kinds of data must be provided to TRAM facilitates automatic editing of TRAM’s input files and its preprocessing tools. by utility software (see e.g. Sect. 4.4.4.1). Apart from keywords, values, and data section headers, an ’.ini’-file (short: ini-file) can contain comments. 3.6.1 General remarks on data input The standard comment character is the semicolon TRAM intentionally does not have a graphical user and the double cross (#) is a non-standard alterna- interface and all data are read from plain ASCII tive. An example of an ini-file is given in Fig. 3.6. text files or they are passed as command line arguments. This might seem old-fashioned but, apart from the saving in development time and platform 3.6.2 Model units independence, there is a number of great advantages. For example, TRAM can be run in batch In principle, the units for expressing quantities of jobs which integrate all modeling steps: prepro- length, area, volume, and mass can be freely cho- cessing, simulation, and postprocessing. Also, sen as long as all input data are consistent. It is however recommended to keep to the convention 7As happened in 2003 in the Spree River, Berlin. 2 8 that all lengths are given in meters, areas in m , and Flow reversal occurred at the mouth of the Havel River in volumes in m3. The suggested base unit of mass is August 2002 when the sluice gate at Neuwerben was opened to divert up to 700 m3 s−1 of water from the Elbe River into gram (g). This avoids confusion and possible unit the Havel floodplain and polder system. conversion errors. 3.6 TRAM’s input data 41

The basic time unit in TRAM is the second ______which is also the highest possible resolution for ;This data section declares a stirred ;tank reactor (str) called 'SampleLake'. time-variable in- and output. The internal time step ;Because its stream order is 1, the list ;of upstream sections is empty. Also, no in numerical computations may be even less than a ;parallel sections which share the same second but such small increments often indicate er- ;inflow can exist. ;The highest stream order of an adjacent rors in the input data or the user-supplied equations ;downstream reactor equals the current ;stream order incremented by one. of the turnover model. [SampleLake] reactor= str stream_order= 1 upstream_sections= - 3.6.3 Network description parallel_sections= - maxorder_recipient= 2 The ’network description file’ declares the num- ;This data section declares a plug-flow 9 ;reactor (reactor = pfr) with the name ber and names of the simulated reactors and pro- ;'PFR_3'. The PFR is located downstream vides information on the network’s topology (see ;of the confluence of two other reactors ;called PFR_1 and PFR_2, i.e. it has two Sect. 3.1.3). Details of a sample file are shown in ;upstream neighbors. [PFR_3] Fig. 3.6, which also demonstrates the ini-format in- reactor= pfr stream_order= 10 troduced in Sect. 3.6.1. Most of the items shown in upstream_sections= PFR_1,PFR_2 Fig. 3.6 are sufficiently commented upon but the parallel_sections= - maxorder_recipient= 11 last three keywords need some further explanation. ;The reactor 'PFR_5' is located just After the key ’upstream_sections’ the names of the ;downstream of a flow split. It shares reactor(s) located directly upstream of the current ;the input from reactor 'PFR_4' with the ;parallel reactor 'PFR_6'. reactor (given in the section header in brackets) [PFR_5] reactor= pfr must be listed. During simulation, the output loads stream_order= 12 of the referenced reactors are summed for each upstream_sections= PFR_4 parallel_sections= PFR_6 component and the resulting load hydrographs are maxorder_recipient= 13 used as boundary conditions for the current reactor. ______The key ’parallel_sections’ is followed by a list Figure 3.6: Extract of an extensively commented net- of those reactors which receive input loads from work description file showing the declaration of a the same upstream reactor(s) as the current reac- stirred tank and two plug-flow reactors. tor (given in the section header). This list is non- empty only if the current reactor is located down- drographs of the current reactor can be removed stream of a split flow junction. During simulation, from memory. the load from the upstream reactors is distributed As mentioned in Sect. 3.1.3, the network de- over the receiving reactors weighted by their indi- scription file can be generated from GIS data using vidual inflow rates (Eq. 3.1 & Eq. 3.2). a utility program. For simple networks it can also The entry ’maxorder_recipient’ contains the be prepared manually. highest stream order of the adjacent downstream reactor(s). Usually, this number will be the stream order of the current reactor incremented by 1, but 3.6.4 Preprocessing of hydrodynamic in- a higher number may appear in a looped river formation net. Based on this information, TRAM decides at It follows from the description of the transport al- which stage in the computation the output load hy- gorithms in Sect. 3.5.2 and Sect. 3.5.3 that TRAM 9The number of reactors that can be simulated with TRAM must be supplied with time series of some ba- is not limited. sic hydrologic variables. For conservative trans- 42 Chapter 3 The water quality simulation tool TRAM port modeling, only hydrographs of the inflow rate and the storage volume of a reactor need to be Bottomfottom2elevtion elevation

(m@mslA above s.l.) known. However, a reasonable formulation of

IW2E2PH many turnover processes requires information on PH2E2PI

PI2E2PP

additional hydrologic variables to be available (see PP2E2PQ Table 3.3). PQ2E2PR PR2E2PS

It is the task of the preprocessing utilities PS2E2PT

PT2E2PU

TRAMP-STR and TRAMP-PFR to create and PU2E2PV

PV2E2PW

store time series of all hydrologic variables which PW2E2QH

b2QH can be accessed later on by TRAM. Basically, the preprocessors merge two types of information: Figure 3.8: Bathymetric map of three lakes on the 1. Hydrographs of water surface elevation and Lower Havel River. ﬂow.

2. Data on channel and lake geometry (cross- input data for both TRAM and the hydrodynamic 11 sections, streamlines, bathymetry). model in order to avoid inconsistencies . As Fig. 3.7 shows, the preprocessor TRAMP- The benefit from using the preprocessors is STR temporarily creates lookup tables from which twofold: First of all, it is guaranteed that TRAM the derived hydrologic variables can be queried for can be used independently from any specific hy- each water surface elevation in the observed range. drodynamic model. Because only the hydrographs This speeds up the preprocessing significantly if of flow and water surface elevation are required input hydrographs contain many values or if the as input data and derived variables (such as water elevation grid has a high ground resolution. depth, wet perimeter, etc.; see Table 3.3) are com- In the current version of the preprocessor, only puted by the preprocessors, TRAM could also be the reactor’s surface area, not the true bottom area run using raw observation data. is computed. For shallow lakes (and only those The second benefit from the use of the prepro- can be represented as STR), the two values can- cessors TRAMP-STR and TRAMP-PFR emerges not differ very much. Also, in the formulation of if a large number of simulations with the same sediment-water interactions the projection of the hydrological boundary conditions are carried out bottom area must be used which is identical to the for a system. In this case it would be a consider- surface area. able waste of time if TRAM was computing values of the derived hydrologic variables from geometry Hydrograph preprocessing for PFR and stage/flow data anew for each model run. The preprocessor TRAMP-PFR is similar to Hydrograph preprocessing for STR TRAMP-STR as it computes time series of derived The functioning of the hydrograph preprocessor hydrologic variables from geometry information for stirred tank reactors is illustrated in Fig. 3.7. and stage/flow hydrographs. However, the spec- The required digital elevation model of the lake trum of the derived variables is different and cross- bottom can be derived from a bathymetry map section data form the geometric data base. The (Fig. 3.8) by vector to raster conversion using overall functioning of the TRAMP-PFR utility is GIS10. It is absolutely essential that the same illustrated in Fig. 3.9. bathymetric map is used for deriving the geometric 11Disregard of this may cause inaccurate results in transport 10The elevation model is read using the ASCII-grid format. modeling, but it does not affect the overall mass balance. 3.6 TRAM’s input data 43

Hydrographs of all upstream & lateral inflows and water surface elevation Lake bathymetry map (from hydrodynamic model) (bottom elevation grid) Input

Determination of minimum and Creation of lookup tables maximum w.s.elevation - volume (w.s. elev.) once - surf. area (w.s. elev.) - avg. depth (w.s. elev.) Processing for every Calculation of derived hydrologic variables time step

STR hydrograph input file for TRAM containing: - inflow - w.s. elevation - volume Output - surf. area time series of derived - avg. depth hydrologic variables

Figure 3.7: Generation of TRAM’s hydrograph input ﬁle for a stirred tank reactor by the preprocessing utility TRAMP-STR.

Hydrographs of all upstream & lateral inflows and water surface elevation at Streamlines file defining up-/downstream end of the reactor Collection of cross- the center-streamlines (from hydrodynamic model) section files of all PFR Input

Calculation of storage volume & reactor-averaged Assignment of cross-sections to derived hydrologic variables for each time step all PFR based on intersections (Values at individual cross-sections are weighted between streamlines and cross- by their contribution to the total reactor lenght.) section cutlines Processing

PFR hydrograph input file for TRAM containing: - inflow - upstream w.s. elev. - downstream w.s. elev.

Output - storage volume - avg. depth (hydraulic depth) * - wet perimeter * time series of derived - top width * hydrologic variables - avg. flow velocity * * reactor-averaged values reactor-averaged *

Figure 3.9: Generation of TRAM’s hydrograph input ﬁle for a plug-ﬂow reactor by the preprocessing utility TRAMP-PFR. 44 Chapter 3 The water quality simulation tool TRAM

Fig. 3.10 shows a detailed sketch of a plug-flow Jungfernsee%Westteil reactor with a definition of cross-section parame- 3368854.940765, 5810239.634686 ters. For each time step in the input hydrographs, 3368769.867465, 5810346.498471 the water surface elevation at the individual cross- ... 3368175.421993, 5811112.224841 sections is estimated by linearly interpolating be- 3367899.609369, 5811273.430393 tween the given values at the upstream and down- END stream reactor end. Then, the stage-dependent pa- SPK%BrueckeDesFriedens rameters A (cross-section area), W (top width) 3367899.609369, 5811273.430393 and P (wet perimeter) can be determined for each ... 3367183.050019, 5811609.579583 cross-section. Using the information on A, the END storage volume V of the PFR with n cross-sections END can be computed from Eq. 3.16. Figure 3.11: Shortened sample of a streamlines file describing the spatial position of two plug-flow reactors. n The reactor names are followed by a sequence of x,y V = Ai · ∆xi (3.16) coordinates that define the streamlines’ nodes (here in a Xi=1 UTM system). This notation complies with the ’gener- Reach-averaged values of the derived hydrologic ate format’ of ESRI’s GIS. variables P and W as well as the average depth D and the average flow velocity U are estimated according to Eq. 3.17–Eq. 3.20. In these equations #Xcoord Ycoord Elevation 3348829.37 5814123.39 29.27 Q is the reactor’s inflow rate and ∆xT is the to- 3348827.89 5814125.99 28.99 tal length of the PFR, i.e. ∆x = ∆x . It is T i 3348826.40 5814128.60 28.94 pointed out that one must not derive further com- P ...... posite variables from P , W , D, and U since their 3348168.01 5815282.83 28.41 interrelationships are non-linear for natural cross- 3348166.52 5815285.43 28.44 sections. For example, W · D does not yield a proper value for the reach-averaged cross-section Figure 3.12: Shortened sample of a three-column area if the reactor’s geometry is non-uniform. cross-section file with UTM coordinates assigned to each elevation record.

n ∆xi P = Pi · (3.17) ∆xT the streamline file (see Fig. 3.11) defines the spatial Xi=1 n position of all simulated PFR. Also, the program is ∆xi W = Wi · (3.18) supplied with a list of all cross-sections covering ∆x i=1 T the entire simulated river network. Xn ∆x D = A /W · i (3.19) For cross-sections to be automatically assigned i i ∆x i=1 T to the correct reactors and for determining their ex- Xn ∆xi act positions within the PFR, the cross-section data U = Q/Ai · (3.20) ∆xT should have a full spatial reference (3D data) as Xi=1 shown in Fig. 3.12. The use of spatially referenced Similar to the utility for stirred tanks, TRAMP- cross-sections is advisable anyway if a hydrody- PFR is able to process the information for all sim- namic model with GIS data import capabilities like ulated plug-flow reactors at once. This is because HEC-RAS (USACE, 2002) is applied. 3.6 TRAM’s input data 45

Water surface elevation at upstream reactor boundary

∆x1 Water surface elevation at elevation i=1 downstream reactor boundary

∆x2 i=2 Cross-section area

Top width ∆x3 i=3

flow Wet axis perimeter

Figure 3.10: Schematic of a plug-ﬂow reactor whose channel geometry is described by three cross-sections. The symbol ∆xi denotes the reach length corresponding to the cross-section with index i.

Fig. 4.17 shows the plug-flow reactor’s stream- simulation. As described in Sect. 3.3.3, TRAM lines as well as the corresponding cross-section knows different types of variables such as com- cutlines for a part of a river network. ponent concentrations, hydrologic variables, and boundary conditions. Variable names for com- 3.6.5 Definition of the turnover model ponents and boundary conditions can be chosen freely but in case of components, the first character As mentioned in Sect. 3.3.3.2 the preprocessing of the name has a special meaning. If it is a dol- program TRAMP-CODEGEN is used for translat- lar sign ($), the component is implicitly declared ing the user-supplied description of the turnover immobile but it is mobile otherwise. If used in ex- model into FORTRAN code. This program must pressions, variable names must always be enclosed be supplied with two ini-files (one for PFR, another in brackets. Valid variable names are for exam- one for STR) containing all information of the pro- ple ’[aVariable]’, ’[NO3]’, ’[NO3_N]’, or ’[$Sed- cess matrix, i.e. component and process names, imentP]’. The names of the hydrologic variables stoichiometry coefficients and process rate expres- are fixed according to Table 3.3 (page 34). The sions (see Table 3.1 on page 31). These files are simulation time can be assessed by the variable called the ’process definition files’. In order to fa- name ’unixtime’. cilitate their preparation, especially with respect to Constants the writing of the process rate expressions Rj (see Sect. 3.3.2), a special but simple language is used. The values of constants do not change during simulation. Constant names may consist of any characters but within expressions, their names must be 3.6.5.1 Language elements enclosed in angle brackets. Valid constant names The language comprises variables, constants, are for example ’’ or ’’. terms, numbers, functions, operators, and condi- Numbers tional expressions. Constant values can alternatively be written as Variables common floating point numbers. This may be use- Generally, variables change their values during ful for converting units but the use of constants 46 Chapter 3 The water quality simulation tool TRAM

______(see above) should be preferred. While constants [Processes] are read from input files and their values can easily Namelist= Growth, Settling, Recycling [SimulationVariables] be modified, hard-coded numbers are fixed once Xdis = 0.1 # the dissolved nutrient (g/m3) Calg = 2.0 # phytoplankton carbon (g/m3) TRAM is compiled. $Cset = 1.0 # settled phytoplankton (g/m2)

[Constants] Terms g = 1.0/86400 # pot. growth rate (1/s) k = 0.02/86400 # decay rate (1/s) A ’term’ can be used to temporarily store the result Hx = 0.01 # half saturation (g Xdis /m3) theta = 1.045 # temp.-depend. of growth (-) value of an expression. The result value should Qxc = 0.025 # X:C ratio in plankton (-) u = 0.1/86400 # settling velocity (m/s) always be assigned to a term name if an expres- ______sion appears in multiple other expressions. This prevents the waste of computer time as the orig- Figure 3.13: Part of a process definition file contain- inal expression is evaluated only once and there- ing the declaration of process names, components and constants. The corresponding process matrix is given in after the term name can be used as a substitute for Table 3.2. it. Also, the storage of intermediate results by the use of terms makes complicated process rates easier to write and to debug. A term name may con- tors, the double asterisk for power operations, and, sist of the characters a–z, the digits 0–9, and the of course, parenthesis. underscore only. In other words, term names must be legal FORTRAN identifiers. If terms are refer- 3.6.5.2 Sections of a process definition file enced in expressions, their names must be enclosed In the following paragraphs, the contents of the in curly braces, e.g. ’{aTerm}’. The expression process definition file are discussed. All presented which is assigned to a term name may contain ref- examples (Fig. 3.13–3.17) refer to the sample pro- erences to previously defined terms but no cyclic cess matrix from Table 3.2 in Sect. 3.3.2. references. Part 1 Conditional expressions The first sections of the process definition file In some cases it is desirable to use conditional ex- (Fig. 3.13) are used for declaring the names of the pressions in the formulation of the process rates. considered turnover processes and simulated com- A conditional expression always starts with the se- ponents as well as the names of constants. As the quence ’|if| (condition) |then| (expression)’ which example in Fig. 3.13 illustrates, numeric values can be followed by an ’|elseif| (condition) |then| must already be assigned to the component con- (expression)’ branch and/or by ’|else| (expression) centrations and constants. It is important to realize |endif|’. The conditions must be enclosed by that these values are in no way used in later sim- parenthesis and the old-style FORTRAN compari- ulations. The values become relevant onlywhen son operators like .lt., .ge., .eq. must be used. It is the mass balance of the process matrix is automati- important to realize that functions containing con- cally checked as described in Sect. 3.6.5.3. Hence, ditional expressions are often discontinuous and guessed values of the correct magnitude are suffi- the convergence of numerical integrations may be cient. hindered. Part 2 Functions and operators The names of boundary condition variables are de- In addition to the items listed above, expressions clared in the next sections of the process definition may contain all FORTRAN intrinsic functions file (Fig. 3.14) and – in order to enable mass bal- with a scalar return value (e.g. log, exp, min, ance checks – values are assigned to them. Also, max). Also supported are the four standard opera- test values are set for the hydrologic variables 3.6 TRAM’s input data 47

______[BoundaryConditionVariables] [ReferenceVolumes] T = 20.0 # water temperature (°C) Xdis = [VOLUME] # water body volume (m3) Calg = [VOLUME] # water body volume (m3) [HydrologicVariablesEstimates] $Cset = [SFAREA] # bottom area (m2) INFLOW = 10.0 # inflow rate (m3/s) WSELEV = 30.0 # water surf. elevation (m) [ReferenceElementFactors] ADEPTH = 2.0 # average depth (m) Xdis = 1.0 SFAREA = 1.0e+06 # surface area (m2) Calg = 0.025 # the stoichiometry factor Qxc VOLUME = 2.0e+06 # volume in reactor (m3) $Cset = 0.025 # the stoichiometry factor Qxc ______

Figure 3.14: Part of a process definition file in which Figure 3.15: Part of a process definition file contain- boundary condition variables are declared and values ing information which is used in the calculation of mass used in mass balance checks are assigned. The corre- balances. The corresponding process matrix is given in sponding process matrix is given in Table 3.2. Table 3.2. which are automatically provided by TRAM rather of the sediment layer and the porosity. The refer- than being defined by the user (see Table 3.3). As ence volume could then be defined as ’[SFAREA] the names of the hydrologic variables suggest, the * *

’ where ’

’ is again a reference example of Fig. 3.14 corresponds to a stirred tank to a user-defined constant. Alternatively, both the reactor. thickness of the sediment layer and the porosity could be declared as variables. Part 3 Since sediment-bound components are currently The information in the ini-file sections shown in not supported in plug-flow reactors, the reference Fig. 3.15 is only evaluated in mass balance checks volume for all simulated components in a PFR is of the user-supplied process matrix. In the sec- equal and should be set to unity. tion ’ReferenceVolumes’ a domain must be speci- In the section ’ReferenceElementFactors’ fac- fied for each of the simulated components. In gen- tors can be supplied with which the component eral, the domain of dissolved or suspended mobile concentrations are multiplied when mass balances components is the volume of the water column. are computed. By means of these factors, certain This means that the total mass of such a compo- components can be excluded from a balance (fac- nent in a STR can be calculated as the product tor zero), they may be included (unity) or stoichio- of the concentration and the volume of the water metric characteristics can be accounted for (e.g. column, given by the hydrologic variable ’[VOL- − − NO could be converted to NO -N). In the exam- UME]’. For immobile components, different do- 3 3 ple shown in Fig. 3.15, the mass balance is com- mains are possible: For instance, a component’s puted for the nutrient X and the conversion fac- mass can be related to the surface area (g m−2) as tor Q must be used in order to account for the this is the case with component C in Fig. 3.15. XC set nutrient mass stored in the phytoplankton and the Then, the required entry is a reference to the vari- settled material which are both defined in units of able ’[SFAREA]’ (see Table 3.3). Alternatively, − g carbon m 3 (C , C ). the component’s mass may be related to the vol- alg set ume of a certain sediment layer (g m−3 of sedi- Part 4 ment) and the entry would be ’[SFAREA] * ’, In this section of the process definition file, the pro- where ’’ represents a user-defined constant for cess rates Rj (j = 1 ...m) for all m processes are the layer’s thickness. In the case of non-particulate assembled and assigned to named terms. An ex- components which are dissolved in the sediment’s ample corresponding to the process matrix of Ta- pore water, concentrations are also given in g m−3 ble 3.2 is shown in Fig. 3.16. Syntax highlighting but the calculation of the total mass requires the greatly facilitates the distinction of variables, con- concentration to be multiplied by both the volume stants, and terms. 48 Chapter 3 The water quality simulation tool TRAM

______[Terms] [Growth] Xdis = {R_grw} * * (-1.0) # Nutrient limitation of growth rate (-) Calg = {R_grw} limit_x = [Xdis] / ([Xdis] + ) $Cset = 0.0 # Temperature limitation of growth rate (-) limit_t = ** ([T]-20.0) [Settling] # Process rate for growth (g Calg / s) Xdis = 0.0 R_grw = * {limit_t} * {limit_x} * [Calg] Calg = {R_set} * (-1.0) $Cset = {R_set} * [VOLUME]/[SFAREA] # Process rate for settling (g Calg / s) R_set = * [SFAREA] / [VOLUME] * [Calg] [Recycling] Xdis = {R_rec} * * [SFAREA]/[VOLUME] # Process rate for recycling (g Cset / s) Calg = 0.0 R_rec = * [$Cset] $Cset = {R_rec} * (-1.0) ______

Figure 3.16: Part of a process definition file illustrating Figure 3.17: Part of a process definition file contain- the use of ’terms’ for storage of the process rates and ing the differential equations which describe the change intermediate results. The corresponding process matrix in the components concentrations over time due to the is given in Table 3.2. considered turnover processes. The corresponding process matrix is given in Table 3.2.

Part 5 In the final sections of the process definition file, constants names only, the other contains the func- the derivatives of the component’s concentrations tion to compute the derivatives of the components’ with respect to time are supplied (Fig. 3.17). Note concentrations. that the differential operator d/dt is not written but In addition, the preprocessor generates a third only the components’ names appear as key names FORTRAN source file. The compilation of this on the left hand sides. As Fig. 3.17 shows, the auxiliary file yields a small separate executable contribution of each process is written down sep- which computes the mass balance for each of the arately for all components. This has several ad- processes using the test values that were assigned vantages over writing the full differential equation to the various variables and constants. It is rec- for each component as a single expression. First of ommended to always build and run this executable all, it brings clarity and makes error tracing easier. as it provides the best means to verify the consis- Secondly, the mass balance can be computed sep- tency of the complex information contained in the arately for each of the conversion processes which process definition file. If this auxiliary source file again facilitates debugging. For a process ma- is not compiled without errors, the compilation of trix involving n simulated components and m pro- TRAM will fail too, and the process definition file cesses, n × m equations must be supplied, each must be revised. Automatic comments in the gen- taking a separate line. erated source file make it rather easy to locate the As mentioned in Sect. 3.3.2, the transport terms cause of the error in the process definition file. must not appear in the right hand side expressions The output of the mass balance verification pro- supplied here as TRAM takes them into account gram for the sample file shown in Fig. 3.13– automatically if necessary. Fig. 3.17 is illustrated in Fig. 3.18. The zero values in the summary rows ’balance’ indicate that 3.6.5.3 Preprocessing of the process definition the presented process definition file (and thus the file process matrix from Table 3.2) provides a closed description of the mass balance of the nutrient X. As mentioned in Sect. 3.3.3, the preprocessor The booleans in the rightmost column indicate for TRAMP-CODEGEN basically creates two FOR- which components the calculation of a mass bal- TRAN files which become part of TRAM’s source ance is activated by setting the reference element code. While one source file declares variables and factor (see Fig. 3.15) to a non-zero value. 3.6 TRAM’s input data 49

______*********************************************** [u] Process | Variable | Derivative | Active? * = 0.1 *********************************************** @Group1 = 0.15 growth | xdis | -0.10522E+01 | TRUE Reactor1 = @Group1 growth | calg | 0.10522E+01 | TRUE Reactor2 = @Group1 growth | $cset | 0.00000E+00 | TRUE Reactor3 = 0.2 ------+------+------+------______| Balance | 0.00000E+00 *********************************************** settling | xdis | 0.00000E+00 | TRUE settling | calg | -0.57871E-01 | TRUE Figure 3.19: Part of an ini-file defining the values of settling | $cset | 0.57871E-01 | TRUE a constant u. The name of the constant is given in the ------+------+------+------| Balance | 0.00000E+00 section header. In the example, u is individually set to a *********************************************** recycling | xdis | 0.57870E-02 | TRUE value of 0.2 for a reactor called ’Reactor3’. A common recycling | calg | 0.00000E+00 | TRUE value of u=0.15 is assigned to the reactors ’Reactor1’ recycling | $cset | -0.57870E-02 | TRUE ------+------+------+------and ’Reactor2’. This is achieved by specifying another | Balance | 0.00000E+00 *********************************************** key name at the right hand side instead of a numerical value. For all reactors whose names do not explicitly Figure 3.18: Output of the preprocessor-generated pro- appear as key names, the default value of 0.1 is used. gram which verifies the process definition file by com- The reserved key name for specifying the default is the puting mass balances for the simulated processes (see asterisk. Table 3.2 for the corresponding process matrix).

3.6.6 Values of constants and initial con- Time series of external loads centrations The passing of loads from a reactor to its downstream neighbors is managed by TRAM automat- In principle, the values of constants appearing in ically. Apart from these loads, every reactor may the turnover model can be set individually for each experience the input of additional external loads, reactor in order to reflect spatial heterogeneities in e.g. due to a discharge of waste water. In a ’bound- the river system. However, often it is desirable to ary condition management file’, which is just an- set or to modify a parameter for a whole group of other ini-file, times series of such additional loads reactors with similar characteristics or even for all can be assigned to each reactor for all mobile com- reactors in the model domain. How this can be ponents. conveniently accomplished by the use of ini-files is The specification of additional input loads is op- demonstrated by Fig. 3.19 which shows a sample tional in general, but load hydrographs must nec- part of a constants file. Initial concentrations of the essarily be assigned to reactors with a stream order components are specified in a separate ’initial con- of 1, i.e. for the upstream boundaries of the model ditions file’ whose structure is identical with the domain. If multiple load hydrographs for the same ’constants file’. component are assigned to a reactor, their values are added up. 3.6.7 Time series data Time series of other boundary conditions TRAM reads all time series files in a standard In addition to the specification of load boundary ASCII format with date and time being formatted conditions for the simulated components, non-load in the form ’DD.MM.YYYY hh:mm:ss’. Hence, boundary conditions can also be defined in the time series files can conveniently be created by the boundary condition management file. For ex- spreadsheet software or data base queries. Apart ample, time series of water temperature must be from the preprocessor-generated hydrographs ad- assigned to each reactor if the water temperature dressed in Sect. 3.6.4, different groups of time se- was declared a boundary condition variable in the ries data can be distinguished. turnover model (see Sect. 3.6.5). 50 Chapter 3 The water quality simulation tool TRAM

Time series of observed concentrations plied boundary condition time series (e.g. For each simulated component and each reactor, preprocessor-generated hydrographs or time time series of observed concentrations can be spec- series of additional user-defined variables). ified. If this is done, TRAM computes a set of This is to make sure that TRAM actually con- goodness-of-fit measures and writes them to an siders all fluctuations in the boundary condi- output file which can be used for assisting model tions. calibration. • The value must not be greater than the desired resolution of the model output but it can be 3.6.8 Simulation control parameters smaller of course. Information on TRAM’s basic simulation control • A rather small value should be chosen if large parameters is provided below. The parameters can changes in the numerical values of simulation either be read from a keyword-driven text file or variables are expected within short periods of they can be specified at the command line. Com- time (highly dynamic systems). Choosing a mand line arguments always take precedence. If too large value may result in unnecessary ad- both alternatives are not used, TRAM prompts for justments of the time increment in the numer- interactive input. ical integration routines. Simulation time window It is a good idea to compare model results gener- Dates have to be specified when the simulation ated with a different simulation time step in order should start and end. Furthermore, a date can be to find the optimum value for a specific applica- supplied from which point onwards TRAM out- tion. puts detail intermediate results. This is useful when errors in the user-supplied description of Numerical solver parameters turnover processes or other input data need to be Several parameters have an influence on the nu- tracked down. merical solution. First of all, the required accu- 12 Simulation time step racy in numerical integrations must be specified . The simulation time step (given in seconds) deter- Another input parameter is the smallest tolerable mines at which time interval the values of the sim- time step (seconds) to be used in the numerical in- ulated variables are actually computed. It also con- tegration. TRAM terminates with an appropriate trols the temporal resolution of a reactor’s export- error if the automatically adjusted step size falls load hydrograph and it sets the maximum possi- below this threshold. In many applications, step ble temporal resolution of TRAM’s standard out- sizes in the order of seconds or less indicate input files. consistencies in TRAM’s input data (bad process It is important to know that this parameter also formulation, inappropriate units of constants, etc.). determines the time interval at which the values of Finally, there is an option to switch between the user-specified boundary condition variables are up- Runge-Kutta solver which is fast but not stiffly sta- dated. Furthermore, the supplied value serves as an ble and the Rosenbrock integration algorithm. initial estimate of the time increment to be used by River network data TRAM’s numerical integration routines. Along with the name of the network description Useful guidelines for selecting the appropriate simulation time step are: 12The accuracy is equivalent to the maximum fractional error in a simulated variable for an integration step. Depending • The value should be equal or less than on the application and the chosen time step, a value of 10−4 the temporal resolution of all user sup- or less may be appropriate. 3.7 TRAM’s output 51

file (see Sect. 3.6.3) stream orders must be sup- ables, boundary conditions). In the case of multiplied at which TRAM should start and finish the component simulations for large networks and simulation. As discussed in Sect. 3.6.3, specifying long time periods, the amount of output data be- an initial stream order k which is greater than 1 comes large and the writing of detailed output files requires that the export load hydrographs of all up- can therefore be optionally suppressed for selected stream reactors (stream orders < k) were written (or even all) reactors. As all output files are plain, to disk in a previous model run. tab-separated ASCII text they can readily be im- Hydrograph directories ported into spreadsheets, a data base, or visualiza- The names of two directories must be specified, tion and statistics software such as GNUPLOT or where TRAM searches for the reactor’s hydro- ’R’. graph files prepared by the preprocessors TRAMP- Hydrographs of export loads PFR and TRAMP-STR (see Sect. 3.6.4). In general, the output load hydrographs of a reactor Further input files are stored in memory and the corresponding mem- File names need to be specified for the initial con- ory is not deallocated before all downstream reac- ditions file, the constants file (Sect. 3.6.6) and the tors which receive these loads as input have been boundary condition management file (Sect. 3.6.7). processed. However, TRAM can be asked to save In addition, the name of an ini-file must be given the output loads of selected reactors to disk, which by which the output of simulation details is con- is useful if the simulation is to be restricted to a trolled. For example, the output of hydrographs part of the river network. If the output load hy- can be switched on and off and the creation of a drographs of all reactors with stream orders < k detailed logfile can optionally be suppressed. Also, were written to disk, subsequent simulations can debug printouts may be enabled for a specific reac- be started at stream order k instead if 1. tor and time through settings in this file. Validation results Finally, the name of another ini-file is required As mentioned earlier, TRAM automatically out- which assigns time series of observed concen- puts goodness-of-fit parameters if observation data trations to specific reactors and components for are provided for certain simulated components and goodness-of-fit computations. reactors. These measures include the mean error Output files and directories ME, the mean absolute error MAE, and the mean A directory must be given where all model re- absolute percentage error MAPE, the root mean sults (Sect. 3.7) and TRAM’s log file are to be square error RMSE, the square of Pearson’s cor- stored. The file names for simulation results and relation coefficient RSQR, and the efficiency af- export load hydrographs are automatically gener- ter Nash & Sutcliffe (1970). In order to assess ated based on the reactor’s names. In addition, the quality of a simulation, it is worth looking at a file must be specified where goodness-of-fit pa- a multiple of these parameters as their sensitivity rameters are written to. to certain types of errors is different (Legates & McCabe Jr., 1999). The author personally prefers to inspect the ME as it shows the bias in the unit 3.7 TRAM’s output of the variable, the MAPE because it is instructive, and the efficiency (Eq. 4.34) since the values are Standard output easy to interpret. As standard, TRAM creates a detailed output file TRAM also outputs statistics of the observed for each reactor containing time series of all vari- data and the corresponding simulated values such ables (component concentrations, hydrologic vari- as variances, percentiles, and extremes. 52 Chapter 3 The water quality simulation tool TRAM

3.8 Possible enhancements of the mated from information on water temperature and current model version wind speed, possibly using stochastic methods. Time series processing This section addresses some known limitations of As pointed out in Sect. 3.4, time series are ap- the current version of TRAM which should be re- proximated by step functions in the current ver- solved in the future. sion of the model, i.e. each time interval is repre- Immobile components in plug-flow reactors sented by a constant average value. In order to en- Currently, the only way to simulate immobile com- hance the accuracy of turnover computations (non- ponents in river or channel sections is to represent linear terms) and to reduce numerical diffusion in these sections by a tanks-in-series model, i.e. by the transport schemes, an alternative representation a series of properly sized STR. In the future, the should be considered. The use of piecewise linear plug-flow routine (Sect. 3.5.3) should be enabled functions is a possible option. to also handle immobile components. In this con- XML-based input files text, an internal spatial discretization of plug-flow The ’.ini’-format (Sect. 3.6.1), which is used for reactors needs to be introduced in order to account most of TRAM’s input, is excellent for storing tab- for the fact that the water travels downstream while ular data. For managing more complex data struc- the bottom sediment, to which immobile compo- tures, such as the definition of the turnover model nents are attached, remains in place. (Sect. 3.6.5.2), the XML format would be a better alternative. Distributed input loads from non-point sources At present, load boundary conditions are generally attached to the upstream end of a plug-flow reactor. 3.9 The source code Thus, for approximating the lateral input of mass from non-point sources, a plug-flow reactor needs TRAM is written entirely in FORTRAN 95 which to be subdivided into a series of PFR, each receiv- is an up-to-date procedural computer language ing a fraction of the total load. A revised version which is comparable to C with respect to execution of TRAM should be able to account for spatially speed. The current model version was built using distributed load boundary conditions. the free G95 compiler (www.g95.org, 2006). The use of this compiler guarantees that TRAM can be Withdrawal of water used and modified by anyone, without restrictions. A future version of TRAM should be able to sim- As this is supported by G95, the source code ulate the withdrawal of water due to seepage or makes use of some features of the FORTRAN 2003 pumping. Currently, only makeshift solutions are standard which is available as a draft (ISO/IEC, available, like the use of parallel branches with 2003). Most important is the availability of allo- near zero return flow. catable arrays in derived types. This allows for Stratified stirred tanks the creation of fully dynamic record types with- Very recent investigations on the Havel River out using pointers. If such types are stored in mod- (Becker, 2006) emphasized the relevance of tem- ule files together with specific constructor, destruc- porary stratification even in lakes of small depth. tor and data handling routines, a well structured, In a future revision of TRAM, a concept for rep- rather object-oriented source code is obtained. The resenting such events should be considered. For contents of such a FORTRAN module is virtually example, stirred tank reactors might be discretized equivalent to a class definition. into a variable number of layers. The vertical In order to facilitate maintenance and to al- fluxes between the layers would need to be esti- low for uncomplicated modifications of TRAM’s 3.9 The source code 53 sources, some coding conventions were introduced.

• Subroutines and functions were grouped into modules. In this way, non-specific parts of the code were well separated from the actual TRAM code. • All input and output data of functions and subroutines are passed as arguments. Global variables are not used at all. This guarantees that the effect of local code modifications remains predictable. • Explicit declaration was used for all variables. • All involved arrays are dynamically allocated. • No reliance is given to compiler options (e.g. variable initialization is always done explicitly, the kind of real data is always specified, etc.) • The code contains comments at crucial locations and identifiers were given associative names. • Compiler specific language extensions were used in rare cases only, such as for retriev- ing command line arguments or for executing shell commands. Wrapper routines were used in order to facilitate a replacement of these non-standard FORTRAN features when another compiler is used or TRAM is ported to another platform. 54 Chapter 3 The water quality simulation tool TRAM Chapter 4

Model application to the Lower Havel River

4.1 Objectives of modeling representation of the river network and the geometric features of individual water bodies with a grade The ultimate motivation for applying TRAM to of detail that is adequate to the spatial scale. It was the Lower Havel River follows from the outline of the task of the underlying hydrodynamic model to the research project presented in Fig. 4.1. A con- describe the flow distribution in the river system densed overview of this project has already been and the apparent influence of river regulation. given in the introduction to this thesis. In sum- Previous studies on nutrient dynamics of the mary, it was aimed at analyzing the efficiency of Havel River (Hoffmann, 1999; Schettler, 1995) different options for controlling eutrophication of have shown that the process of phosphorus remo- the Havel River and its tributaries. These options bilization from sediments strongly influences the were focused on the elements nitrogen and phos- observed P concentrations. Furthermore, Behrendt phorus and different scenarios with reduced emis- & Opitz (2000) demonstrated the significance of sions from point and non-point sources were set nitrogen retention in river systems. Based on the up. Since the Water Framework Directive requires results of Kozerski et al. (1999), significant reten- the ’good ecological status’ to be achieved until tion effects must also be expected for the Lower 2015, the time frame for scenario analyses was Havel River. 2003–2015. Consequently, a first objective of modeling was The impact analysis involved the simulation of to carry out a system analysis in order to find out the river basin’s water and nutrient budgets for the to what extent nutrient concentrations are actually scenarios with the mesoscale hydrological catch- controlled by internal turnover. The focus of the ment models SWIM (Krysanova et al., 2000) and analysis was on the magnitude and the seasonal ArcEGMO-Urban (Biegel et al., 2005). By nature, pattern of nitrogen retention and phosphorus re- these models are not capable of adequately repre- tention/release. Because TRAM does not consider senting hydrological features of individual water any conversion processes by default, an appropri- bodies and nutrient turnover in the aquatic envi- ate description of N and P turnover in the Lower ronment. Havel River had to be established first (Sect. 4.4). Consequently, TRAM was set up as an extension In this context, an ’appropriate description’ is one to the above mentioned catchment models for the which balances simplification and reliability. The river section introduced in Sect. 4.2. Using the research for an acceptable compromise was certainly actor concept (Sect. 3.1.1) TRAM allowed for the the crux of modeling.

55 56 Chapter 4 Model application to the Lower Havel River

Management scenarios Catchment modeling Alternative options for reducing Simulation of hydrology nutrient emissions from and nitrogen/phosphorus different sources export from catchments

Integrated assessment Water quality modeling Assessment of management Simulation of nutrient trans- options in a WFD-conformal port and turnover in the way river system

Figure 4.1: Basic outline of the research project ’Management Options for the Havel River Basin’.

As shown in Fig. 4.1, the suggested manage- North ment options had to be evaluated using a WFD- Sea conformal water quality classification. Thus, the Elbe Havel final objective of modeling was to provide esti- River mates of the N and P concentrations in the study Berlin river under moderately changed management as a Germany Elbe Spree basis for inter-scenario comparison and the assessment of management options. Havel River Basin Last but not least, the model study presented Land Brandenburg in this chapter was a baptism of fire for the concepts underlying the newly developed simulation Figure 4.2: Location of the Havel River Basin within tool TRAM. Germany.

4.2 Introduction to the study site River, the Spree River has its source in the hills of Upper Lusatia at an altitude of 478 m. 4.2.1 Hydrology On their course both the Havel and the Spree River pass a number of flushed lakes. Downstream With a catchment size of about 24100 km2, the of the Berlin metropolitan area, the Havel enters a Havel River is one of the major tributaries of the river-lake system with a total length of about 40 km Elbe River (Fig. 4.2). The total river length is 325 (Fig. 4.3). This river section is marked in Fig. 4.4 km but the difference in elevation between source as ’study area’ and a more detailed map is given in and mouth is only 41 m. The average slope of the Fig. 4.5. Most of the lakes are shallow (mean depth riverbed is 0.13h (Naumann, 1995). 3.5 m) and polymictic, i.e. they experience contin- The largest tributary is the Spree River which uous mixing of the water column. The large cross- joins the Havel just downstream of Berlin. At the sections in lakes and widened channels as well as confluence, the catchment size of the Spree River the lowland topography cause flow velocities to be exceeds that of the Havel by a factor of about three generally low. Due to additional river regulation by (Fig. 4.4). While the Havel itself is a pure lowland weirs, the slope of the water surface at mean flow 4.2 Introduction to the study site 57

water table (so-called seiches) are caused by wind action. For most of the time water levels are exclusively controlled at the weirs and gates (see Fig. 4.4). Target water levels and their seasonal variation are negotiated every year by the navigation and water authorities, fishermen, farmers, environmentalists, and residents. Since the Havel is a waterway under federal control, the navigation requirements usually have priority. The difference in target water levels for the summer and the winter period is only 10–15 cm at the Brandenburg weir (shown at the very left of Fig. 4.5). Figure 4.3: View over a lake-like bay of the Havel Due to the lowland topography as well as re- River in Potsdam city. tention in lakes and impoundments, flood waves of the Havel River are generally smooth. Even at the mean annual flood the water surface slope − is about 0.006h only and flow velocities may drop is only 0.9 cm km 1 (period 1993-2002; WSA, to near-zero values in dry periods during summer. 2003). The estimated discharges with return periods of 5, 20, 50, and 100 years are 151, 167, 202, The mean specific discharge of the Havel River − and 216 m3 s 1 at the Ketzin gage. at the Ketzin gage (see Fig. 4.4) is 4.6 l s−1 km−2. The Havel River is low in suspended sediments Compared with other German river basins (Nau- since erosion rates are low and much of the trans- mann, 1995) this is a rather low value (e.g. Elbe ported material is deposited in lakes. With respect at Wittenberge: 5.6, Oder at Eisenhüttenstadt: to the Spree tributary, the inland delta ’Spreewald’ 6.1, Rhine at Kaub and Danube at Ingolstadt: is an additional huge sediment trap. The bottom 15.1 l s−1 km−2). The low value of 4.6 l s−1 km−2 sediments of the Havel River are mainly sandy. In appears even more remarkable if one takes into ac- lakes and dead zones organic mud is deposited on count that the discharge of the Havel River was ar- top of the fluvial substrate. tificially increased in the past due to drainage of the Lusatian coal mining areas (Spree catchment). 4.2.2 The eutrophication problem Basic hydrologic parameters for the Ketzin gage (label 6 in Fig. 4.4 and label Hv0195 in Fig. 4.5) The major water quality problem of the Lower are summarized in Table 4.1. Like at many other Havel River with its many interconnected lakes is gages along the Havel River, discharges are mea- eutrophication. Consequently, water quality classi- sured by the ultrasonic method. Nevertheless, even fication focuses on the trophic state rather than on the gage operators regard measurements as highly saprobity. uncertain during periods of low flow. Attempts to Various methods of trophic state classification measure flow rates at some ungaged locations of for lakes exist (e.g. Carlson, 1977; Vollenweider & the Havel River by induction sensors had to be Kerekes, 1982) and comparable schemes for plank- given up by the author. However, the measure- ton dominated rivers have recently been developed ments led to the insight that, at low discharge, the (Behrendt & Opitz, 1996; Behrendt & Mischke, flow is heavily controlled by wind and navigation. 2002; LAWA, 2002). Most of them use either sta- In some lakes, low-frequency oscillations of the tistical values of phosphorus and nitrogen concen- 58 Chapter 4 Model application to the Lower Havel River

Havel Main gauges: River 3480 km² 1 Borgsdorf Elbe 1 Spree 2 Berlin Sophienwerder 2 3 Berlin Spandau River 10140 0 4 Kleinmachnow 3 Teltow- km² 9 5 Potsdam Babelsberg 4 kanal 6 Ketzin (16170 km²) Dosse Nuthe 1780 km² 7 Brandenburg (16530 km²)

Rhin PHV SPK 5 8 Rathenow 9 Havelberg 8 6 Emster 210 km² 0 Gnevsdorf (24100 km²) Kanal Weirs & gates Stremme Study area EHK 7 PHV: Potsdamer Havel Buckau Plane SPK: Sacrow-Paretzer-Kanal EHK: Elbe-Havel-Kanal

Figure 4.4: Schematic representation of the Lower Havel River between Berlin Spandau and its mouth at Gnevs- dorf. Shown are the main tributaries (italic names; all are heavily regulated) and the major gaging stations (numbers) with their corresponding catchment areas. Highlighted is the study area between Berlin Spandau and the city of Brandenburg. The ’Potsdamer Havel’ represents the original course of the river while the ’Sacrow-Paretzer- Kanal’ is an artiﬁcial shortcut channel designed for navigation.

Table 4.1: Flow statistics of the Havel River at the Ketzin gage for the hydrological years 1988–2004. All values in m3 s−1. Source of raw data: Wasser- & Schifffahrtsamt Brandenburg.

Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Year Year of appearance ’88 ’94 ’94 ’94 ’88 ’88 ’94 ’94 ’95 ’02 ’94 ’02 HQ 120 139 176 179 185 170 185 122 105 123 103 97 185 MHQ 77 93 101 110 116 97 75 60 51 43 55 60 130 MQ 58 69 82 91 93 82 53 40 31 26 38 45 59 MNQ 41 45 59 64 67 55 27 18 14 10 19 29 7 NQ 11 21 14 21 40 20 3 0 0 1 1 12 0 Year of appearance ’04 ’97 ’97 ’97 ’04 ’93 ’00 ’03 ’03 ’03 ’03 ’04 4.2 Introduction to the study site 59 pree fvxITHGITIGQPH 6 fvxRPSGRPI fvxQPS 6 6 IH um fvxQQH 6 uvisxwegrxy fvxQQS fivsx2exxii eltowknl e 6 fvxQSH

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Figure 4.5: Detailed map of the river-lake system between Berlin Spandau and the city of Brandenburg. Names of major tributaries are given in bold, city and village names in capitals. 60 Chapter 4 Model application to the Lower Havel River

Table 4.2: WFD-conformal water quality assessment ) 1000 -1 scale for the Havel Lakes with respect to phosphorus Station Bad High Poor from LUA (2005a). The total phosphorus concentra- Good HV0120 Bad −1 HV0130 tions are annual average values in µg l . The class Moderate ’high’ represents the reference conditions without an- HV0140 HV0160 100 thropogenic disturbance. Poor HV0170 Moderate HV0180 State High Good Moderate Poor Bad HV0195 Good TP <97 97– 172– 305– >538 172 305 538 High

Chl-a (May-Sep. average, µg l 10 10 100 1000 TP (May-Sep. average, µg l -1 ) trations (focus on the causes) or parameters like Figure 4.6: Water quality of the Potsdamer Havel Secchi-depth or chlorophyll-a (focus on the conse- Lakes according to the classification suggested by quences). Some classification schemes make use Behrendt & Mischke (2002) based on average values of of both kinds of indicators. The application of the chlorophyll-a and total phosphorus over the vegetation assessment scale suggested by Behrendt & Mis- period (May–September). Each data value represents a chke (2002) is illustrated in Fig. 4.6. According to single year of the period 1998–2004. Mietz & Vietinghoff (1994) all lakes of the Pots- damer Havel shown in Fig. 4.5 belonged to the hy- Influence of hydrological conditions pereutrophic type at the beginning of the 1990s. According to the classification of German rivers With the implementation of the European Wa- and lakes (LAWA, 2003), the lakes of the Lower ter Framework Directive in 2000, the standards for Havel River belong to type 12 (calcium-rich, non- trophic state evaluation changed. It became nec- stratified lakes of the lowland with residence times essary to assess the ’ecological state’, i.e. the de- of 3–30 days and a volume quotient1 V Q greater viation between the current ecological conditions than 1.5 m−1). and reference conditions, expected in the absence The volume quotient is a key parameter in the of anthropogenic disturbances. For the lakes of identification of potentially natural nutrient con- the Havel catchment, an assessment scale for the centrations (LUA, 2005a). The larger V Q is, the chemical parameter phosphorus was developed by higher is the expected in-lake nutrient concentra- LUA (2005a). It is based on a paleolimnetic anal- tion in steady state. In the case of the lakes shown ysis of diatom skeletons in undisturbed sediment in Fig. 4.5, the volume quotients are large. For ex- cores. This scale (see Table 4.2) is less stringent ample, V Q is 550 m−1 for Lake Schwielowsee. than the one from Fig. 4.6 and the recent qual- In addition, short residence times of the water (see ity state of the Havel Lakes downstream of Berlin Fig. 4.7) guarantee a constant delivery of nutrients is classified as ’moderate’. Within the research from the catchment. project ’Management Options for the Havel River Finally, elevated nutrient concentrations over the Basin’ the scale from LUA (2005a) was used for its vegetation period are favored by the absence of compatibility with the requirements of the WFD. summer stratification. While settled nutrients are temporarily trapped in the hypolimnetic layer of It is important to understand that the high stratified lakes, an efficient recycling throughout trophic level of the Havel River is favored by sev- the whole year occurs in shallow water bodies. − eral factors: River morphology, catchment hydrol- 1The volume quotient VQ (m2 m 3) relates the size of a ogy, and the excessive input of nutrients. water body’s catchment area to its volume. 4.2 Introduction to the study site 61

100 Although external nutrient loading decreased in the 1990s, the concentrations of phosphorus and Schwielowsee nitrogen are still elevated (Fig. 4.8). In the case of Werdersche Havel 20 Zernsee phosphorus, this is not only due to ongoing emissions from point and non-point sources but also 10 due to internal loading. A large phosphorus pool 5 has accumulated in the lake sediments over the past decades (Rohde, 1995). Today, after partial reduc- 2 Templiner See Süd tion of external P loads, the excess phosphorus of Templiner See Nord the sediments is slowly being released. The result- Water residence time (d) 1 ing delay in the recovery of water quality is a com- 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 mon phenomenon (Jeppesen et al., 1991; Kozerski Empirical non-exceedence probability & Kleeberg, 1998; Søndergaard et al., 2003). Figure 4.7: Period-of-record duration curves of the water residence time (days) in lakes of the Potsdamer 4.2.3 Water quality overview Havel. Period: 1995-2004. At the monitoring stations shown in Fig. 4.5, surface water quality is inspected fortnightly by the state environmental authorities (LUA Branden- Impacts of nutrient loading burg, Berliner Senat). At selected locations, daily Anthropogenic eutrophication is primarily caused data from automatic samplers are available; these by elevated nutrient emissions from the catchment. samples are analyzed for a greater number of phys- As Table 4.3 illustrates, both point and non-point ical, chemical, and biological parameters includ- sources contribute to the pollution of the Havel ing water temperature, pH, conductivity, oxygen, River in significant shares. Because many sewage total N and P, organic nutrient fractions, nitrogen treatment plants were equipped with enhanced P species, and chlorophyll. elimination in the mid of the 1990s, the contri- Fig. 4.8 illustrates the trend in the annual aver- bution of P emissions from non-point sources is ages of nutrient and chlorophyll-a concentrations expected to have increased compared to Table 4.3 at four monitoring stations in the period 1990– while the total emissions declined. 2004. Obviously, there was a continuous decline The predominant non-point sources of phospho- in all nitrogen species with a remarkable drop in rus are polluted groundwater, erosion, and wash- ammonium until the mid of the 1990s. This is cer- off from urban areas (LUA, 2002). In the case of tainly the result of a successful sewage treatment nitrogen, groundwater pollution as well as emis- policy. The latter focused on a reduction in total N sions from drained areas are most important. The emissions but also aimed at emitting the residual − total emissions have declined by about 66 % (P) inorganic N in oxidized form (NO3 ) rather than as + and 47 % (N) when comparing the periods 1993– oxygen-depleting and potentially toxic NH4 . Pos- 1997 and 1983–1987 (LUA, 2002). With respect sibly, the disproportional drop in ammonium con- + to phosphorus, point-source emissions were pri- centrations is intensified by the NH4 preference in marily reduced due to the use of phosphorus-free phytoplankton uptake. detergents and enhanced elimination in wastewa- Compared to nitrogen, phosphorus concentra- ter treatment. With regard to nitrogen, the shift in tions show a different pattern. At the mouth of the the contribution of different sources is much less Teltowkanal and downstream monitoring stations pronounced. a significant drop in TP could be observed in the 62 Chapter 4 Model application to the Lower Havel River

Table 4.3: Phosphorus and nitrogen emissions (EP , EN ) at selected gages after LUA (2002). Presented are total emissions (t a−1) and the estimated contribution of non-point sources (%) in the period 1993–1997.

−1 −1 River Gage EP (t a ) % diffuse EN (t a ) % diffuse Obere Havel Henningsdorf 78 65 1766 88 Spree Berlin Mühlendammschl. 413 53 10498 54 Nuthe Potsdam Babelsberg 58 59 1181 84 Havel Ketzin 712 55 17984 50 Havel Brandenb. Mühlendamm 722 55 18186 51

first half of the 1990s. Thereafter, no clear trend in two other factors: self-shading and possibly tem- TP can be identified and also the concentrations of porary nitrogen shortage. dissolved phosphorus do not exhibit a continuous The effect of self shading can be illustrated by decline. The above mentioned process of internal P a simple numerical example. According to the loading provides a reasonable explanation for this law of Lambert-Beer, light extinction is a first or- observation. der process and the underwater light intensity I at depth is related to the intensity at the water sur- In eutrophic rivers there are strong seasonal dy- z face by Eq. 4.1, with being the extinction co- namics in many water quality parameters. Fig. 4.9 I0 E efficient. displays the seasonality of both nutrients and chlorophyll based on data from 1998–2004. −Ez With respect to water quality management, I(z)= I0 · e (4.1) knowledge of the factors that currently control In eutrophic waters low in suspended solids, the aquatic primary production is fundamental. A look extinction coefficient for photosynthetically active at Fig. 4.9 reveals that the concentration of dis- radiation (PHAR) is primarily controlled by the solved phosphorus, which limits algal growth in chlorophyll concentration. Various empirical for- many rivers and lakes, is too high to restrict the mulas exist for estimating E as a function of the annual maximum in primary production. In fact, chlorophyll concentration and a widely used model there is a significant increase in the concentra- is presented by Ambrose et al. (2001). A rule tions of soluble reactive phosphorus (SRP) due of thumb for estimating E from the Secchi-depth to internal loading and weak dilution in summer z is provided by Witter (2002) with E = rather that a depletion by algal uptake. Dissolved Secchi 1.5/zSecchi. If the background extinction E0 is as- phosphorus concentrations are much lower in early − signed a value of 0.7 m 1 both cited approaches spring only. Though Sas (1989) states that only av- fit the data from the Havel River quite well. Using erage SRP concentrations lower than 0.01 mg l−1 Eq. 4.1, the function I(z) can now be plotted for over the entire vegetation period can indicate different chlorophyll levels. Fig. 4.10 reveals that phosphorus limitation of the phytoplankton, half- about 90% of the available light is extinguished in saturation concentrations used in Michalis-Menten the first meter of the water column at a chlorophyll- type growth models (Bowie et al., 1985) suggest − a concentration of 75 µg l 1 (recall the data from that the low SRP levels in spring might have some Fig. 4.9). Thus, in a well-mixed shallow lake of minor effect. 2.5 m depth, the algae are exposed to less than According to the present understanding, peak 10% of the near-surface light intensity for 3/5 of primary production is likely to be controlled by the daylight period. 4.2 Introduction to the study site 63

Hv0195 BLN345 TN DIN TN DIN 4.00 NH4 NO3 4.00 NH4 NO3 3.00 3.00 2.00 2.00 1.00 1.00 0.00 0.00 CHLA CHLA 0.40 TP 0.40 TP SRP SRP 0.30 0.30 0.20 0.20 0.10 0.10 0.00 0.00 1990 1992 1994 1996 1998 2000 2002 2004 1990 1992 1994 1996 1998 2000 2002 2004

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Figure 4.8: Trend in the annual average concentrations (medians; mg l−1) of the key variables chlorophyll-a (CHLA), total and dissolved phosphorus (TP, SRP), total nitrogen (TN), dissolved inorganic nitrogen (DIN), and − + the DIN species NO3 -N and NH4 -N (given as N) at four monitoring stations near Potsdam. The stations are Krughorn (BLN345), Teltowkanal - Paschabrücke (TK0030), Potsdamer Havel - Baumgartenbrücke (HV0160), and Ketzin - Fähre (HV0195). Note that the scale for nitrogen concentrations at station TK0030 differs from the other charts. The phosphorus peak in 2000 is a result of extraordinarily high P remobilization from sediments (see Fig. 4.36 in Sect. 4.5.1.) 64 Chapter 4 Model application to the Lower Havel River

Hv0195 BLN345 4.0 4.0 TN DIN TN DIN 3.0 NH4 NO3 3.0 NH4 NO3 -1 -1 2.0 2.0 mgN l mgN l 1.0 1.0

0.0 0.0

Jul 120 Jul 120 Oct Oct Jun Jun Apr Apr Jan Jan Nov Nov Mar Mar Aug Aug Feb Feb Dec Dec Sep Sep 0.8 CHLA May TP 0.8 CHLA May TP SRP 100 SRP 100 0.6 80 -1 0.6 80 -1 -1 -1 60 60 0.4 0.4

mgP l 40 mgP l 40 0.2 µgChl-a l 0.2 µgChl-a l 20 20 0.0 0 0.0 0 Jul Jul Oct Oct Jun Jun Apr Apr Jan Jan Nov Nov Mar Mar Aug Aug Feb Feb Dec Dec Sep Sep May May

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Jul 120 Jul 120 Jun Jun Oct Oct Jan Jan Apr Apr Mar Mar Nov Nov Feb Feb Aug Aug May May Dec Dec 0.8 CHLA TP Sep 0.8 CHLA TP Sep SRP 100 SRP 100 0.6 80 -1 0.6 80 -1 -1 -1 60 60 0.4 0.4

mgP l 40 mgP l 40 0.2 µgChl-a l 0.2 µgChl-a l 20 20 0.0 0 0.0 0 Jul Jul Oct Oct Jun Jun Jan Jan Apr Apr Nov Nov Mar Mar Aug Aug Feb Feb Dec Dec Sep Sep May May

Figure 4.9: Seasonality in the concentrations of chlorophyll-a (CHLA), total and dissolved phosphorus (TP, SRP), − + total nitrogen (TN), dissolved inorganic nitrogen (DIN), and the DIN species NO3 -N and NH4 -N (given as N) at the four monitoring stations shown in Fig. 4.8. In order to illustrate the variability in the data, the upper and lower quartiles are shown for each parameter and each month, i.e. 50% of the observed values fall in the range between the two corresponding lines or within the grey band, respectively. Note that the scale for nitrogen concentrations at station TK0030 differs from the other charts. 4.2 Introduction to the study site 65

Light intensity (% of surface value) Hv0110 30 0 20 40 60 80 100 100 25 0.0 80 20 ) 60 0.5 -1 15 25 µg/l 40

1.0 10 ) 50 µg/l -1 5 20 1.5 75 µg/l 0 0 2.0 100 µg/l

Chl-a Jul Hv0180 Water depth (m) Oct Apr Jun Jan Mar Feb Chl-a (µg l (µg Chl-a Nov Aug Dec Sep Abundance (mg l (mg Abundance 125 µg/l 30 Total phytoplanktonMay 100 2.5 25 Green algae Diatoms 80 20 Figure 4.10: Underwater PHAR intensity as percent- Cyanobakteria 60 15 age of the intensity just below the air-water interface at 40 10 different levels of chlorophyll-a in µgl−1. 5 20 0 0 Jul Apr Oct Jun Jan Mar Feb Nov Aug Dec Sep As mentioned above, shortage of available ni- May trogen (DIN) in summer might contribute to the Figure 4.12: Concentration of chlorophyll-a (shaded limitation of phytoplankton growth. According to area; µgl−1) and abundance of the major phytoplank- present understanding, two mechanisms of DIN ton groups (lines; mg l−1) upstream and downstream of + removal are important: the depletion of NH4 and the Potsdamer Havel Lakes (see Fig. 4.5 for the loca- − tions Hv0110 and Hv0180). Shown are monthly medi- NO3 by algal uptake as well as denitrification. While the first process obviously changes the ans for the period 1996–2002. pelagic DIN concentrations, the effect of denitrification is believed to be more indirect. As den- relation to some extent. A possible explanation itrification is largely bound to oxic–anoxic inter- is a gradient in DIN (in space and/or time) that faces in the sediment, it attenuates the recycling of correlates with a shift in the species composition settled organic N rather than affecting the pelagic − of the phytoplankton, with different species ex- NO concentration directly. 3 hibiting different cellular chlorophyll levels. Addi- For the time being, N limitation of algal growth tional disturbing factors must be considered when can only be suspected based on correlation anal- analyzing DIN-chlorophyll correlations such as ysis. Fig. 4.11 (left chart) indicates that, dur- loss processes (e.g. grazing) or nitrogen fixation ing the vegetation period, higher concentrations of by cyanobacteria. Nutrient addition experiments chlorophyll-a correspond to higher concentrations (e.g. Weithoff & Walz, 1999) or an analysis of of dissolved inorganic N. A similar relationship be- cyanobacteria species would be required in order tween chlorophyll and available phosphorus (SRP) to verify the hypothesis of temporary nitrogen lim- could not be found (Fig. 4.11, right chart), under- itation. pinning the oversupply of P. Though N limitation seems plausible when The share of different taxonomic groups and the looking at the left chart of Fig. 4.11, one must seasonal pattern of phytoplankton abundance in the not confuse correlation and causality. In fact, the Lower Havel River are illustrated in Fig. 4.12. Di- average DIN concentrations during the vegetation atoms and cyanobacteria dominate the phytoplank- period are quite high (recall Fig. 4.9) when com- ton throughout the year. Both groups are well pared to the threshold of 0.1 mg l−1 which is re- adapted to low light intensities which makes them quired for N-limitation according to Sas (1989). competitive at high chlorophyll levels (e.g. Schef- Hence, Fig. 4.11 is likely to show a spurious cor- fer et al., 1997). 66 Chapter 4 Model application to the Lower Havel River

300 ) -1 250

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Chl-a (May-Sep. Quantile, 0.75 µgChl-a l 0 0.0 2.0 4.0 6.0 0.0 0.2 0.4 0.6 0.8 DIN (May-Sep. 0.75 Quantile, mg l -1 ) SRP (May-Sep. 0.75 Quantile, mg l -1 )

Figure 4.11: Correlation between the upper quartiles of dissolved nutrient concentrations (nitrogen as DIN, phosphorus as SRP) and the upper quartiles of chlorophyll-a (Chl-a) in the vegetation period May–September. Note that each data point represents one single year of the period 1992–2004. Also note that the data for each monitoring station Hv0120–Hv0195 are plotted with a different style in order to show that (for nitrogen) correlations exist at each single monitoring station and not only due to a spatial trend of decreasing DIN and Chl-a concentrations from upstream to downstream monitoring stations.

The time lag between the peaks of cyanobacteria light limitation might be more severe in the lakes and diatom biomass (Fig. 4.12, upper chart) prob- than in upstream reaches like the Teltowkanal be- ably results from higher growth rates of the latter cause of increased depth and less turbulence2. A group which typically dominates the phytoplank- spatial gradient in zooplankton abundance3 or a de- ton in early spring (Sommer et al., 1986). A high crease in DIN (denitrification, settling) and/or sil- proportion of cyanobacteria in summer is also typ- ica (settling) along the flow path could be alterna- ical for eutrophic lakes due to the ability to cope tive explanations. with high grazing pressure, high turbidity, and pos- The shift in the ratio of chlorophyll-a and al- sibly shortage of nitrogen. gal biomass is pointed out again in Fig. 4.13. The The significant drop in phytoplankton concen- significant increase in this ratio from monitoring trations in May–June is a phenomenon observed in station Hv0110 to Hv0180 shows an adaptation many lakes (e.g. Noges et al., 1998). Grazing of of the phytoplankton’s cell-internal pigment con- zooplankton and an adaptation of the algae com- centration to low underwater irradiance. This fact munity to decreased flushing and higher tempera- corroborates that light availability is in fact a ma- tures are believed to be of importance. Fig. 4.12 displays a decrease in phytoplankton 2The effect of turbulence on primary production was concentrations through the Potsdamer Havel Lakes demonstrated by Gervais et al. (1997). 3 from monitoring station Hv0110 to Hv0180 which Since zooplankton growth rates are low compared to al- is contrary to observations in many other river-lake gal growth rates, the buildup of zooplankton takes more time. Hence, one should expect an increase in zooplankton and a systems (e.g. Kneis, 2002). For the time being one reduction in its prey (algae) during the passage of a river-lake can only speculate on the causes. For example, system. 4.3 Hydrodynamic submodel 67

the creation of hydrodynamic models for spa- -1 12 Hv0180 tially large domains. 10 Hv0110 • The program’s GIS capabilities allow for a 8 visual overlay of the river network as repre- 6 sented in RAS with TRAM’s network of re- 4 actors. This makes it easy to identify those cross-sections where stage and flow data must 2 be recorded for later use by TRAM’s prepro- 0 cessors. µg Chl-a (mg Phytoplankton) Jul Apr Oct Jan Jun Mar Feb Nov Aug Dec Sep May • Software utilities are available for extracting the desired information (e.g. hydrographs at a Figure 4.13: Chlorophyll-a content of total phyto- − certain location) from large model outputs in plankton biomass (µg Chl-a (mg Phytoplankton) 1) upstream (Hv0110) and downstream (Hv0180) of the a very efficient manner (see Sect. 4.3.4). Potsdamer Havel Lakes. Shown are monthly medians • HEC-RAS is widely used because it is freely for the period 1996–2002. available and well documented. jor controling factor for primary production in the 4.3.1 Sources and preprocessing of geom- Lower Havel. etry data For creating the geometry input file for HEC-RAS, 4.3 Hydrodynamic submodel data from different sources were combined. Sonar data As mentioned in Sect. 3.2, the hydrodynamics Cross-section data from sonar measurements were are not computed by TRAM but by an external made available by the navigation authorities for the software. In the presented application, the one- river section between the mouth of the Havelkanal dimensional, unsteady hydrodynamic model HEC- near Ketzin and the downstream model boundary RAS (USACE, 2002) was used for computing flow at the city of Brandenburg. Since sonar and GPS rates and stages in the river-lakes system shown in measurement were combined, the data have a full Fig. 4.5. This model proved to be particularly suit- spatial reference (3D coordinates). able for several reasons: Data from an existing model For the river section between Berlin-Spandau (up- RAS handles looped systems, i.e. river net- • stream boundary at the confluence of Spree and works with multiple parallel flow paths. Havel) and Krughorn (flow split upstream of Pots- • The software supports the import of geome- dam), cross-section data were extracted from the try data via an ASCII-interface. It is there- geometry data base of the water surface profile cal- fore possible to import the geometric data culation model HYDRAX5. Some further param- (cross-sections, stream lines, channel junc- eters such as the cross-sections of bridges were tions, bank lines, levee positions, ineffective also taken from this source. Since HYDRAX uses areas4) in an automated way using GIS and/or 2-dimensional cross-section data (offset from left other external utilities. This greatly speeds up bank, bottom elevation), the data had to be geo- referenced, based on river station information. To 4Areas which are inundated at high stage but which do not contribute to the active cross-section area (ponding areas). 5The data were provided by the federal water authorities. 68 Chapter 4 Model application to the Lower Havel River do so, the cross-section cut lines6 were drawn in the GIS and the 3-dimensional position of each el- Bottom elevation evation measurement was recomputed by a utility Bank lines program Xs2dToXs3d. Cross-section extent Levee Topographic maps x Ineffective area For the lakes of the Potsdamer Havel, contour lines of the bottom elevation were digitized from topographic maps with a scale of 1:10000 (see Fig. 3.8). The information was converted into cross-section data by another utility (ContourToXs3d). Data on levee positions were also taken from digital topographic maps. Digital elevation model Figure 4.15: A section of the Lower Havel River dis- Since the sonar data were recorded from a boat, played in HEC-RAS’s 3D-viewer. shore line positions as well as elevation data for the floodplain are (with some exceptions) lacking. In the same way, bank and levee positions etc. are In order to allow for a continuous unsteady simu- identified for all cross-sections. lation including floods, floodplain elevations were The final result of the geometry preprocessing is extracted from a digital elevation model (DEM) shown in Fig. 4.15 and Fig. 4.16. The complete which was made available by project partners. The HEC-RAS data base for the studied section of the location and extent of ineffective areas were also Lower Havel River comprises about 1100 cross- estimated based on the DEM. sections and 27 junctions. How all data (bed elevations, floodplain eleva- It should be noted that the Havel Lakes are rep- tions, levee positions, stream lines, etc.) from dif- resented in the model by ordinary channels, not by ferent sources were merged into one single data reservoirs. This is necessary because the lakes’ base is exemplified in Fig. 4.15. Basically, the water surface elevation is entirely controlled by sonar data which are present as point data (x, y, the weir at Brandenburg (see Fig. 4.5). Hence, elevation) were automatically converted to short no stage-discharge relationships exist at the lakes’ line segments oriented parallel to the channel. The outlets which could be used for computing the out- same was done with floodplain elevation data ex- flow rates as a function of stage. In the existing tracted from the DEM along specified paths. Fi- HYDRAX model, the lakes are also represented by nally, the cut lines, defining the position of the channel sections. individual cross-sections, were drawn in the GIS. Other information like streamlines, bank positions 4.3.2 Boundary conditions and levees were drawn as lines as well. The dif- The two major upstream boundary conditions of ferent line data sets were then exported from the the hydrodynamic model are the inflows at Berlin- GIS and preprocessed by a FORTRAN program Spandau (Upper Havel, Spree) and Kleinmach- Gis2Ras developed by the author. This utility col- now (Teltowkanal). The smaller tributaries (Nuthe, lects all elevation data along the cross-section cut Havelkanal, Emster Kanal) appear as lateral in- lines by looking for intersections with the men- flows as does infiltrating groundwater. Because of tioned line segments representing elevation data. the large water surface area, evaporation was also 6Top view of a cross-section. taken into account. Compared to the discharges of 4.3 Hydrodynamic submodel 69

Cross-section Elevation data from Sonar ele- cutlines DEM vation data

Main channel

Side branch

Bank positions Streamlines Interpolated Levee countours

Figure 4.14: Illustration of geometry data preparation for HEC-RAS in the GIS. Elevation information is shown by arrays of short line segments oriented parallel to the channel. At this site, they were taken from two sources: the DEM (brown lines) and from the sonar data (colored lines, elevation increases from dark blue to red). The original elevation data (point data) were converted to the line segments shown in order to facilitate the digitizing of the intersecting cross-section cutlines (bold black lines perpendicular to the river).

ilevtion2dt2from2hiw genter2strem2lines

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Obere Havel km 0 Spree Berlin-Spandau

Ketzin Havelkanal Kladower Mittlere Havel 11 10 Seestrecke Sacrow- 9 13 Paretzer- 12 8 Kanal km 55 15 14 1817 16 30 2019 21 23 22 24 25 Emster Kanal Griebnitz- 3 Brandenburg Potsdam kanal 6 1

27 Potsdamer 2 4 Havel Nuthe Teltowkanal

Figure 4.16: The Havel River between Berlin-Spandau (river station 0) and the city of Brandenburg (km 55) as represented in HEC-RAS. The distance refers to the connection via the Sacrow-Paretzer-Kanal. The total length of the Potsdamer Havel side branch is approximately 29 km. Blue lines represent the reaches, green lines show the cross-section cut lines and red numbers identify channel junctions. The names of tributaries (inﬂow boundary conditions) are printed in bold. Note the smaller distance between adjacent cross-sections in the western part of the model domain resulting from better data availability (sonar data). Also note how the large dead zone of Lake Schwielowsee (in the very south of the Potsdamer Havel) was represented in the model by a dummy tributary. 4.4 Water quality submodel 71

Table 4.5: Error statistics for the hydrodynamic sim- identify the cross-sections (or external inflow lo- ulation in the calibration period (1988–2000) and four cations) in the hydrodynamic model which corre- subsequent years (Q: flow rate, W : water surface el- spond to boundaries of TRAM’s reactors. Prac- evation). ME: mean error, MAE: mean absolute error, tically, this was done by loading a projected im- NSE: Efficiency after Nash & Sutcliffe (1970). age or shape file of the TRAM reactor network Variable Period ME MAE NSE into HEC-RAS’s geometry viewer as a background (see Fig. 4.17). For each of TRAM’s reactors, Q at Ketzin 88–00 2.79 7.68 0.92 the river stations of the most upstream and down- (m3 s−1) 01–04 1.48 6.23 0.94 stream cross-section were determined. At these lo- W at Spandau 88–00 0.00 0.03 0.96 cations, stage and flow hydrographs were extracted (m) 01–04 0.03 0.04 0.87 from the HEC-RAS output. Because a long-term hydrodynamic simulation the tributaries Upper Havel, Spree, Teltowkanal, produces a huge amount of output data, HEC- and Nuthe, the other boundary conditions are of RAS saves its simulation results in a binary for- minor importance only. At the downstream bound- mat using a specially designed data storage system ary of the model domain, the time-variable water (HEC-DSS). For exporting the required stage and surface elevation at the Brandenburg weir was pre- flow hydrographs at the identified ’linking’ cross- defined. Table 4.4 summarizes how the bound- sections from the HEC-DSS, the DSSUTL util- 7 ary condition values were obtained for the calibra- ity was used. The exported data served as in- tion period and scenario simulations in the period put for TRAM’s preprocessing utilities TRAMP- 2003–2015. PFR and TRAMP-STR which were introduced in Sect. 3.6.4. It should be noted that the comparison of model 4.3.3 Model calibration geometries for creating the list of HEC-RAS’s output locations is the only manual step. The actual Estimates of the channel roughness were adopted transfer of information between the two models is from the existing HYDRAX model. A compari- fully automated. Data exchange is a matter of min- son of observed flow rates and water surface ele- utes once all preprocessing tools are configured. vations with simulated values at the gages Ketzin, Potsdam, and Berlin-Spandau indicated satisfying agreement. After some further calibration runs, a 1/3 −1 value of Kst=37 m s was chosen for the main 4.4 Water quality submodel channel and the floodplain was assigned a value of 1/3 −1 Kst=15 m s with Kst being the reciprocal of For a proper understanding of the nutrient turnover Manning’s n. Some goodness-of-fit parameters for model presented in Sect. 4.4.3, some basic knowl- the calibration period and four additional years are edge of the fate of nitrogen and phosphorus in shown in Table 4.5. the aquatic environment is indispensable. Con- sequently, a short introduction to nutrient cycles (Sect. 4.4.1) and P turnover (Sect. 4.4.2) is given, 4.3.4 Exchange of data between the hy- before the model approaches taken are addressed drodynamic model and TRAM in Sect. 4.4.3. Before data can be exchanged between HEC-RAS 7DSSUTL allows for very efficient retrieval and format- and TRAM, the geometry of both models needs to ting of mass data from the HEC-DSS when run in batch mode. be overlayed first. This is necessary in order to The program is freely available from the HEC. 72 Chapter 4 Model application to the Lower Havel River

Table 4.4: Boundary conditions of the hydrodynamic model (Q=ﬂow, W=water surface elevation) and corresponding data sources.

Boundary Type Observation period Scenario simulations condition (1988–2004) (2003–2015) Obere Havel Q Measured flow rates at the gages Flow rates simulated by the catch- Berlin-Freybrücke und Berlin- ment model SWIM d Stößenseebrücke (sum of Obere Havel and Spree) a Spree Q see above Observed flows for the period 1988–2000 at the Sophienwerder gage corrected for a trend as predicted by BfG (2003) Teltowkanal Q Measured flow rates at Klein- Observed flows for the period machnow b 1988–2000 at the Kleinmachnow gage corrected for a trend as predicted by BfG (2003) Nuthe Q Measured flow rates at Potsdam- Flow rates simulated by the catch- Babelsberg c ment model SWIM d Havelkanal Q Estimated seasonally variable Flow rates simulated by the catch- flow rates e ment model SWIM d Emster Kanal Q Monthly averages, estimated from Flow rates simulated by the catch- the discharges of the Nuthe River ment model SWIM d and the ratio of catchment sizes Groundwater infil- Q Estimated from model error at Flow rates simulated by the catch- tration gage Ketzin ment model SWIM d Evaporation Q Computed by the PENMAN- see left Equation (DVWK, 1996) Weir Brandenburg W Observed data Monthly target values after WSA (upper gage) (2003) taking into account ex- ceedance during floods

a Provided by Senatsverwaltung für Umwelt und Stadtentwicklung Berlin b Provided by Wasser- und Schifffahrtsamt Brandenburg c Provided by Landesumweltamt Brandenburg d Provided by A. Habeck, Potsdam Institute for Climate Impact Research e Löper, G. (WSA Brandenburg), pers. comm. 4.4 Water quality submodel 73

331.919 267.649 163.423 1 km 19 20492.077

385.043 666.314 198.583 586.921 21 498.344 433.404 310.258 290.324 23 93.007 446.27722 251.164 178.061 66.636 565.850 1222.659 1347.224 697.426 24 822.800 505.407

385.486 768.964 1120.1081323.186 1741.969 263.444 675.184 971.558 1892.909 295.227 427.231 169.263421.478 25 237.105 375.413 355.611 993.850 891.382 53.419 199.972 742.930 HEC-RAS cross-section cut lines with river station 644.013 Center streamlines of TRAM's plug-flow reactors (individual reactors are shown in different colors) 490.130 Bank lines 438.878 594.252 392.903 293.764

Figure 4.17: Visual overlay of the geometry of the HEC-RAS model with TRAM’s network of reactors. The comparison of the model’s geometries is necessary in order to identify those ’linking’ cross-section, for which stage and ﬂow hydrographs must be exported from the hydrodynamic model as input for TRAM’s preprocessors TRAMP-PFR and TRAMP-STR.

4.4.1 Simplified outline of the aquatic nu- • Nutrients are lost from all trophic levels. Ex- trient cycle cretion of nutrients from the consumers (zooplankton, fish) recharges the pool of dissolved A generalized chart of the nutrient cycle in a shal- inorganic nutrients directly. Particulate or- low, non-stratified aquatic environment is shown ganic matter and released dissolved organic in Fig. 4.18. While many processes equally affect matter must first be mineralized before the nu- both N and P, there are also some important differ- trients they contain become available for up- ences. take again. Common features of the N and P cycle • Particulate organic matter from the plankton Both nitrogen and phosphorus are affected by the and detritus pool is subject to settling which major turnover processes listed below. results in a transfer of nutrients to the sed- • Dissolved inorganic nutrients (ortho- iment. In the active upper layer with high + − phosphate, NH4 , NO3 ) are taken up by the microbial activity, nutrients recycling takes autotrophic phytoplankton but also by bacte- place and the dissolved forms (mainly ortho- + − ria feeding on dissolved organic material low phosphate, NH4 , NO3 ) are released to the in nutrients. pore water. They may be transfered back into • The phytoplankton is the basis of a food chain the pelagic zone via diffusion and other trans- with zooplankton and fish as higher trophic port processes (see Sect. 4.4.2.2). levels. Due to filtration by zooplankton, bac- • Some part of the settled organic matter is lost terial and particulate organic matter (detritus) to the deeper sediment layers by bioturba- is partly included in this food chain. tion and continued deposition of fresh mate- 74 Chapter 4 Model application to the Lower Havel River

Atmosphere Fishes

Pelagic Zooplankton Gas (N only) zone

Phytoplankton dissolved inorganic Bacteria

particulate dissolved inorganic (only P)

Non-living particulate organic matter

Upper sediment particulate dissolved dissolved particulate inorganic inorganic organic organic (only P)

Deep sediment

Figure 4.18: Generalized ﬂow chart of the nutrient cycle in a non-stratiﬁed water body. It should be noted that the biological groups are not homogeneous but each of them consists of multiple species with different stoichiometry which is not necessarily constant in time. The organisms in the sediment are not explicitly considered in the chart. 4.4 Water quality submodel 75

rial (burial). This fraction forms the muddy the DIP concentration in equilibrium with sorbents sediments which can be found in many eu- and phosphate minerals is low. As discussed in trophic lakes. Sect. 4.2.3 the situation in the Havel River has shifted towards nitrogen limitation due to high ex- Major differences between N and P ternal P input and phosphorus remobilization from Since nitrogen does not form insoluble minerals, sediments. the particulate inorganic fraction can usually be ne- + glected unless the sorption of NH4 to suspended Options for simplification clay minerals is of special relevance. In the case For modeling, it is important to know which of the of phosphorus however, sorption to mineral sur- nutrient pools and interactions shown in Fig. 4.18 faces and the formation of insoluble minerals is are most relevant and which may be neglected. At important (see Sect. 4.4.2.1). This partly explains this point, some knowledge of the energy flow in why phosphorus is usually stored in sediments in food webs is helpful. After Begon et al. (1996), the greater quantities compared to nitrogen. Particu- transfer efficiency TE which relates the biomass late inorganic phosphorus may also be subject to production P of the consumers at the trophic level settling when P is sorbed to suspended sediment or n to the production of the prey at trophic level precipitation with calcite occurs. n − 1, i.e. TE = Pn/Pn−1, is typically in the In contrast to phosphorus, nitrogen may not only range 0.02–0.24. Often 0.1 is used as a standard es- be removed from the system by burial in the sed- timate. This figure indicates that only about 10% iment but also by gaseous losses in the form of of the energy is transfered from one trophic level N2 or N2O. This process is believed to be mostly to the next while 90% is lost due to respiration bound to the sediment, as oxic conditions (ni- and incomplete assimilation. In consequence, the trification) and anoxic conditions (denitrification) biomass – and thus the amount of stored nutrients – must be present in close vicinity (Scheffer, 1998). at the level of the primary producers (phytoplank- Ammonia volatilization is believed to be less sig- ton) is much higher than that at higher trophic lev- nificant in natural waters as long as the pH is not els. Exceptions may occur only if the system is far very high. from steady state, e.g. during mass developments Another interesting feature of the nitrogen cy- of zooplankton. cle is that some groups of cyanobacteria are able In many models, the nutrient pools associated to take up dissolved N2 and to reduce it to form organic N. Thus, the atmosphere is not only a ni- with consumers are consequently neglected. The trogen sink but it can also become a source. dynamics of the zooplankton is of interest only, be- Unlike phosphorus, dissolved inorganic nitro- cause it controls algal mortality. The three major gen appears in different oxidation levels (NH+, pools of nutrients in the water column are there- − − 4 fore (1) phytoplankton, (2) dead organic material NO2 , NO3 ). Dissolved inorganic phosphorus however is always present as phosphate, with the and (3) dissolved inorganic nutrients (e.g. Am- actual species depending on the pH. In chemical brose et al., 2001). While nutrient storage in mi- analysis, ortho-phosphate can hardly be isolated crobial biomass is often not explicitly modeled, it from colloidal phosphates (Lampert & Sommer, must be taken into account in systems with large 1993) and the concentration of molybdate reactive input of allochtonous organic matter such as pol- phosphorus (SRP) is commonly used as an esti- luted rivers (see e.g. Reichert et al., 2001). mate of DIP. In the sediment, the nutrients stored in organic In most freshwater systems, the nutrient which matter, dissolved nutrients in the pore water as well limits primary production is phosphorus, because as the inorganically bound fraction are all relevant. 76 Chapter 4 Model application to the Lower Havel River

Whether the burial of nutrients in the deep sedi- amount of the total phosphorus is bound to iron. ment must be taken into account depends on many For the Lower Havel River, this has been proven criteria. The error from neglecting this process will by Knösche (2006a). Two different kinds of bind- probably be small if, ing need to be distinguished: adsorption and the • the simulation time period is short compared formation of minerals (Jacobsen, 1978; Smits & to the quotient of the burial velocity (m s−1) Van der Molen, 1993). and the thickness of the considered upper sed- Most important for phosphate sorption are iron- iment layer (m), III hydroxides and oxides, e.g. Fe(OH)3 and FeOOH (goethite). The sorption capacity of amor- • the majority of the organic matter is mineral- phous, freshly precipitated Fe hydroxides is higher ized in the top sediment due to high microbial than that of aged, crystalline forms such as goethite activity, due to a greater specific surface area. In sorption • the formation of inorganic compounds which experiments, Jacobsen (1978) found a strong neg- are stable in the reduced environment of the ative correlation between the sorption capacity of deep sediment is less important. iron hydroxides and the pH. This was explained In principle, the burial velocity can be assessed by the positive charge of hydroxide surfaces at low + + from data on the rate of sediment net growth but pH ([Fe]-OH + H → [Fe]-OH2 ) which favors it is difficult to measure under natural conditions. the sorption of the phosphate anion. At low redox potentials, iron-III is reduced and 4.4.2 Phosphorus storage in sediments the hydroxides dissolve, e.g. according to Eq. 4.2. The formerly adsorbed phosphate is released into and remobilization the pore water. Because the release of phosphorus from the bottom sediments of the Havel Lakes is of great sig- − − F e(OH) + e ↽⇀ F e2+ + 3OH (4.2) nificance for its present and future water quality, 3 the theory of storage and remobilization is pre- After Jacobsen (1978) the released Fe2+ may presented in more detail in the following paragraphs. cipitate again, e.g. as siderite (FeCO3) or hydrox- The topic is rather complex because multiple frac- ysiderite after reaction with bicarbonate, as pyrite tions of sedimentary P exist, each exhibiting differ- (FeS2) if sufficient sulfide is present, or vivian- ent stability with respect to the ambient conditions ite (Fe3(PO4)2). The formation of vivianite might such as pH, redox-potential, and temperature. In explain why the phosphorus binding capacity of Sect. 4.4.2.1 only those P fractions are addressed iron-rich but reduced sediments is still significant which are known to be of relevance in the sedi- (e.g. Kozerski, 1977; Schellenberger et al., 1983). ments of the Havel Lakes (see Schettler, 1995). Vivianite was shown to be present in the Havel The situation is further complicated by the oc- sediments (Schettler, 1995) and in their SWITCH currence of multiple mechanisms which govern the model Smits & Van der Molen (1993) attributed transport of sedimentary phosphorus back to the the phosphorus binding in deeper (reduced) sedi- water column. The mechanisms of P remobiliza- ment layers to Fe3(PO4)2 precipitation. The min- tion are covered by Sect. 4.4.2.2. eral strengite (FePO4) is presumably irrelevant as it is only stable at high levels of Fe3+ which corre- 4.4.2.1 Phosphorus fractions spond to oxic conditions and low pH. The following hypotheses summarize the Iron-bound phosphorus present understanding of how phosphorus is bound In many freshwater sediments, a considerable to iron in sediments and how it is released. 4.4 Water quality submodel 77

• Under oxic conditions, Fe-III hydroxides pro- Fe-III hydroxides should increase due to the vide a very high phosphate sorption capacity. pH effect. The sorption capacity decreases with increas- • The deeper layers of the sediment are perma- ing pH and the mineral age. In sediments, nently reduced. Information on the binding oxic conditions can be found in the uppermost of phosphorus to iron in this zone is scarce. layer due to the penetration of oxygen and ni- The formation of vivianite provides a possible trate from the water column. explanation for correlations between the iron • The thickness of the oxidized surface layer and phosphorus content even in the deeper is subject to seasonal variation. In winter, sediment. when microbial activity is low and oxygen solubility is high, the critical depth, where the The above discussion revealed how complex the consumption of electron acceptors equals the binding of phosphorus to iron in natural sediments supply from the water column, moves down- is. All of the relevant P-Fe compounds are sen- wards. Thus, the oxidized layer with high P sitive to the microbial activity via pH and redox sorption capacity increases to a thickness of a potential. This results in a considerable temporal few millimeters to centimeters. variation in the P-binding capacity, especially in the uppermost sediment. Simple sorption models, • In eutrophic lakes, the oxidized layer may e.g. built on the Langmuir isotherm, are therefore vanish altogether during summer. After the inappropriate. Furthermore, one must not over- electron acceptors O , NO , and manganese- 2 3 look that concurrent reactions such as the precipi- IV (see Stumm & Morgan, 1996, for the se- tation of calcium phosphate may interfere with the quence of consumption) become depleted, the iron related mechanisms (see Golterman (1988) iron-III hydroxides are reduced and Fe2+ to- and Golterman (1995a) for a revised view). gether with the formerly sorbed phosphate is released. Some part of the Fe2+ may pre- Calcium-bound phosphorus cipitate in different minerals. If vivianite is Stumm & Morgan (1996) list the follow- formed, a fraction of the released phosphate ing minerals composed of calcium and phos- may be rebound. After Kozerski (1977) the phate: CaHPO4, Ca4H(PO4)3, Ca10(PO4)6(OH)2, reduction is a rather slow process. It is there- Ca10(PO4)6F2, CaHAl(PO4)2 but further forms fore hypothesized that sudden peaks in the such as Ca3(PO4)2, Ca(H2PO4)2 are known to ex- 3− sediment P release originate from fast miner- ist. Furthermore, the chemosorption of PO4 to alization of fresh organic matter under anoxic surfaces of calcite (CaCO3) is possible, but Jacob- or near-anoxic conditions rather than from the sen (1978) has shown that the formation of hydrox- dissolution of P-sorbents. yapatite is quantitatively more important. 8 • The pH, which modiﬁes the sorption capacity, In calcium rich natural waters , hydroxyap- is subject to seasonal and spatial variation as atite (Ca10(PO4)6(OH)2) is the predominant form well. Whereas high values (up to 9 and above) because it is less soluble than other calcium may occur in the water column of eutrophic phosphates. The existence of apatite minerals lakes during the day, the pH decreases in the in sediments of the Lower Havel River (Lake Breitlingsee) was proven by x-ray spectrometry sediment due to CO2 production. Thus, mineralization causes two opposite effects: On (Schettler, 1995). The formation and dissolution the one hand, the P sorption capacity is re- of hydroxyapatite is often written as duced due to iron reduction. On the other 8Commonly called hard waters because Ca2+is often a hand, the sorption capacity of the remaining major species contributing to hardness. 78 Chapter 4 Model application to the Lower Havel River

temperature dependence of both constants are presented in Golterman & Meyer (1985). Eq. 4.3, ⇀ 2+ 3− − Ca5(PO4)3(OH) ↽ 5Ca +3PO4 + OH Eq. 4.4, and Eq. 4.5 can now be combined to yield the DIP concentration in equilibrium with hydrox- where all stoichiometric factors have been di- yapatite at a given pH and Ca2+ concentration. If vided by two. The solubility product KL of DIP is defined as −50 −60 Ca5(PO4)3(OH) is in the range 10 to 10 . Golterman (1995b) suggests to use pKL=50, since − 3− 2− − 3 [DIP] =[PO ] + [HPO ]+[H2PO ] it best explains the PO4 concentrations found in 4 4 4 the hardwater rivers Rhine and Rhone (Golterman one arrives after some algebra at Eq. 4.6. & Meyer, 1985). Using the known pKL, the concentration of 3− PO in equilibrium with hydroxyapatite can be 3 KL 4 [DIP] = − · (4.6) computed according to Eq. 4.3 9. To do so, the con- s[Ca2+]5 · [OH ] centration of Ca2+ as well as OH− must be given. [H+] [H+]2 1+ + Kd 2− Kd 2− · Kd − Ã HPO4 HPO4 H2PO4 ! 2+ 5 3− 3 − [Ca ] · [PO4 ] · [OH ] − + KL = (4.3) The concentration of [OH ] and [H ] in Eq. 4.6 [Ca5(PO4)3OH] can be substituted by the pH according to − − Eq. 4.3 predicts the equilibrium concentration of (pH pKd H2O) − [OH ] =10 PO3 only. At pH values normally occurring in 4 [H+]= − log (pH) surface waters, the dissociation equilibrium of the 10 − 2− species H2PO4 and HPO4 must be taken into with pKd H2O being the dissociation constant account if the total concentration of dissolved in- or ’ionic product’ of water (pKd H2O≈ 14; see organic P (DIP) is to be computed. As indicated Golterman & Meyer (1985)). by Eq. 4.4 and Eq. 4.5, these equilibria are pH- The solution of Eq. 4.6 for a range of com- dependent as well. mon pH and Ca2+ concentrations is presented in Fig. 4.19. As the figure shows, the concentration of DIP in equilibrium with hydroxyapatite is very 2− + [HPO4 ] · [H ] sensitive to changes in the pH. A change in the pH Kd − = − (4.4) H2PO4 + [H2PO4 ] by one degree (∆[H ]= 10), changes the DIP con- 3− + centration by up to two magnitudes. [PO4 ] · [H ] Kd 2− = − (4.5) HPO4 [HPO2 ] Due to the strong influence of the pH, a consid- 4 erable seasonal variation of DIP in the sediment The second and third dissociation constants of pore water must be expected. This is because the phosphoric acid are approximately − = sediment pH is affected by the activity of microor- pKd H2PO ◦ 4 − ganisms which release acidifying substances such 7.2 and pKd HPO2 = 12.3 at 25 C(Stumm & Mor- 4 + gan, 1996; Hellmann, 1999). Polynomials for the as CO2 and NH4 into the pore water. The biological activity increases with temperature and the sup- 9 The brackets around the species symbols stand for the re- ply of freshly settling organic matter10. Thus, the spective activities but in solutions of low ionic strength (low total ion concentration), activities and concentrations are vir- 10The dependence of the pH on freshly settled degradable tually equal. Note that molal concentrations (mol kg−1 of organic matter is corroborated by an observed negative corre- H2O) are used here. Also note that the activity of the solid lation between pore water pH and chlorophyll-a in the Havel phase in the denominator of Eq. 4.3 equals unity. sediments (data from R. Knösche). 4.4 Water quality submodel 79

9.0 DIP (mg l-1) 20 DIP obs. 8.5 0.01 DIP mod.

) 15 -1 8.0 0.1 7.5 10 pH 1

7.0 DIP l (mg 5 10 6.5 100 0

6.0 6.8 6.9 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 20 40 60 80 100 120 140 160 pH Ca2+ (mg l-1) Figure 4.20: pH and DIP concentration in the pore wa- Figure 4.19: Total concentration of dissolved inorganic ter of 19 sediment samples (0–6 cm) from the Pots- phosphorus (DIP) in equilibrium with hydroxyapatite as damer Havel collected in November 2005 (R. Knösche). a function of pH and dissolved calcium. The plot was The solid line shows the theoretical DIP concentration drawn with an apatite solubility constant of pKL=50 in equilibrium with hydroxyapatite as a function of the ◦ and a temperature of 20 C. Note that concentrations are pH at a Ca2+ concentration of 90 mg l−1 and a tem- −1 given in mg l , not as molalities as in Eq. 4.6. perature of 10 ◦C. A good model fit would require the Ca2+ concentration to be as low as 18 mg l−1. dissolution of hydroxyapatite at lower pH in summer provides another potential explanation for the Organic phosphorus observed sediment P release in the Havel Lakes. The organic matter (OM) in sediments is spread The application of the apatite model (Eq. 4.6) over different fractions. The largest part consists to 19 sediment samples from the Potsdamer Havel of particulate OM which originates from the set- collected in November 2005 by R. Knösche is pre- tling of organisms and detritus. Some degradable sented in Fig. 4.20. Sediment pH values in the part of the OM is taken up by consumers and het- range 6.9–7.7 (x= 7.45, SD= 0.16, n= 19) were erotrophic bacteria. In the latter case, the particu- observed in the top 6 cm. The corresponding cal- late OM must first be hydrolyzed to dissolved OM cium concentrations were not measured. However, with the help of enzymes. Another smaller part of in the Potsdamer Havel, Ca2+ levels of about 90 the sedimentary OM is stored in the biomass of au- mg l−1 are found with only little seasonal varia- totrophic organisms such as nitrifying bacteria or tion and it seems reasonable to assume a similar autrotrophic denitrifiers. value for the pore water of the upper sediment. The recycling of organic carbon to CO2 (oxic According to Fig. 4.20, the DIP concentration conditions) or CO2 and reduced compounds such in the pore water of the sediments is apparently as methane (anoxic conditions) is the result of the (at least not permanently) in equilibrium with hy- organisms’ respiration. Respiration and hydrolysis droxyapatite. Because the overall existence of of OM as well as excretion finally result in the re- Ca5(PO4)3OH was proven by Schettler (1995), it lease of nutrients according to the stoichiometry of seems likely that its formation is a rather slow the organic matter. process which is unable to buffer massive release To illustrate how the concentration of organic of P from mineralization. The simulation re- phosphorus (and nitrogen) in the sediment is regu- sults of Hoffmann (1999) indicate that the apatite- lated, a simple mass balance equation can be writ- equilibrium can explain the observed P concentra- ten. If the organic P concentration in the sediment −3 tions during the winter period. (g P m sediment) is denoted Porg,sed and Porg,pel 80 Chapter 4 Model application to the Lower Havel River

(g P m−3) is the corresponding concentration in the over the different fractions. A number of sequen- pelagic zone, one arrives at Eq. 4.7. tial extraction procedures are discussed in the literature (Hieltjes & Lijklema, 1980; Psenner et al., 1984). Unfortunately, these procedures are costly dPorg,sed u · Vw = ·Porg,pel −k·Porg,sed (4.7) and their applicability to a large number of sam- dt z · V w s ples is therefore limited. Less complicated extrac- The first term describes the input to the pool of sed- tion schemes distinguishing acid and base solu- − imentary organic P by settling, where u (m s 1) ble phosphorus only have been developed therefore is the settling velocity, zw (m) is the depth of the (Schettler, 1995). The drawback of simplification water column, and Vw/Vs is the ratio of the vol- is a loss of information. For example, treatment ume of the water column and the considered sed- with 0.24 N H2SO4 is believed to extract all inor- iment layer, respectively. The second term de- ganically bound P from a mud sample without dis- scribes the conversion of sedimentary organic P criminating the calcium and the iron bound frac- to dissolved inorganic P by microbial degradation tion (Keizer & Sinke, 1992; Knösche, 2006b). with k (s−1) being a temperature dependent rate constant. Solving Eq. 4.7 for steady state and as- 4.4.2.2 Remobilization processes suming Vw/Vs ≈ zw/zs with zs being the thickness of the sediment layer yields Eq. 4.8. Diffusion If phosphorus is released from one of the above u mentioned fractions due to desorption, dissolution, Porg,sed∗ = · Porg,pel (4.8) k · zs excretion, or hydrolysis, an increase in the phosphate concentration of the pore water occurs. If Thus, the steady state concentration of sedimen- the released phosphate is not sorbed, precipitated, tary organic P (P ∗) increases linearly with org,sed or taken up by organisms again, it is subject to dif- the concentration of organic P in the pelagic zone fusive transport. The specific vertical flux rate J and the settling velocity u. An inverse dependence − − (g m 2 s 1) is related to the spatial concentration on the decay rate k and the thickness of the (mixed) − − gradient (g m 3 m 1) by the sediment bulk diffu- sediment layer is predicted. − sion coefficient D (m2 s 1) according to Fick’s It should be kept in mind that the sediment con- s first law (Eq. 4.9). The reduction in diffusion area ditions turn from oxic to anoxic at high concen- by the sediment particles is taken into account by trations of degradable organic matter (including the dimensionless porosity φ. Porg,sed). If such a change in the redox milieu occurs, the value of k will change too. Hence, Eq. 4.7 dC is actually a nonlinear equation because the value J = −φ · D · (4.9) s dx of k is not independent from Porg,sed. Furthermore, the use of a single rate constant k The bulk diffusion coefficient Ds in Eq. 4.9 is re- does not take into account the heterogeneity of or- lated to the more fundamental molecular diffusion ganic matter. Differences in the degradability of coefficient in pure water Dm by the turtuosity θ the OM fractions are better reflected by a multi- and the constrictivity δ (Eq. 4.10). rate approach (e.g. Carignan & Lean, 1991) which, however, requires more parameters to be known. δ Ds = Dm · (4.10) Identification of the P fractions θ2 In practice, it is difficult to determine how the total The turtuosity is the ratio of the true travel length amount of sedimentary phosphorus is distributed between two locations (around particles) and the 4.4 Water quality submodel 81

straight distance between the locations. According 0.05 Pelagic DIP 2 -1 to Maerki et al. (2004), the squared turtuosity θ ) 0.04 0.0 mg l -1 mg l -1 can be estimated from the porosity φ by Eq. 4.11. d 1.0

-2 0.03 0.02 θ2 = F · φ (4.11) 0.01

P flux (g m 0.00 The factor F is the ’formation factor’ and repre- -0.01 sents the ratio of the bulk electric resistivity and 0 5 10 15 20 the resistivity in the pore water alone. Often, F is DIP in pore water (mg l -1 ) simply estimated from the porosity by F = a·φ−m where a is a constant close to one and m is a value Figure 4.21: Calculated phosphorus flux rates at the − − in the approximate range 1.2–3. For sandy fresh- sediment water interface (g m 2 d 1) according to water sediments Maerki et al. (2004) fitted the em- Eq. 4.12. The molecular diffusion coefficient of phos- −10 2 −1 −1.21 phate was set to 7.17·10 m s which is the average pirical relation F = 1.04 · φ . 2− of the individual coefficients for the species HPO4 Combining Eq. 4.10 and Eq. 4.11 and inserting −10 2 −1 ◦ − −10 (7.34·10 m s at 25 C) and H2PO4 (8.46·10 into Eq. 4.9 finally yields Eq. 4.12 for calculating m2 s−1 at 25 ◦C) after transformation to 20◦C. The mix- the diffusive flux rate in the sediment. ing length dx was set to 0.01 m, the porosity φ was assigned a value of 0.85, δ = 1, and F = 1.04 · φ−1.21. δ dC J = −D · · (4.12) m F dx it can be used for calculating the vertical phospho- The diffusion coefficient in water Dm is specific rus flux in a sediment column. Eq. 4.9 can further- for each molecule and the influence of temper- more be used for estimating the flux through the ature can be described by the Einstein equation sediment-water interface (Ramm & Scheps, 1997; (Eq. 4.13). Lavery et al., 2001). In the latter case, the gradient can either be computed from measured pore water concentrations at two different depths near below Dm(T1) Va(T1) Dm(T2) Va(T2) = (4.13) the sediment surface or one can directly use the T1 + 237 T2 + 237 difference between the pelagic and the pore water ◦ In this relation, T ( C) is the temperature and concentration at a certain depth. −1 −1 11 Va(T ) (kg m s ) is the dynamic viscosity . Fig. 4.21 shows calculated phosphorus flux rates Molecular diffusion coefficients for a number of at the sediment-water interface for pore water con- ions can be found in Li & Gregory (1974) or Carig- centrations between 0 and 20 mg l−1 DIP and nan & Lean (1991). pelagic DIP levels of 0 and 1 mg l−1. One must If Eq. 4.9 is combined with the one-dimensional not forget that the computed flux rates account for mass balance equation to yield Fick’s second law, molecular diffusion only. In natural sediments con-

11 curring processes occur which may be – at least The dynamic or ’absolute’ viscosity Va(T ) is related to 2 −1 the kinematic viscosity Vk (m s ) by Va(T ) = Vk · ρ(T ) temporarily – much more effective (see following − with ρ being the density of water (kg m 3). For temper- paragraphs). Also, porosities are highly variable in ◦ atures T in the range 0–30 C the density is approximately both space and time. ρ = 1000 − 0.00653 · (T − 3.98)2. The kinematic viscosity 2 −1 −6 In models which approximate the upper sed- Vk (m s ) can be approximated by Vk = 1.78 · 10 /(1 + − ◦ 0.0337 · T + 0.221 · 10 3 · T 2) with T in C after Preissler iment as a single layer (e.g. Hoffmann, 1999), & Bollrich (1985). the concentration gradient dC/dx for sediment- 82 Chapter 4 Model application to the Lower Havel River

stirring up the sediment on their search for benthic prey. The term bio-irrigation refers to the activ- Cpel Vw ity of invertebrates (e.g. chironomides) who pump z w oxygen-rich water into the sediment through bio- A J genic macropores. These mechanisms of random z mixing in the upper sediment layer can be modeled s Cpw Vs φ in the same way as molecular diffusion but values of the related exchange coefficients are hard to es- Figure 4.22: Schematic representation of a water- timate. It is reasonable to assume a dependence sediment system. C : concentration in the water col- pel on those variables which control benthic bioactiv- umn, Cpw: corresponding concentration in the sediment ity, e.g. temperature, redox conditions, settling pore water, zw,zs: depth of water and sediment, Vw,Vs: volume of the water column and total volume of the sed- rate of organic matter, occurrence of macrophytes, iment layer, A: interface area, φ: sediment porosity. etc.. According to Mermillod-Blondin et al. (2003) bioactivity also modifies the sediment’s porosity. In the SWITCH model (Smits & Van der Molen, water flux calculations is approximated by (C − pw 1993) bioturbation was assumed to be a very im- C )/(0.5 · z ). Here, z is the thickness of the pel s s portant process, causing the transport of vivian- considered layer, C is the average pore water pw ite from deeper, reduced layers to the upper sedi- concentration, and C is the concentration in the pel ment where the mineral dissolves and the formerly water column (see Fig. 4.22). bound P is released. Using Eq. 4.12, the change in the water col- The deeper sediment zones of eutrophic waters umn’s concentration may be expressed by Eq. 4.14 are usually anoxic, since the consumption of elec- and the change in the pore water concentration fol- tron acceptors by the decay of organic matter ex- lows Eq. 4.15. ceeds the rate of supply from the water column. Under these conditions, methane is produced from d incomplete organic carbon oxidation or the use of Cw =J · A/Vw dt CO2 as electron acceptor. In times of high micro- =J/zw (4.14) bial activity, methane bubbles are formed. When 2 · Dm · δ bubbles move toward the sediment surface driven = · (Cpw − Cpel) by buoyancy, a directed mixing of particles and zs · zw · F pore water takes place. Resuspension of sediment by wind-induced d Cs = − J · A/(Vs · φ) shear stress contributes significantly to nutrient re- dt mobilization in shallow water bodies (Hellstrøm, = − J/(zs · φ) 1991; James et al., 1997; Welch & Cooke, 2005). 2 · Dm · δ Again, several control factors are involved such as = − 2 · (Cpw − Cpel) zs · φ · F wind speed and direction, the lake geometry, as (4.15) well as the sediment composition, compaction, and its spatial distribution. Sediment mixing Compared to molecular diffusion (Eq. 4.9 & Bioturbation refers to the mixing of the sediment 4.12), the effects of bioturbation, bubbling, and matrix by organisms (Lampert & Sommer, 1993). resuspension are much harder to predict. This is These might be organisms living in the sediment a problem, because the apparent dispersion coef- and feeding on detritus and bacteria, but also fish, ficients (m2 s−1) related to these processes may 4.4 Water quality submodel 83 be large compared to molecular diffusion coeffi- effect of different processes and also compensate cients. In present models, remobilization of sedi- for unknown mechanisms. They can only be de- mentary substances by turbation and resuspension termined by calibration and it is therefore hard to is either neglected or a lumped dispersion coeffi- transfer parameter values from one study site to an- cient is used that covers all mechanisms, including other. diffusion (e.g. Smits & Van der Molen, 1993). In order to identify the key processes and param- Further difficulties for modeling sediment phos- eters of a process-oriented model of nutrient cy- phorus release arise from the fact that the environ- cling in the sediments of the Havel Lakes, a mon- mental condition in the sediment and the pelagic itoring of physical and chemical parameters over zone are different with respect to the pH and the many years with high spatial and temporal resolu- redox potential. For example, dissolved phosphate tion would be required. Since sediment samples which is brought to the sediment surface may pre- are – compared to surface water – hard to col- cipitate there as calcium phosphate due to the in- lect and more difficult to analyze without disturb- crease in pH from deeper sediments toward the ing the natural conditions (e.g. anoxia), this is not surface. Also, if reduced iron (Fe2+) and phos- practically feasible without significant investment phate are transported to the sediment surface, in- of time and money. At present, the only available stantaneous refixation by sorption or precipitation data on sediment characteristics and nutrient con- may occur when iron is oxidized in contact with centrations are those collected by Knösche (2006a) the oxygenated pelagic water. and those published by Schettler (1995). Finally, the catchment models SWIM and ArcEGMO-Urban, to which TRAM must be cou- 4.4.3 Representation of nutrient retention pled for scenario simulations, compute the emis- and remobilization in TRAM sions of total N and P only. Thus, the catchment 4.4.3.1 Required degree of simplification models cannot provide the required boundary conditions for a complex process model, such as frac- From the discussion in Sect. 4.4.1 and 4.4.2, sev- tions of organic P and N, or species of dissolved eral conclusions for modeling of the Lower Havel inorganic N. River can be drawn: Consequently, a nutrient turnover model for the Many different mechanisms are responsible for Lower Havel River must primarily focus on a the storage of nutrients in sediments and their re- closed description of the nitrogen and phosphorus mobilization. This is especially true for phospho- mass balance. The bottom sediment and the dy- rus, which – in contrast to nitrogen – forms insolu- namics of nutrient exchange with the pelagic zone ble inorganic compounds (minerals). Transforma- need to be approximated by a strongly simplified tion, binding and remobilization processes are all approach. mediated or indirectly affected by microbial activity. How poorly the true mechanisms behind phos- 4.4.3.2 Nitrogen retention phorus release, after several decades of research, are actually understood was shown by Golterman According to Jensen et al. (1992b) and Seitzinger (1995a). (1988), permanent losses of nitrogen may result Because existing models describe the sediment from two different processes: denitrification and diagenesis in a greatly simplified, conceptual man- burial of organic N in deeper sediment layers. This ner (Van der Molen, 1991; Smits & Van der Molen, is also reflected by Fig. 4.18. Jensen et al. (1992b) 1993; Hoffmann, 1999), they use lumped param- found denitrification in lakes to be closer related eters. Hence, the parameter values integrate the to the pelagic concentration of total nitrogen than 84 Chapter 4 Model application to the Lower Havel River

d PON =+ k · PON dt grw Growth − k · PON dw1 (4.17) DIN DON PON Water − kdw2 · PON column u − nset · PON

Denitrification Degradation zw

Upper

Settling d Exchange Degradation sediment DON =+ k · PON dt dw1 (4.18) SDIN SDON SPON − kdw3 · DON Deep Burial sediment d 2 · D · φ DIN =+ · (SDIN − DIN) Figure 4.23: Simplified nitrogen cycle with the two dt zs · zw sinks, denitrification and burial. The considered nitro- + kdw2 · PON gen fractions in the water column are: PON (particulate + kdw3 · DON organic N), DON (dissolved organic N) and DIN (dissolved inorganic N). The corresponding sediment frac- − kgrw · PON tions carry the initial letter ’S’. (4.19) The symbols in Eq. 4.16–Eq. 4.19 have the following meaning: kgrw: growth rate of organisms − −1 to the pelagic NO3 concentration. This indicates taking up DIN (s ), kdw1: PON to DON con- −1 that denitrification is in fact mainly bound to the version rate (s ), kdw2: PON to DIN conversion −1 sediment (see Sect. 4.4.1) since it is the particulate rate (s ), kdw3: DON to DIN conversion rate −1 −1 fraction of TN which is responsible for the transfer (s ), unset: settling velocity of PON (m s ), of N from the pelagic to the benthic compartment. zw: depth of the water column (m), zs: depth of the considered, well mixed sediment layer (m), Fig. 4.23 shows a simplified chart of the N cycle D: bulk diffusion coefficient for dissolved inor- that distinguishes three fractions of N in the water ganic N in the sediment pore water (SDIN) given column and the sediment. The total nitrogen in the in (m2 s−1), φ: sediment porosity (–). The concen- water column (TN) is the combination of particu- trations of all fractions are expressed in the stan- late organic N (PON), dissolved organic N (DON) dard unit (g m−3). Note that Eq. 4.19 contains and dissolved inorganic N (DIN). The actual dis- − − the implicit assumption that the volume of the wa- tribution of DIN over the species NO , NO , and 3 2 ter body V divided by the volume of the consid- NH+ is neglected. In accordance with Fig. 4.23, w 4 ered sediment layer V can be approximated by the dynamics of these N fractions can be expressed s V /V = z /(z · φ), i.e. the areal extent of the by Eq. 4.16–Eq. 4.19 if the exchange of mass via w s w s water body and the sediment are assumed to be inflow and outflow is left unconsidered (closed sys- equal. tem). Eq. 4.16–Eq. 4.19 can be summarized to yield the TN balance (Eq. 4.20).

d 2 · D · φ d d d d TN =+ · (SDIN − DIN) TN = PON + DON + DIN dt zs · zw (4.20) dt dt dt dt u (4.16) − nset · PON zw 4.4 Water quality submodel 85

Obviously, the DON fraction no longer appears in SDIN pool due to denitrification (s−1). Note that the TN balance (Eq. 4.20), because the conversion some of the constants cancel out but the possible of PON to DON and the decay of DON to DIN is simplifications have not been adopted for the sake just a redistribution of nitrogen in the water col- of transparency. As in Eq. 4.19, the assumption umn’s total N pool. Only the diffusive exchange Vw/Vs = zw/(zs · φ) was used in the derivation of dissolved inorganic N with the sediment (first of Eq. 4.23. Furthermore, for Eq. 4.19 to be valid, − term in Eq. 4.20) and the settling of the particulate some part of SDIN must be present as NO3 be- fraction (second term) control the dynamics of TN. cause this is a precondition for denitrification. Similar to Eq. 4.17–4.19, three differential equa- If the sediment is in steady state, i.e. no accu- tions can be written for the N fractions in the sedi- mulation of nitrogen occurs and no decay of a for- ment (Eq. 4.21–Eq. 4.23). While the concentration merly accumulated nitrogen excess takes place, the of the sedimentary particulate organic N (SPON) left hand sides of Eq. 4.21, Eq. 4.22, and Eq. 4.23 −3 has the unit mass per sediment volume (g m ), can be set to zero. The steady state equations ob- the dissolved fractions in the sediment pore water tained can, after some rearrangement, be inserted (SDON, SDIN) are given in mass per volume of into the TN balance equation (Eq. 4.20) to yield −3 pore water (g m too). Eq. 4.24.

d u z SPON =+ nset · w · PON dt zw zs d − kds1 · SPON TN = (4.21) dt − kds2 · SPON zs unbur unbur · φ · kdenit · SDIN − · SPON − · SPON zw zs zs µ ¶ (4.24) d k SDON =+ ds1 · SPON dt φ (4.22) This equation expresses that, as long as the con- − kds3 · SDON centrations of the sedimentary N fractions are in steady state, the change in the water column’s TN d k mass must balance the N losses from the system SDIN =+ ds2 · SPON dt φ caused by burial and denitriﬁcation. This becomes clear when both sides of Eq. 4.24 are multiplied by + kds3 · SDON 2 2 · D · φ zw · A with A (m ) being the areal extent of the − · (SDIN − DIN) sediment layer and the water body. zs · zs · φ In a next step, the values of SDIN and SPON − kdenit · SDIN (4.23) in Eq. 4.24 can be substituted by steady state solutions of Eq. 4.23 and Eq. 4.21, also making use of Most symbols used in Eq. 4.21–Eq. 4.23 were Eq. 4.22. If this is done (not presented here), the already declared above. The new symbols are: right hand side of Eq. 4.24 turns out to be a linear −1 kds1: SPON to SDON conversion rate (s ), kds2: function of PON and DIN only. Under the assump- −1 SPON to SDIN conversion rate (s ), kds3: SDON tion that that PON and DIN are constant fractions −1 to SDIN conversion rate (s ), unbur: burial veloc- of TN, i.e. PON = α1 · TN, DIN = α2 · TN, −1 ity of SPON (m s ), kdenit: rate of N loss from the DON = α3 · TN, αi = 1, Eq. 4.24 simpliﬁes P 86 Chapter 4 Model application to the Lower Havel River

−1 to Eq. 4.25, with the single parameter kTN (s ) 1.00 1.06 Value of ΘTN integrating all remaining constants. 1.04 1.10 20 1.5 d 15 TN = −k · TN (4.25) 1.0 dt TN 10 0.5 5 The result of the above calculations can be summa- Temperature (°C) 0 0.0 rized as follows: Given a constant distribution of Temp. correction term (-) Jul Jan Mar Nov Sep TN over the fractions PON, DON, and DIN as well May as steady state conditions, the total nitrogen loss in Figure 4.24: Seasonal variation of the average water the water column (dTN/dt) can be expressed as a temperature at Ketzin (bars) and corresponding values (T −20) linear function of TN. That is, the effects of both of the temperature correction term θT N (lines) at burial and denitrification losses can be merged into different values of θT N . a single rate constant as in Eq. 4.25. The value of k is likely to be dominated by the effect of den- TN where T is the water temperature and θ is an itrification. TN empirical coefficient. It must be noted that the approach described is built completely on linear differential equations, i.e. dependencies of the rate constants on the con- d − TN = −k · θ (T 20) · TN (4.26) centrations were neglected. The linear approach of dt TN TN Eq. 4.25 will certainly be invalid if the TN con- The influence of the temperature correction term centration varies over a very wide range. If for ex- − θ (T 20) on the rate of nitrogen retention is il- ample, the sediment shifts from fully oxic to com- TN lustrated by Fig. 4.24. pletely anoxic conditions due to very high settling of PON or vice versa, this would not only alter mineralization constants but also effect nitrifica- 4.4.3.3 Phosphorus retention and release tion and thus denitrification. A simplified scheme of the aquatic phosphorus cy- The advantage of the first order equation 4.25 cle is presented in Fig. 4.25. The major contrast lies in the fact that it describes the retention of ni- to the N cycle (Fig. 4.23) is the appearance of the trogen in terms of TN only and no information on particulate inorganic species PIP and SPIP. the quantity of the different N fractions or DIN The discussion of nutrient cycles in Sect. 4.4.1 species is required. This simplifies the targeted and phosphorus storage in Sect. 4.4.2 revealed how coupling of TRAM to catchment models, which complex the processes of P retention and remobi- can only deliver TN loads as boundary conditions. lization actually are. Because data on sediment The drawback of simplicity, however, lies in the qualities and their seasonal variability are scarce, fact that calibrated values of the rate constant kTN the processes and parameters of a detailed sedi- are related to a certain constitution of the TN pool. ment model cannot be identified. Thus, similar to For example, kTN is expected to show a seasonal the description of nitrogen retention, a greatly sim- variation not only because of the temperature de- plified, conceptual modeling approach was chosen pendence of denitrification but also due to changes with two premises in mind: in the PON/TN ratio which controls nitrogen settling. In the application presented, the seasonal • The model must reflect the apparent decay variability in N retention was accounted for by ex- of the phosphorus excess which has accumu- tending Eq. 4.25 with an Arrhenius term (Eq. 4.26) lated in the Havel sediments in past decades. 4.4 Water quality submodel 87

Growth ) -1 0.2 PIP DIP DOP POP Water column Degradation 0.1

Upper Settling Settling Exchange Degradation sediment

Particulate P l (mg 0.0 SPIP SDIP SDOP SPOP Jul Jan Mar Nov Sep May Deep Burial sediment Figure 4.26: Seasonality of the concentration of partic- Figure 4.25: Simplified phosphorus cycle with burial in ulate phosphorus based on pooled data from the Pots- the deep sediment as a sink. The phosphorus fractions damer Havel Lakes (1991–2002). The box shows the in the water column are: POP (particulate organic P), interquartile range with the median as horizontal line DOP (dissolved organic P), DIP (dissolved inorganic and the 10–90 percentile range as whiskers. P) and PIP (particulate inorganic P). The corresponding sediment fractions carry the initial letter ’S’. requires the particulate fraction of total phosphorus and its effective settling velocity upset to be • The model must take into account the pro- known. As illustrated in Fig. 4.25, there are partic- nounced seasonal variability in P remobiliza- ulate forms of both organic P (plankton, detritus) tion and retention in order to yield realistic P and inorganic P 12. In the model presented, these concentrations during the growth period of al- two fractions are not distinguished but the lumped gae. variable PP (particulate P, see Table 4.6) is used, for which measured data are available. Table 4.6 presents the phosphorus model using the compact matrix notation introduced in Sect. 3.3.2. In the model, PP is a boundary condition vari- Only two components are simulated: total phos- able, not a simulated component as in more com- phorus in the water column (TP) and the total phos- plex ecological models. Average values of PP are phorus in a finite layer of the upper sediment (PS). computed from the day of the year using a non- Because PS integrates the particulate and the dis- linear regression model fitted to the data shown in solved fraction, it is defined in g P per sediment Fig. 4.26. volume. The fact that the values of PP are predefined, not The three processes considered are the settling simulated, must be kept in mind when the model of particulate P, the immobilization of sediment P is driven by altered boundary conditions (scenario (burial) and the return of sedimentary P into the simulations). The use of PP as a boundary con- water column (remobilization). These three pro- dition is appropriate as long as nutrient concen- cesses also form the basis of the ’tier 5’ phospho- trations change only slightly and do not result in rus retention model developed by Kronvang et al. limitation of plankton growth. In the case of heav- (2005). Details of the process description are dis- ily altered external loading, however, there might cussed in the following paragraphs while the esti- be a significant drop in phytoplankton abundance. mation of parameters and initial values is covered Then a much more complex nutrient-algae model by Sect. 4.4.4. has to be used which includes PP as a simulated component. As a makeshift, the boundary condi- Settling of phosphorus Settling of particulate matter is the dominant pro- 12The settling of organic and inorganic P is often linked cess of phosphorus transfer from the pelagic zone because organic particles are involved in the process of calcite to the sediment. Its representation in the model precipitation. 88 Chapter 4 Model application to the Lower Havel River

Table 4.6: Process matrix of the phosphorus model for the Havel Lakes (see Sect. 3.3.2 for details on this kind of notation). TP: total phosphorus in the water column (g m−3), PP: particulate phosphorus (g m−3), PS: total phos- −3 −3 phorus in the upper sediment layer (g m sediment), P0: P-binding capacity of the sediment (g m sediment), −1 −1 upset: settling velocity of particulate P (m s ), upbur: velocity of sediment growth (m s ), zw: water depth (m), −2 −1 zs: thickness of the active sediment layer (m), kprem: ﬂux constant (g P m sediment surface s (g P excess · m−3 sediment)−1), f(T ): a nonlinear function of temperature T (–).

Process Components Process rate TP PS

Settling −1 zw/zs upset/zw · P P Burial 0 −1 upbur/zs · P S kprem · f(T ) · (P S − P0) if PS>P0 Remobilization 1/zw −1/zs (0 if P S ≤ P0 tion variable PP is corrected for the effects of nu- ticulate organic N (PON) from the simulated TN trient limitation as described below. concentration is PON ≈ 0.5 · TN. It is implicitly First, the concentration of PP is restricted to 4/5 assumed that PP is mostly particulate organic P. of the simulated total P concentration (TP) if the The pragmatic correction of the average con- difference T P − P P is smaller than 0.02 mg l−1. centration of particulate P (P P ) for phosphorus The threshold used for the occurrence of P limita- and nitrogen limitation is summarized in Eq. 4.27– tion is based on Sas (1989), who found that phos- Eq. 4.28. Here, P PcP is the concentration of phorus is likely to limit algal growth at DIP con- P P corrected for P shortage and the ﬁnal estimate −1 centrations below 0.01 mg l during the vege- P PcPN accounts for both P and N shortage. tation period. The increased threshold value of −1 0.02 mg l takes into account that the residual of 4 5 · T P if T P − PP < 0.02 T P − P P does not consist of DIP only but also P PcP = contains unavailable dissolved organic P. The fac- (P P else tor P P/T P ≈ 4/5 was derived from observation (4.27) − 1 · · data at TP concentrations lower than 0.1 mg l . 0.5 TN if 0.5 TN < 6 6 PPcP Second, PP is corrected for possible N limitation P PcPN = (4.28) P P else based on the N:P ratio. According to the Redﬁeld- ( cP Ratio (Lampert & Sommer, 1993), the optimum Phosphorus burial stoichiometric ratio of N and P in algal biomass is approximately 16 which is equivalent to a mass The driving force behind phosphorus burial is the ratio of about 7 (see Klausmeier et al., 2004, for in- continuous settling of fresh material at the sedi- formation on the variability). Data from the Havel ment surface. In the model, it is assumed that Lakes indicate a decline in phytoplankton concen- no phosphorus from the deep sediment is trans- trations at a N:P ratio in particulate organic matter ferred back into the upper layer or the water col- of about 6 (see Fig. 4.27). umn. Thus, burial irreversibly removes phosphorus from the system. In order to correct the average concentration of If vertical heterogeneities in the uppermost sed- PP for N shortage, it was multiplied by the factor iment layer with high bioactivity13 are neglected, 1/6 · P ON/P P if PON/PP < 6. A reasonable upper-limit estimate for the concentration of par- 13This is also called the active layer. 4.4 Water quality submodel 89

the rate constant of P loss equals the quotient of the effective sediment growth velocity and the thickness of the active layer upbur/zs (Chapra, 1997, see Table 4.6). Phosphorus remobilization In accordance with Table 4.6, the inﬂuence of phosphorus remobilization on the water column’s total P concentration is described by Eq. 4.29

· 25 kprem f(T ) d (P S − P0) PS>P0 T P = zw dt (0 P S ≤ P0 0 (4.29)

-25 and the corresponding change in the concentration of sediment phosphorus PS follows from Eq. 4.30. Anomaly of Chl-a (%) -50

0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 d z d

9-10 w

10-11 11-12 P S = − · T P (4.30) part. N : part. P (mass ratio) dt zs dt

Figure 4.27: Anomaly in the concentration of A fundamental assumption of this approach is the chlorophyll-a against the N:P ratio in particulate or- linear dependence of P remobilization on the sedi- ganic matter. The anomaly Ak for each class k of the ment phosphorus excess. The excess is defined as N:P ratio was computed as Ak = Median(Ck,m/Cm− the difference between the current total P concen- . In this definition, is the observed chloro- 1) · 100 Ck,m tration PS and a threshold value denoted P0. Con- phyll concentration at a date in month m where the N:P ceptually, P0 represents the capacity of the sedi- ratio falls into class and is the average concentra- k Cm ment for permanent phosphorus storage. As long tion of chlorophyll in month m. For example, a value of −25% indicates that – with an empirical probability of as P S ≤ P0 no remobilization of phosphorus oc- 50% – the chlorophyll concentration is lower than 75% curs in the model. In other words, it is assumed that of the monthly average. The figure is based on pooled all settling phosphorus is being adsorbed or precip- data from the Potsdamer Havel. The number of data in itated after mineralization or it is stored in organic each class is >100 for N:P≤7, it is >50 for N:P≤10, and form. Phosphorus remobilization begins after the ≥25 for the remaining two cases. surface sediment has become supersaturated, i.e. if PS>P0. From sediment analysis (Knösche, 2006a; Schettler, 1995) it is known that a considerable fraction of the sedimentary phosphorus in the Havel Lakes is iron-bound (see Sect. 4.4.2.1). The dependence of the P remobilization from aerobic lake sediments on the iron content was demonstrated by Jensen et al. (1992a). It is therefore ad- −3 visable not to define P0 in absolute units (g P m sediment), but to use a threshold iron:phosphorus 90 Chapter 4 Model application to the Lower Havel River

ratio rp:fe instead. P0 is then computed from After extensive model tests, the term f(T ) was Eq. 4.31 deﬁned as in Eq. 4.32

(T7−20) f(T )= θprem (4.32) P0 = rp:fe · F e (4.31)

with T7 being the average water temperature of the with Fe being the iron content of the sediment previous seven weeks and θ being a dimen- −3 prem (g m ) and rp:fe being the P binding capacity in sionless calibration parameter. gP(gFe)−1. In this way, differences in the P binding capacity of sediments with different Fe content In summary, the model of phosphorus release is can be taken into account. a conceptual one. It assumes a linear dependence The constant kprem in Eq. 4.29 is a scaling pa- of P release on the sediment phosphorus excess rameter that controls how much P is released per which is derived from the phosphorus:iron mass square meter and time at a given sediment P ex- ratio. The seasonality of the phenomenon is access. The corresponding unit is (g P m−2 surface counted for by an empirical model based on the av- area s−1 (g P excess)−1 m3 sediment). erage temperature over of a previous time period. The dimensionless term f(T ) was introduced to The behavior of the complete phosphorus model account for the seasonality of P remobilization. As can best be understood by analyzing a steady state discussed in Sect. 4.4.2, the intensity of P release situation. If the full expression dPS/dt according should increase with rising bioactivity due to a de- to Table 4.6 is set to zero, solving for PS yields the cline in the redox potential, lowered pH, and turba- steady state P concentration in the sediment PS∗ tion effects. Because the activity of microbes and (Eq. 4.33; see Table 4.6 for declaration of sym- other organisms in the sediment is not simulated, bols). the temperature, which is the ultimate control variable, was chosen as a surrogate parameter. ∗ upset · P P + kprem · f(T ) · P0 Model tests revealed that best simulation results P S = (4.33) upbur + kprem · f(T ) can be obtained if P release rates are linked to the average temperature of a previous time period From Eq. 4.33 two important conclusions can be rather than to instantaneous values. In other words, drawn: the maximum of sediment P release lags behind the 1. The steady state concentration of sediment P maximum of water temperature. There are multi- increases linearly with the intensity of phos- ple plausible explanations for this observation. phorus settling. Firstly, the maximum abundance of microbes 2. If burial is neglected (u = 0) and the set- should lag behind the maximum of sediment pbur tling rate is non-zero, the value of PS∗ will temperature according to the laws of population exceed the storage capacity of the sediment growth. Secondly, a delay between the tempera- P . That means, even in steady state, there ture of sediment and water is to be expected. In 0 is a sediment phosphorus excess. This excess consequence, the minimum values of redox poten- will be greater, the less effective P remobi- tial and sediment pH should also be retarded with lization is. Only at large values of k does respect to the temperature maximum. The time lag prem PS∗ approach P and the excess (P S∗ − P ) may be further ampliﬁed by the fact that reductable 0 0 goes down to zero. substances other than iron must ﬁrst be depleted in the sediment before iron itself is reduced and Fe- For proper application of this phosphorus model, bound P is released. attention should be paid to the following points: 4.4 Water quality submodel 91

• The computed rate of sediment P release is growth and settling rates of functional algae independent from the concentration of dis- groups, solved phosphorus in the overlying water 2. observation data for all simulated variables of (DIP). Consequently, the model should only a plankton model are available with higher be applied in its existing form if the flux temporal resolution (e.g. daily values), rate is insensitive to altered values of DIP. 3. the catchment models to which TRAM is In the sediment of the Havel Lakes, an aver- linked are able to provide the complete set of age pore water concentration (SDIP) of about − required boundary conditions instead of total 4 mg l 1 was found in November 2005 (data N and P loads only, from Knösche, 2006a). Because the corresponding concentration in the pelagic zone is 4. more information on the phosphorus fractions much lower (about 0.3 mg l−1), the diffusive in the Havel sediments and their remobiliza- flux rate would increase by only 4% if DIP tion mechanisms is available. was be reduced by as much as 100%. • The seasonal pattern of P release, which is ap- 4.4.4 Parameter estimation and model proximated by the function f(T ), is assumed performance to be unaffected by changes in the sediment P excess. This is reasonable since the bioac- 4.4.4.1 Calibration strategy tivity in the sediment – and thus P release – are basically bound to the seasonal cycle of Because the parameters of the nutrient turnover temperature. models described above cannot be measured directly, their values had to be determined by cal- • As mentioned above, the particulate P frac- ibration. The goal was to achieve a good agree- tion is a boundary condition variable, not a ment of observed and simulated TN and TP con- simulated component. If the model is applied centrations at the water quality monitoring stations to situations in which a significant reduction shown in Fig. 4.5 given the model structures out- in plankton biomass is likely (e.g. due to se- lined in Sect. 4.4.3.2 & 4.4.3.3. vere nitrogen shortage or manipulation of the With the PEST software (Doherty, 2004), a flex- pelagic foodweb), the particulate P concen- ible and free optimization tool for automatic cali- tration needs to be introduced as a simulated bration of nearly any simulation model is available. state variable. Notice that, at present, a mod- The use of such nonlinear optimization algorithms erate reduction in total P is unlikely to cause a is certainly superior to ’manual calibration’ with reduction in particulate P because the system respect to objectivity and efficiency but its suc- is currently not limited by phosphorus. cessful application may be less easy (see e.g. Do- It is obvious that the applicability of the cho- herty, 2004; Beven, 2000). A complicated shape sen model formulation is limited to scenarios with of the objective function or insufficient precision moderately altered boundary conditions. From of the model output may slow down convergence this point of view, the use of a complex, more or cause the algorithm to fail completely. A fur- process-oriented ecological model would be de- ther drawback of unsupervised optimization algo- sirable. However, such a model will only be of rithms lies in the fact that the value of the objective greater value if, function cannot be tracked over the full parameter space but only along a more or less random path. 1. key parameters of a phytoplankton model Another, more practical, problem is the automatic have been determined in the field, such as editing of the model’s input files by the optimiza- 92 Chapter 4 Model application to the Lower Havel River

PARVAR • A list of values covering the parameter range Read a simulation control file. to be scanned15.

Execute for each parameter set: • Optionally, the name of another parameter 1. Save the number of the model run and the can be specified to which the present parame- current value of each parameter to a file. ter is tied. In this way, parameter interdepen- 2. Update the parameter values in the model input file(s) according to the control file. dencies can be accounted for and the number 3. Call a batch file. of simulations is reduced since the values of both parameters are altered synchronously. In BATCH FILE true Monte Carlo simulations with np param- 1. Call the model executable. 2. Save selected results (e.g. simulation errors eters, it may be useful to tie np − 1 of them to at specified stations) to a file along one single master parameter. with the number of the model run. After some postprocessing, the information TRAM recorded by PARVAR can easily be inspected for Perform a simulation run using the updated input file(s). the parameter set(s) which gave the best agreement of simulated and observed data. Figure 4.28: Functionality of the PARVAR utility pro- In the application presented, the model perfor- gram used for automatic calibration. mance was quantified using the efficiency after Nash & Sutcliffe (1970) which is defined as the ratio of the mean square error MSE(p,o) and the ob- tion tool. While PEST’s low level editing routines servation variance VAR(o) subtracted from unity provide great flexibility, the risk of producing cor- (symbol NSE; Eq. 4.34). In order to improve the rupt data files without notice is quite high. sensitivity of the objective function at low concen- In order to avoid these difficulties, a utility trations, the efficiency was divided by the mean program PARVAR was designed that facilitates a absolute percentage error (MAPE) which is com- model calibration by the method of combinato- puted from observations (o) and predictions (p) ac- rial scenarios (Granger Morgan et al., 2003) or the cording to Eq. 4.35. Further parameters such as Monte Carlo approach14 . The principle of its ap- the mean error (bias) were logged for each model plication is illustrated by Fig. 4.28. run. The performance measures were averaged over multiple monitoring locations if necessary. In a simulation control file, the names of the calibration parameters are specified. For each of 1 n 2 the parameters, the following information must be (oi − pi) NSE = 1 − n i=1 supplied: 1 n 2 n P i=1 (oi − o) MSE(p,o) = 1 − P (4.34) • The name of the ini-file from where TRAM V AR(o) n reads the parameter value. 1 |o − p | MAPE = i i · 100 (4.35) • The name of the corresponding data section n o i=1 i and keyword (see example in Fig. 3.19). X If the number of independent calibration parameters is denoted np and nv(i) values are to be tested 14In a Monte Carlo simulation, the parameters are sampled randomly from their distributions. In combinatorial scenario 15No automatic sampling is performed, i.e. for Monte analysis, the test values are predefined instead of being ran- Carlo simulations, the array of random values must be gen- domly chosen. erated externally, based on distribution information. 4.4 Water quality submodel 93 for the i-th parameter, the total number of simulations ns follows from Eq. 4.36. Zone 5 Zone 1

Zone 6 np n = n (i) (4.36) 3 s v Zone 4 Zone 2 Yi=1 Considering the required time for a single simula- tion16, the possible number of test values for each parameter is strictly limited. In the case of spatially Figure 4.29: Subdivision of the river network for calibration of the nitrogen retention model. Zone 1: Lake variable parameters, it is a good idea to split the section downstream of Berlin Spandau, 2: Teltowkanal study area into subsections and to calibrate each of with high sewage load, 3: Zone downstream of the Tel- them separately. towkanal’s mouth, 4: Lake section downstream of Pots- dam, 5: Channel section of the Sacrow-Paretzer Kanal, 6: Middle Havel River downstream of Ketzin. 4.4.4.2 Nitrogen submodel

Parameter estimation variable model parameters shown in Table 4.7 im- The model was calibrated using observed TN data proved the model performance (Fig. 4.30) signifi- of the period 1997–2000 and the subsequent 4 cantly. years served as validation period. Although to- According to Table 4.7, nitrogen retention ap- tal nitrogen is measured since 1995, observation pears to be higher in the lakes (zone 1 & 4) com- data from before 1997 were excluded because of pared to the more typical river/channel sections inconsistencies which might result from a change (zone 5 & 6). The high value of kTN for the Tel- in measurement practice. towkanal (zone 2) seems to contradict this finding In a first attempt, global values were used for the at first glance. However, the increased N retention parameters kTN and θTN . Since the goodness-of- is likely to result from the high and constant ni- fit measures obtained were poor, the study area was trate load (see Fig. 4.9). It is presumed that the divided into zones as shown in Fig. 4.29. The goal nitrogen retention determined in the lakes of the of subdividing the model domain was to allow for Berliner Havel (zone 1) is somewhat too high be- the use of different parameter values for water bod- cause it compensates for an overestimated inflow ies which were expected to show a different reten- of water from the polluted Teltowkanal into Lake tion behavior. The primary criterion for defining Wannsee. the zones shown in Fig. 4.29 were the hydrological Furthermore, the θTN values greater than one characteristics. Whereas zone 1 and 4 mainly com- for all separately calibrated zones indicate that ni- prise lakes, the remaining zones represent more trogen retention is more effective during summer typical river conditions. The Teltowkanal and the (recall Fig. 4.24). As explained in Sect. 4.4.3.2, river section downstream of its mouth (zone 2 & 3) this may partly be attributed to the temperature de- were separated because they experience an extraor- pendency of denitrification but a seasonal varia- dinarily high nitrogen load from Berlin’s wastew- tion in the composition of the TN pool (plankton ater treatment plants. The use of the spatially growth) may also be of relevance. One must keep in mind that the model does not take N fixation by 16A single run with 88 reactors (21 STR, 67 PFR) over a period of 10 years with a maximum time step of 1 day takes 2 cyanobacteria into account which also affects the minutes on a 2.8 GHz machine. seasonality of N concentrations. 94 Chapter 4 Model application to the Lower Havel River

Table 4.7: Calibrated parameters of the nitrogen reten- 0.40 tion model for the six regions shown in Fig. 4.29. 0.20 ) -1 0.00 −1 Zone index kTN (d ) θTN (–) -0.20 Krughorn

ME (mg l -0.40 1 0.025 1.04 Potsdam -0.60 Ketzin 2 0.020 1.06 Brandenbg. 3 0.015 1.10 -0.80 4 0.010 1.04 100 Calibration Validation 5 0.006 1.04 80 6 0.006 1.06 60 40 MAPE (%) 20 0 Model performance The performance of the nitrogen retention model 1.0 0.8 during the period of calibration (1997–2000) and 0.6 validation (2001–2004) is illustrated in Fig. 4.30. 0.4

Furthermore, Table 4.9 summarizes the major Efficiency (-) 0.2 goodness-of-fit indicators of the nitrogen and 0.0 phosphorus model. 1997 1998 1999 2000 2001 2002 2003 2004 After Fig. 4.30 and Table 4.9, the model perfor- Figure 4.30: Deviation between simulated and mance measured by the Nash-Sutcliffe efficiency observed TN concentrations at Berlin Krughorn (NSE) is only moderate since the values are gen- (BLN435), Potsdam Humboldtbrücke (HV0110), Ket- erally smaller than 0.6. However, other statis- zin (HV0195) and Brandenburg (HV0200). The figure tics such as the mean absolute percentage error shows the mean error or bias (ME), the mean absolute (MAPE) show that the performance of the very percentage error (MAPE) as well as the efficiency after simple nitrogen model is quite satisfactory. In Nash & Sutcliffe (1970). terms of the NSE and the bias (ME), the simulation error during the validation period is even lower rameter set however cannot exist from a theoretical than during the calibration period. point of view. With respect to the deviation between observed Another potential source of error are badly esti- and simulated concentrations, several sources of mated nitrogen loads at the systems’s inflows. In- errors must be distinguished (refer to Sect. 4.6.5.1 accuracies in discharge values are expected to be for a comprehensive overview). The nitrogen rather low, since daily averages were used. Larger retention model is certainly subject to structural errors are likely to result from the coarse tempo- deficits because many different processes were ral resolution of concentration time series requir- lumped in Eq. 4.26. Model errors are likely to re- ing interpolation17. Finally, errors in individual sult from variabilities in the composition of the TN observations are not unlikely in highly eutrophic pool, unconsidered boundary conditions and inter- systems because the particulate organic fraction of actions, as well as the total neglect of some pro- TN is not always homogeneously distributed over cesses such as nitrogen fixation. The fact that the a river or lake cross-section. Non-representative model is is not strictly process-oriented has another consequence: Calibration can only identify those 17At some stations, daily concentration data were available, parameter sets that fit the data well. A ’true’ pa- whereas only fortnightly data exist for others. 4.4 Water quality submodel 95

Initial remobilization rate P:Fe mass ratio Remobilization after 21 days 50 0.0 0.1 0.2 -1 40 0 d

-2 30 20 10 10 mg P m 0 20 0 0.05 0.1 0.15 0.2 Depth (cm) 30 P:Fe mass ratio in sediment

Figure 4.31: Relationship between phosphorus remo- Figure 4.32: Vertical proﬁle of the P:Fe mass ratio bilization under aerobic conditions and the phospho- in 11 sediment cores from the Potsdamer Havel Lakes rus:iron mass ratio in the upper sediment (after Jensen (data from Knösche, 2006a). et al., 1992a).

Based on Fig. 4.31, it is reasonable to assume observations distort parameter estimation and (in negligible phosphorus remobilization rates at P:Fe the case of non-systematic errors) result in poor ratios less than ≈ 0.025 g P (g Fe)−1. However, goodness-of-fit measures. because of the small number of data, this is a crude In the system studied, modeling results are be- estimate only and Fig. 4.31 indicates that P remo- lieved to be extra sensitive to errors in the hydro- bilization may also be low at much higher P satura- dynamic simulation. This is because the nitrogen tion. In Saxonian reservoirs, Maassen et al. (2005) load of the Teltowkanal, which is a major source found a sharp decline in the phosphorus concentra- of N, may be differently distributed at two split tion of the pore water at a P:Fe mass ratio of 0.09. −1 flow junctions (see Fig. 4.9 and Fig. 4.38). In addi- The use of rp:fe= 0.025 g P (g Fe) is justi- tion to potential inaccuracies in the flow distribu- fied by the fact that the phosphorus:iron ratio in tion computed by HEC-RAS, one must not forget the deeper layers of the Havel sediments is some- that TRAM is unable to properly handle flow re- what higher than this value (Fig. 4.32). SuchaP:Fe versal (Sect. 3.5.4) which may temporarily occur ratio slightly above the storage capacity rp:fe can in some channels of the looped river network. be expected for sediments which were deposited before the period of extensive phosphorus pollution, i.e. when P settling was still low (recall the 4.4.4.3 Phosphorus submodel steady state consideration from Sect. 4.4.3.3). Al- though the sediment profile data support the pre- Determination of the sediment phosphorus excess sumed magnitude of rp:fe, the absolute numerical A major issue of the phosphorus model is the def- value remains highly uncertain. inition of a reasonable value for the P storage ca- The vertical profile of sediment P can also be in- pacity of the sediment expressed by the parameter terpreted another way: If the P:Fe ratio approaches rp:fe. One option is to derive such a value from a ± constant value in the deep (old) layers of undis- experimental data. Fig. 4.31 shows the relation be- turbed core samples, this ratio reflects the histori- tween the P:Fe ratio and the rate of P remobiliza- cal equilibrium of the sediment with the average tion identified by Jensen et al. (1992a) at values of pelagic P concentration. Consequently, the amount P:Fe which are comparable to those found in the of P which would potentially be remobilized if Havel Lakes. P loading was reduced to the historical level can 96 Chapter 4 Model application to the Lower Havel River

Table 4.8: Average concentrations of phosphorus and Estimation of further parameters iron in the uppermost 20 cm of sediment of the sam- The four real calibration parameters of the phos- pled Havel Lakes (data for the year 2005 from Knösche, phorus submodel are the burial velocity u , 2006a). pbur the settling velocity upset, and the parameters Lake label TP Fe P:Fe kprem and θprem which control magnitude and tim- from Fig. 4.5 (g m−3) (g m−3) (gg−1) ing of P release. A simultaneous calibration of all 4 parameters using the strategy discussed in C 1330 13000 0.103 Sect. 4.4.4.1 was not feasible with respect to the E 482 4408 0.124 computational effort. Furthermore, the identifia- F 543 6706 0.081 bility of all parameters is questionable, given the G 254 3759 0.076 available observation time series which include H 253 4184 0.060 only pelagic concentrations but no sediment samples. Consequently, u was excluded from the cali- be estimated from the increased P:Fe ratio in the pbur bration and set to a fixed value. Based on sediment young deposits at the top of the profile. In a sim- growth rates observed in a shallow, hypertrophic ilar way, Kleeberg (1995) and Kleeberg & Kozer- − Swedish lake (Granéli, 1999) a value of 2 mm a 1 ski (1997) quantified the phosphorus excess in the was chosen. According to investigations in lakes shallow Lake Müggelsee based on the vertical pro- of Brandenburg (LUA, 2005a) 1 cm slices of sed- fileofgP(gDW)−1. iment core samples integrate the deposits of 2–20 Apart from setting the value of rp:fe, choosing years. The assumed value of 2 mm a−1 agrees well an appropriate value for the thickness of the active with this observation18. sediment layer z (m) is important if the model is s The remaining three parameters were globally applied to a simulation period of several years or calibrated. Reasonable start values for k and decades. After Søndergaard et al. (2003), phospho- prem θ were determined from P-balances of the rus can be remobilized from a depth of up to about prem Havel Lakes (Kneis et al., 2006). An initial guess 20 cm and this estimate was adopted for the model of the P settling velocity was taken from Bowie simulations. The uncertainty of this value is, how- et al. (1985). Calibration was carried out using ever, large because it reflects the efficiency of vari- observed data of the period 1995–2000. As with ous remobilization mechanisms (see Sect. 4.4.2.2) the nitrogen model, data from 2001–2004 were re- which may also be highly variable in space. served for the purpose of model validation. Because both r and z are sensitive parame- p:fe s It soon became apparent that the settling veloc- ters but their values are not well known, it seemed ity u is barely identifiable given the available necessary to carry out multiple simulations with pset observation data. For values of u below ≈ both parameters being varied around the estimated pset 0.1 m d−1, the sensitivity of the model was low values of r = 0.025 g P (g Fe)−1 and z = 0.2 m p:fe s with respect to any of the computed goodness-of- (see Sect. 4.6.5). fit measures. At larger values, the model error The P excess in the top sediment of the Havel increased significantly. Therefore, the parameter Lakes was estimated from measured P and Fe con- −1 upset was fixed like upbur. A value of 0.04 m d centrations (Table 4.8). The initial sediment P con- was chosen based on the model performance dur- centration at the beginning of the calibration period

(1995) was counted back using the values observed 18 For a more reliable estimation of upbur, cores samples in 2005 and average annual net phosphorus export from the Havel should be analyzed for isotopes emitted after rates as computed from long-term mass balances. the Tschernobyl disaster in future research. 4.4 Water quality submodel 97

ing winter (Nov-Apr) considering the full period 0.05 1995–2004. During summer, u was unidenti- pset ) 0.00 -1 ﬁable at all because of the compensating effect of -0.05 P release. The value of 0.04 m d−1 is at the lower -0.10 Krughorn bound of the data published by Bowie et al. (1985). ME (mg l Potsdam -0.15 Ketzin For shallow, turbulent surface waters a low effec- Brandenbg. -0.20 tive sinking rate is plausible. 100 Fig. 4.33 shows the response surface of the Calibration Validation 80 model with respect to the two remaining free pa- 60 rameters kprem and θprem. On the one hand, the 40 plot reveals the existence of an optimum parame- MAPE (%) 20 ter set within the range scanned in the simulation. 0 On the other hand, the location of this optimum set obviously depends on the chosen goodness-of-ﬁt 1.0 0.8 criteria. 0.6 In addition, the response surfaces do not exhibit 0.4 steep peaks but a number of parameter sets pro- Efficiency (-) 0.2 duced almost equally good results near to the ’best’ 0.0

values, e.g. NSE > 0.69 or MAPE < 27%. The di- 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 agonal, ridge-like structure in Fig. 4.33 also shows Figure 4.34: Deviation between simulated and ob- that a low value of k may, to some extent, prem served TP concentrations during the calibration and val- be compensated by an increased value of θprem idation period (same presentation as in Fig. 4.30). and vice versa. According to the GLUE philosophy (Beven, 2000), a larger number of parameter sets, e.g. those with an efficiency above 0.68, are With respect to the model efficiency after Nash equally likely. This must be accounted for when & Sutcliffe (1970), the phosphorus model per- the uncertainty of model predictions is assessed forms well and certainly better than the nitrogen (see Sect. 4.6.5). retention model. The mean absolute percentage er- Based on the maximum quotient of the effi- ror, however, is larger than for the N model, con- ciency and the MAPE obtained, values of kprem firming that a single goodness-of-fit measure is in- = 0.000085 g P m−2 d−1 (g P excess)−1 m3 and sufficient for assessing the quality of a model. In θprem = 1.35 were selected. Using this set, the sed- the present case, the Nash-Sutcliffe efficiency indi- iment P concentrations observed in 2005 were re- cates a good fit because the model is able to repro- produced with a deviation ≤ 14%. That is, the sim- duce the summer peak in the phosphorus concen- ulated cumulative P export from the lakes in 1995– tration. The MAPE, which is less sensitive to the 2004 is consistent with the mass balances used for concordance of peaks, weights the error by the ab- estimating the initial P pool in 1995. solute values and even small errors are considered severe at low P concentrations. Model performance As in case of the nitrogen model (Fig. 4.30), The performance of the phosphorus model dur- a better fit was obtained for the validation period ing the calibration and the validation period is than for the period of model calibration. More re- illustrated by Fig. 4.34 and Fig. 4.36. Further liable observation data for recent years might ex- goodness-of-fit parameters are listed in Table 4.9. plain this fact. 98 Chapter 4 Model application to the Lower Havel River

MAPE (%) 80

50 prem θ 40

30 1.0 1.2 1.4 1.6 1.8 2.0

0.00004 0.00006 0.00008 0.00010 0.00012 0.00014

k prem

Figure 4.33: Response surface of the phosphorus model obtained from 529 simulations with kprem in range −2 −1 −1 3 0.000040–0.000150 g P m surface area d (g P excess) m sediment and values of θprem between 1.0 and 2.1. Both parameters were sampled in regular intervals from the corresponding ranges. Contours show the Nash-Sutcliffe efﬁciency (Eq. 4.34), the mean absolute percentage error (MAPE) is represented by colors.

Table 4.9: Performance indicators of the nutrient turnover model at the four monitoring stations from Fig. 4.30 and Fig. 4.34. ME: mean error, MAPE: mean absolute percentage error, NSE: efﬁciency after Nash & Sutcliffe (1970).

Calibration Validation Station Component ME MAPE NSE ME MAPE NSE (mg l−1) (%) (–) (mg l−1) (%) (–) BLN345 TN -0.20 21 0.45 -0.14 19 0.52 TP -0.016 30 0.69 -0.001 26 0.87 Hv0110 TN -0.29 13 0.33 -0.10 12 0.52 TP -0.013 23 0.69 0.011 24 0.78 Hv0195 TN -0.06 17 0.11 0.10 18 0.59 TP -0.048 27 0.68 -0.016 23 0.85 Hv0200 TN -0.18 21 0.30 -0.03 23 0.37 TP 0.004 34 0.47 -0.021 28 0.55 4.5 Use of the calibrated model for system analysis 99

The more general sources of simulation errors 4.5 Use of the calibrated model for discussed in conjunction with the nitrogen model system analysis (Sect. 4.4.4.2) apply to the phosphorus model too. With respect to the phosphorus model, it is impor- Apart from being used for predicting possible fu- tant to recall that the key process – the remobi- ture trends in water quality, the simulation model lization of P from sediments – is actually highly can also contribute to the understanding of past and variable on different time scales. For example, present conditions (system analysis). For example, sediment resuspension during storm events or the the model allows the retention of N and P to be depletion of electron acceptors at the sediment- quantified and seasonal patterns to be evaluated. water interface during temporal stratification peri- This can be done in alternative ways: ods may cause sudden peaks in P remobilization. The model, however, which approximates the seasonality of P release by an empirical temperature 1. With knowledge of the model’s structure, the function, produces a rather smooth pattern. calibrated parameter values can be interpreted directly. An evident example for the large interannual 2. The simulation results of the calibrated model variability in the magnitude of P remobilization is can be compared to those produced by a con- the year 2000 (see Fig. 4.36). In this year, the servative model, i.e. a model which does not model dramatically underestimates the phospho- take into account any turnover processes. rus concentration at all monitoring stations downstream of the Havel Lakes. No significant corre- 3. Finally, the model can be used for calculat- lations between the annual residuals and possible ing mass balances for individual water bodies control variables such as minimum winter or aver- with high temporal resolution as described in age summer temperatures, chlorophyll concentra- Kneis et al. (2006). In this case, the output of tions, etc. could be identified so far. conservative simulations is directly compared with observations, not model results19. Finally, one must keep in mind the strong simplification which is due to the approximation of the bottom sediment by a single layer. In reality, most 4.5.1 Significance of phosphorus release of the sediment parameters are not homogeneously distributed with depth and this is also true for the A result of mass balance calculations (third ap- P:Fe ratio (Fig. 4.32). It is not unlikely that the proach from the above list) is shown in Fig. 4.35. magnitude of P release is in fact more closely cor- Using measured P loads at a lake’s inlet, the ex- related to the P excess at the very top of the sed- pected hydrograph of the P load at the lake’s outlet iment rather than to the average excess in a layer was computed, assuming that no retention or re- of 20 cm thickness. At present, sediment stratifi- mobilization of P would take place (conservative cation is ignored by the model because processes model). The modeled loads at the lake’s outlet such as turbation or compaction would need to be (Lout,sim) were then compared with observed val- −2 simulated otherwise. The corresponding parame- ues (Lout,obs) and net export rates (rTP , mg P m ters are certainly not identifiable from the available d−1) were computed according to Eq. 4.37. In this data base. 19 A quantitative analysis of the uncertainties asso- Note that a direct comparison of observed loads at a lake’s inlet and outlet is unsuitable for computing short term ciated with the phosphorus submodel is presented mass balances because the residence time of the water is not in Sect. 4.6.5. taken into account. 100 Chapter 4 Model application to the Lower Havel River

60 Templiner See Nord (E) the current P concentrations are increased by this )

-1 Templiner See Sued (F) factor compared to a situation with zero P release.

d Schwielowsee (G) -2 40 This clearly affects not only the studied river sec- Werdersche Havel (H) tion but also downstream waters including several 20 other lakes. The good agreement between observed and 0 simulated P concentrations during winter, when

-20 turnover is less signiﬁcant, veriﬁes that the sys-

Net P export (mg P m tem’s most relevant boundary conditions have been -40 accounted for. Once more, Fig. 4.36 reveals that a significant part of the observed variance in the TP Jul Oct Apr Jan Jun Mar Feb Nov Dec Aug Sep May concentration cannot yet be explained, using the Figure 4.35: Phosphorus balance of four Havel Lakes simple model structure described in Sect. 4.4.4.3. during the period 1995–2000. The columns show monthly median values with the 90% confidence interval as whiskers. Lake labels refer to Fig. 4.5. 4.5.2 Significance of nitrogen retention

By comparing the simulation results of the cal- 2 context, A (m ) denotes the lake’s sediment sur- ibrated model with the output of a conservative face area. model, the effect of nitrogen retention can be quantified. From Fig. 4.37 it can be deduced that a sig- 86.4 · 106 nificant fraction of the total nitrogen which is dis- r = ·(L − L ) (4.37) TP A out,obs out,sim charged into the studied river section is subject to retention upstream of the Ketzin gage. For the pe- The near zero or negative values of in win- rTP riod 1997–2004 total retention efficiencies of 27% ter and early spring indicate that P remobilization and 28% were identified for the gages Ketzin and is either negligible or it is at least balanced by Brandenburg, respectively. This results in a sig- net phosphorus sedimentation. During the season nificant reduction of the N export to downstream of increased P remobilization, is greater than rTP waters. Compared to estimated in-stream retention zero. According to Fig. 4.35, the rate of sediment P in a Danish lowland river (Svendsen et al., 1998) release exceeds the rate of net sedimentation in all nitrogen losses in the Havel River are remarkably of the lakes from July to October. Comparable sea- high. Since denitrification is most effective at the sonal patterns and magnitudes of net phosphorus sediment-water interface (Seitzinger, 1988; Schef- release were reported from Danish lakes (Sønder- fer, 1998), this might be explained by the large sur- gaard et al., 2002). The fact that is greater than rTP face area of the Havel Lakes. zero when averaged over the whole year clearly in- The identified rates of nitrogen retention were dicates that the lakes currently undergo a phase of also compared to predictions based on the empir- recovery, i.e. they are loosing P from the sediment. ical approach of Behrendt & Opitz (2000). They In Fig. 4.36, the simulation results of the cal- estimate the quotient of the observed nitrogen load ibrated P model and the output of a conservative L and the theoretical load due to emissions E using model run are presented together with observed Eq. 4.38 data. Summer phosphorus concentrations predicted by the conservative model deviate from the simulation results with consideration of sediment L 1 = (4.38) P release by 100% or more. One can conclude that E 1+ a · Hb 4.5 Use of the calibrated model for system analysis 101

1.2 Observed 1.0 Conservative model )

-1 0.8 Calibrated model 0.6 0.4 TP (mgTP l 0.2 0.0

4.0 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 ) -1 3.0

2.0

1.0 TP load (t load TP d

0.0 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

Figure 4.36: Observed and simulated TP concentrations (mg l−1) and loads (t d−1) at Ketzin (station HV0195 in Fig. 4.5) with and without consideration of phosphorus retention/release.

where a and b are coefficients and H (m a−1) is the hydraulic load. The latter is calculated from the annual runoff (m3 a−1) divided by the water surface area (m2). With a reference to the EU- ROHARP project (Kronvang et al., 2005), Venohr et al. (2003) present values for the empirical co- 40 Conservative model efficients (a ≈ 7.28, b = −1) applicable to total Calibrated model 30 Observed nitrogen retention in lakes. Using the lake area of the studied river section (43 km2) and a mean 20 flow rate20 of 30 m3 s−1, a hydraulic load of H ≈ −1 10 22 m a is obtained. With this figure, Eq. 4.38 predicts a retention efficiency of 25% (L/E=0.75) 0 Cumulated N-load (kt) which is close to values estimated with TRAM. Apart from showing the magnitude of nitrogen 1996 1997 1998 1999 2000 2001 2002 2003 2004 retention, Fig. 4.37 also illustrates that the devia- Figure 4.37: Cumulated load of total nitrogen (kilo- tion between simulation and observation is rather tons) at Ketzin as computed from observed data and small when looking at cumulated TN loads. How- model simulations. ever, as presented in Sect. 4.4.4.2, the variability in

20This estimate is based on the mean ﬂow rate at Ketzin (Table 4.1) taking into account the system’s split ﬂow junctions. 102 Chapter 4 Model application to the Lower Havel River

TN concentrations is replicated by the model still In contrast to the Teltowkanal, the inflow of the less accurately. Nuthe River at km 20.5 (label ’N’) is of minor relevance only. In the phosphorus plot, one can see a slight increase in the concentrations downstream 4.5.3 Visualization of patterns of the mouth in winter and spring. Finally, a remarkable temporal shift shows up By merely looking at the time series of nutrient at km 30 in the phosphorus plot (upper panel). concentrations observed at individual monitoring This retardation effect is due to the large residence stations, it is hard to get a holistic impression of the time of the water in Lake Schwielowsee (recall spatial and temporal patterns. Using the calibrated Fig. 4.7). model, these patterns can be visualized in space- The most conspicuous fact about the two panels time plots like Fig. 4.38. In this case, the model ba- in Fig. 4.38 is the contrasting temporal pattern of sically serves as an interpolator in space (unmoni- TP and TN concentrations. The peak in the phos- tored sections) and time (dates between sampling). phorus concentration appears in late summer due A common feature of both panels in Fig. 4.38 to internal loading and – to a minor extent – be- are skew structures illustrating the movement of cause point source emissions are subject to weaker water and nutrients through the system. A particle dilution in times of low flow. In the case of nitro- which enters the system at km 0 at day x passes the gen, the impact of point sources is also pronounced downstream model boundary (km 70) at day x+T . during low flow. This is shown by the peak in TN The dependence of the travel time T on flow rates at km 0 and at the confluence with the Teltowkanal is reflected by an increased skewness in the diago- at km 18. However, nitrogen retention results in a nal structures during summer and autumn. This is significant reduction in summer TN levels down- best illustrated by a plot of the characteristics as in stream of the system’s major inflows. The gener- Fig. 4.39. ally higher N concentrations in winter are typical While the temporal fluctuation in the concentra- because non-point emissions are highest and reten- tions of TP and TN is high at the system’s upstream tion is lowest during that time. end (km 0 in Fig. 4.38), the patterns become more smooth in the downstream direction. This is a consequence of dispersion effects in the lakes’ large 4.6 Management scenarios water bodies which are represented in TRAM as 4.6.1 Integration of catchment models completely stirred tanks. and TRAM In both panels of Fig. 4.38, a sharp break in the colors at km 18 attracts attention. This steep spatial During the period of calibration and validation, gradient is caused by the inflow of the Teltowkanal TRAM was supplied with observed data on flow (label ’T’). From the lower panel one can conclude rates and loads at the system boundaries. These that the Teltowkanal is a major source of nitro- boundary conditions had to be simulated for in- gen throughout the year because the concentration vestigating possible future trends (scenario anal- strongly increases at it’s mouth. A different picture ysis). For this purpose, the total nutrient loads shows up for phosphorus (upper panel). While the which enter the system shown in Fig. 4.5 and corre- P concentration increases downstream of km 18 in sponding flow rates were calculated by the project spring, there is a significant dilution effect in late partners using the hydrological catchment models summer when the upstream concentrations are el- SWIM (Krysanova et al., 2000) and ArcEGMO- evated due to P remobilization in the lakes of the Urban (Biegel et al., 2005). SWIM computes the Berliner Havel. discharge of tributaries and the corresponding P 4.6 Management scenarios 103

TP (mg l −1)

0.6

60 0.5 50 K J 0.4 40 0.3 30 S

Station (km) N 0.2 20 T 10 0.1 B 0 0.0 50 100 150 200 250 300 350 TN (mg l −1)

7 60 6 50 K J 5 40 4 30 S 3

Station (km) N 20 T 2 10 1 B 0 0 50 100 150 200 250 300 350

Day of the year

km2H

km2IH

km2SH

km2TH

km2RH

km2PH

km2QH

Figure 4.38: Variation of the concentrations of TP (upper panel) and TN (lower panel) with the season (x-axis) and along a flow path (y-axis). The plots are based on results of the calibrated model for the year 2003 where the simulation matched the observations very well. The selected flow path with major stations is shown in the map below. Label ’B’ marks the entry to the lakes of the Berliner Havel, the mouthes of the Teltowkanal and the Nuthe River are labeled ’T’ and ’N’, respectively. Further labels highlight Lake Schwielowsee (’S’), the confluence with the Sacrow-Paretzer-Kanal at km 47 (’J’), and the Ketzin gage (’K’). See Fig. 4.5 for a more detailed map. 104 Chapter 4 Model application to the Lower Havel River

Figure 4.39: Characteristics illustrating the transport of matter through the river-lake system in the absence of turnover processes. River stations and labels refer to the same flow path as in Fig. 4.38 (see this figure). The plots were produced by simulating the transport of a conservative tracer added at the upstream model boundary (Berlin Spandau) at the first of each month (separate panels). The hydrological boundary conditions correspond to the year 2003 as in Fig. 4.38. In contrast to common plots of characteristics, the spatial distribution of mass is shown for each time step, not only the position of the center of mass. For each river station, concentrations were scaled by the peak concentration at that station in order to eliminate the effect of dilution. The maximum concentration at a station appears in black; white color represents clean water. The arrow labeled ’SPK’ indicates the river section which is bypassed by the Sacrow-Paretzer-Kanal (see Fig. 4.5 or 4.38). It is this short cut which causes the conspicuous break in all concentration profiles at km 47. 4.6 Management scenarios 105

Table 4.10: Average observed nutrient loads (g s−1) Hydrodynamic model Stage at downstream in the major tributaries of the Lower Havel River. HEC-RAS 3.1.1 model boundary Inflow from Spree The evaluated monitoring stations are Sophienwerder Flow and stage hydrographs catchment (Spree), Henningsdorf (Obere Havel), Kleinmachnow for all reactors (Teltowkanal) and Potsdam Babelsberg (Nuthe). QualityTRAM model Nutrient load from Spree catchment Nutrient loads from point & Tributary Period TP TN non-point sources Spree 07/94–12/02 3.7 70 Discharge of tributaries Catchment models Catchment data Teltowkanal 07/94–12/02 1.9 68 (SWIM, ArcEGMO-Urban) Obere Havel 07/94–12/01 1.4 18 Nuthe 07/94–11/02 1.1 10 Figure 4.40: Interaction between the water quality model with the hydrodynamic and catchment model(s). Dotted arrows indicate the exchange of boundary conditions while solid arrows represent external input data. reduced drainage) and the operation of newly con- structed reservoirs. The relevant trend was derived from predictions of monthly average flow rates in and N loads originating from non-point emissions. 2003–17 provided by the GLOWA research project In contrast, the Urban module of ArcEGMO sim- (BfG, 2003). Within this period, the effect of cli- ulates N and P inputs from point sources only mate change underlying the GLOWA simulations (Biegel, 2005). The interaction between TRAM is of minor significance only (Rachimow, 2004). and the catchment models is illustrated in Fig. 4.40 The corresponding loads of phosphorus and (see also Table 4.4 on page 72). nitrogen were estimated using observed load- For simulating the period 2003–2015, the catch- discharge relationships at the gages Kleinmachnow ment models were driven by the meteorological and Sophienwerder. It was assumed that the ob- boundary conditions of the period 1988–2000, i.e. served loads at the two gages reflect both point climate change was not considered in the study. and non-point emissions. Nutrient loads from diffuse sources are usually correlated with discharge because they are mediated by groundwater exfil- 4.6.2 Runoff and nutrient emissions from tration, (sub)surface stormflow, or drainage. In non-modeled subcatchments contrast, nutrient emissions from point sources are Since the Spree catchment was not modeled with expected to be more or less constant. However, SWIM or ArcEGMO-Urban, discharge and nutri- a dependence on discharge may appear due to ent loads of the Spree River and the Teltowkanal combined sewer overflows or less efficient nutri- had to be estimated from other sources (Kneis, ent elimination in WWTP during winter or storm 2005). A proper estimation of N and P loads in events. Finally, observed load-discharge correla- these two tributaries is essential, as can be con- tions at a gage may be influenced by retention cluded from Table 4.10. and/or remobilization effects. For instance, low For consistency with the other subcatchments, flow favors settling and denitrification due to pro- discharges of the Spree River and the Teltowkanal longed residence times while particulate P may be from the period 1988–2000 were extrapolated to remobilized from the river bed at high flow. the scenario period 2003–2015. The hydrographs The basic equation for computing loads (L) were corrected for the predicted negative trend in from flow rates (Q) is Eq. 4.39, with L0 denoting flow rates which is due to the shut down of open- the constant base load from point sources and Q0 pit mines in Lusatia (flooding of former mines & being a discharge threshold above which the effect 106 Chapter 4 Model application to the Lower Havel River

4.6.3 Description of the scenarios c= ∆L/ ∆Q Q0 4.6.3.1 The base scenario S0 ) 200 ∆L -1 The base scenario called ’S0’ assumes that no con- ∆Q

L (g s 100 certed actions are taken to further reduce nitrogen and phosphorus emissions from point or non-point L 0 0 sources. However, the scenario accounts for some 0 25 50 75 important trends such as a rise in the extent of paved areas as well as the anticipated movement of Q (m 3 s -1 ) population from rural to urban areas. The primary Figure 4.41: Application of Eq. 4.39 to the relation be- aim of this scenario is to approximate the current tween total nitrogen loads (L) and discharge (Q) at the situation and to provide a basis for inter-scenario Sophienwerder gage (Spree River). In this figure, the comparison. model was fitted to the collectivity of data from 1995– 2004 for the purpose of illustration. In the actual ap- 4.6.3.2 Scenario S1 plication, the parameters L0, Q0, and c were estimated individually for each month. Scenario ’S1’ assumes a reduction in nutrient emissions from all subcatchments modeled with SWIM and ArcEGMO-Urban due to enhanced of non-point sources becomes visible. The param- emission control. This includes the basin of the eter c represents the slope of the load-discharge Upper Havel River, the Nuthe River, minor tribu- relationship for non-point sources. It is identi- taries downstream of Potsdam, as well as the part cal with the average concentration of diffuse emis- of the catchment which drains into the study sec- sions. An application of this type of model is tion via groundwater exfiltration. Potential man- shown in Fig. 4.41. The choice of a linear model agement actions in the Spree catchment were ex- was supported by the data sets. plicitly excluded in S1, i.e. no enhanced emission control was considered for this part of the river basin. With respect to point sources, S1 includes the completion of all sewage treatment plants currently

L0 + c · (Q − Q0) if Q > Q0 under construction as well as upgrades in P elim- L = (4.39) ination techniques to the highest standards in all (L0 if Q ≤ Q0 large and small-scale plants. The corresponding reduction in emissions was calculated by Markus Biegel at IÖR21 using ArcEGMO’s Urban module (Biegel, 2005). L0 was estimated from the loads at the smallest ob- Contrary to phosphorus, the potential for further served flow rates (Q0) at the cited gages. Because the parameter c shows a significant seasonal vari- enhancement of N-elimination in sewage treatment ation, it was computed separately for each month was considered to be insignificant since the offi- from observation data of the period 1995/97–2002. cially reported grades of purification are very high It was implicitly assumed that the distribution of already. Thus, the scenario S1 considers further re- Berlin’s waste water over the Spree River and the ductions in non-point nitrogen emissions only. In Teltowkanal (including its seasonal modification) 21Institute of Ecological and Regional Development, Dres- remains intact in the future. den 4.6 Management scenarios 107

order to simulate this, digital maps of the present known hydrograph of the total load Ltot,S0 can be land use were reclassified (Jacobs & Jessel, 2003) applied. The basis of this approach is given by to reflect a conversion of arable land to grassland Eq. 4.41–Eq. 4.43. and fallow as well as the growing of intercrops in zones with high N losses. The expected reduction (4.41) in the catchments’ N export rates due to the as- Ltot,S0 = Lpnt,S0 + Ldif,S0 sumed changes in land use was calculated by Anja Lpnt,S0 = L0 + (Ltot,S0 − L0) · k (4.42) 22 Habeck at PIK using SWIM. Ldif,S0 = (Ltot,S0 − L0) · (1 − k) (4.43) Eq. 4.41 simply represents the mass balance. 4.6.3.3 Scenario S2 Eq. 4.42 & Eq. 4.43 state that loads above a con- While management options in the Spree catchment stant base value L0 are a linear combination of were explicitly neglected in S1, the scenario ’S2’ point and non-point loads. The constant load L0 hypothesizes that additional concepts of emission was set equal to the N load observed at low flow control are also implemented in this part of the and the factor k was determined by optimization river basin. Except for the Spree catchment (see in order to bring the long-term average proportion below), the same measures were considered as in of point and non-point sources into agreement with S1. the values reported by LUA (2002). Reduced N exports from the Spree catchment Reduced P exports from the Spree catchment With respect to nitrogen, it was assumed that non- The assumed reduction of phosphorus loads in the point emissions from the Spree catchment are re- Spree River and the Teltowkanal is based on a qual- duced similarly to the Nuthe basin. Thus, an equal ity target proposed by Senat (2001). According to potential for the application of control measures this management plan, an average concentration of − was assumed. The fundamental equation for com- ≈ 80 µg TP l 1 during the vegetation period could puting hydrographs of reduced TN loads in the represent the ’good status’ with respect to phos- Spree River and the Teltowkanal under scenario S2 phorus in the Berlin river system. This value takes (Ltot,S2) is Eq. 4.40. into account a natural background P level as well as the morphological characteristics of the Spree- Havel system. Simulations carried out with the Ltot,S2 = Lpnt,S0 + Ldif,S0 · f(mon) (4.40) MONERIS model (Senat, 2001) support that this target could be achieved if concerted actions were In Eq. 4.40, L is the hydrograph of the N load pnt,S0 taken. Such actions would need to be focused on due to emissions from point sources under the base a reduction of point source emissions due to en- scenario and L is the corresponding N load dif,S0 hanced P elimination in sewage treatment – espe- attributed to non-point sources. f(mon) represents cially in Berlin – and the connection of all house- the average monthly reduction factor of non-point holds to WWTP. According to the MONERIS sim- N emissions in the Nuthe catchment obtained from ulations, the P emissions from soil erosion and sur- a comparison of the scenarios S1 and S0. face runoff are of minor importance, and the poten- From LUA (2002) information on the average tial for further reduction is therefore low. In terms contribution of point and non-point emissions to of P loads, a reduction to about 44 % of the current the total N load is available. Therefore, an ad- level would be required in order to meet the above vanced approach for determining the unknown mentioned quality target. fractions L and L in Eq. 4.40 from the dif,S0 pnt,S0 For the Spree River and the Teltowkanal, hydro- 22 Potsdam Institute of Climate Impact Research graphs of the reduced total P loads Ltot,S2 were 108 Chapter 4 Model application to the Lower Havel River derived from the hydrographs of the base scenario S0 using Eq. 4.44. In this context is the S0 Ltot,S0 S1 hydrograph of the total P load in the base scenario S2

S0 and L is the fraction of L which is ) pnt,S0 tot,S0 1 attributed to point source emissions (base load; see − L0 in Eq. 4.39). According to Eq. 4.44, the proportional reduction in the base load (factor 0.44) TP (mg l equals the desired relative reduction in total annual emissions. The parameter k describes the reduc- 0.1 0.2 0.3 0.4 tion in P loads which exceed the base load. k was high good moderate poor determined by optimization with the objective be- 2003 2006 2009 2012 2015 ing an average P concentration of 80 µg l−1 during April-September (quality target). The values found Figure 4.42: Simulated TP concentrations at the Ketzin for k are close to the reduction factor of 0.44 ap- gage (see Fig. 4.5) for the management scenarios S0–S2 grouped by years. The box represents the interquartile plied to the base load. range with the median value and the full range is shown by whiskers. The classes at the right plot margin refer Ltot,S0 = 0.44·Lpnt,S0 +(Ltot,S0 − Lpnt,S0)·k to the assessment scale from LUA (2005a) but note this (4.44) scale uses annual (arithmetic) mean values which are not shown.

4.6.4 Simulation results 4.6.4.1 Phosphorus seen in Fig. 4.43. Furthermore, Fig. 4.43 reveals that the proportional change in P levels from sce- Fig. 4.42 displays the statistics of the computed nario S0 to S2 is much greater in winter and spring TP concentrations at Ketzin for the 13 simulation than in summer, when internal loading strongly years. The results for scenario S1 differ not very controls the phosphorus concentration. much from the base scenario S0 but a pronounced reduction of the TP level was predicted for sce- According to the assessment scale from LUA nario S2. This can be attributed to the substantial (2005a) presented in Table 4.2, the phosphorus contribution of the Spree catchment to total P load- level in the base scenario can be classified as ing on the one hand (see Table 4.10) and the as- moderate–good. Under scenario S1, the good sta- sumed drastic decrease in loads on the other hand tus is reached in 8 of the 15 simulated years and (Sect. 4.6.2). under scenario S2 the criterion is fulfilled in every single year (see Fig. 4.44). When looking at a single scenario, for example S0, Fig. 4.42 indicates a significant variability be- However, Fig. 4.42 and Fig. 4.43 show that a tween years which is clear evidence for the strong clear cut classification is difficult because of the dependence of nutrient levels on the hydrological large interannual and seasonal variability. Fig. 4.43 conditions. The highest interannual differences ap- gives a good impression how representative the an- pear in the upper quartiles and maximum values. nual mean value of the total phosphorus concentra- This fact corroborates that phosphorus concentration actually is. From this point of view, an assess- tions in summer are particularly sensitive to flow ment scale which is not built on the annual average variability. alone but takes into account distribution informa- The increased variability in P concentrations in tion (e.g. quantiles) would permit a more reliable summer as compared to winter times can also be classification. 4.6 Management scenarios 109

The results presented so far all refer to the single station Ketzin located downstream of the Havel Lakes. However, the reduction in phosphorus concentrations is spatially variable as shown by the S0 maps in Fig. 4.45. For example, the change in S1 S2 the TP concentration at Ketzin under scenario S1 (upper map) is almost exclusively attributed to re- ) 1 − duced phosphorus loads in the Nuthe River. The lower map shows a very different situation for sce-

TP (mg l TP (mg l nario S2. Here, the assumed reduction in P loads in the Teltowkanal and the Spree River explains most

0.1 0.2 0.3 0.4 of the changes observed at Ketzin.

Jan Mar May Jul Sep Nov 4.6.4.2 Nitrogen

Figure 4.43: Seasonality of the simulated TP concen- Fig. 4.46 and Fig. 4.47 present the statistics of tration at the Ketzin gage for the three management sce- the simulated TN concentration at Ketzin for the narios S0–S2. three scenarios. The absolute reduction in the TN level from scenario S0 to S2 appears to be somewhat larger than twice the decrease predicted for S1. For both scenarios a greater impact was predicted during winter where the median concentration changes by about 0.2 and 0.5 mg l−1 for S1 and S2, in comparison to the base scenario S0. During summer, the predicted changes drop to about 0.1 and 0.2 mg l−1 for scenario S1 and S0 S2, respectively.

S1 These seasonal differences primarily result from the significant share of non-point emissions in total S2 N loading. During low flow in summer, the proportional impact of this type of emission declines in 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 favor of point source pollution. The significance Figure 4.44: Application of the water quality assess- of diffuse N sources can also be gathered from ment scale from LUA (2005a) to the simulated phos- the large interannual variability shown in Fig. 4.46 phorus concentration at Ketzin for scenario S0, S1, and which is greater than the corresponding variability S2 in 2003–2015. In accordance with the WFD convention, green color indicates the targeted ’good sta- in phosphorus concentrations (Fig. 4.42). tus’ while yellow stands for ’moderate’. The small in- As with phosphorus, the predicted proportional set rectangles show the status which results from apply- decline in the average N concentration is spa- ing the scale to the average TP concentration during the tially variable. As indicated by the upper map in vegetation period (April–September) rather than to the Fig. 4.48, the simulated shift in the concentration annual average. at Ketzin under scenario S1 (Fig. 4.46 & Fig. 4.47) is mainly caused by reduced nitrogen loads in the Nuthe and the Upper Havel River. In scenario S2, the impact of scenario S1 is superimposed by the 110 Chapter 4 Model application to the Lower Havel River

Obere Havel ∆TP (%) Spree Ketzin Berlin 0 7 Spandau 14 20 27 Branden− Potsdam 34 burg 41 Teltow− 48 kanal 55 Nuthe 61

10 km 5800000 5810000 5820000

3340000 3350000 3360000 3370000 3380000

Obere Havel ∆TP (%) Spree Ketzin Berlin 0 7 Spandau 14 20 27 Branden− Potsdam 34 burg 41 Teltow− 48 kanal 55 Nuthe 61

10 km 5800000 5810000 5820000

3340000 3350000 3360000 3370000 3380000

Figure 4.45: Simulated proportional reduction of the TP level during the vegetation period (Mai–Sep; average over 2003–2015) for scenario S1 (upper panel) and S2 (lower panel) compared to the base scenario S0. 4.6 Management scenarios 111

Obere Havel ∆TN (%) Spree Ketzin Berlin 0 2 Spandau 4 7 9 Branden− Potsdam 11 burg 13 Teltow− 16 kanal 18 Nuthe 20

10 km 5800000 5810000 5820000

3340000 3350000 3360000 3370000 3380000

Obere Havel ∆TN (%) Spree Ketzin Berlin 0 2 Spandau 4 7 9 Branden− Potsdam 11 burg 13 Teltow− 16 kanal 18 Nuthe 20

10 km 5800000 5810000 5820000

3340000 3350000 3360000 3370000 3380000

Figure 4.48: Simulated proportional reduction of the TN level during the vegetation period (Mai–Sep; average over 2003–2015) for scenario S1 (upper panel) and S2 (lower panel) compared to the base scenario S0. 112 Chapter 4 Model application to the Lower Havel River

assumed reduction of nitrogen emissions from the Spree catchment (Spree and Teltowkanal). An assessment of the simulation results which is conformal with the requirements of the Water S0 Framework Directive cannot be presented so far. S1 As opposed to phosphorus, a well founded as- S2

) sessment scale for nitrogen concentrations in the 1 − Lower Havel River does not yet exist. The approach of deriving a scale for nitrogen from that

TN (mg l TN (mg l for phosphorus (Table 4.2) using a critical N:P ratio (Kneis, 2005), was not adopted because the lat-

1.0 1.5 2.0 2.5 3.0 ter is merely a pragmatic and makeshift solution.

2003 2006 2009 2012 2015 4.6.5 Uncertainty of predictions

Figure 4.46: Simulated TN concentrations at the Ket- If any inferences are to be drawn from the results zin gage (see Fig. 4.5) for the management scenarios of the scenario simulations presented in Sect. 4.6.4 S0–S2 grouped by years. The box represents the in- one must keep at least two things in mind: terquartile range with the median as horizontal line. The First, the scenarios are subject to certain as- full data range is shown by whiskers. sumptions with respect to the reduction in nutrient emissions from different sources and subcatchments (Sect. 4.6.3). The scenarios must therefore not be confused with actual predictions of the future. Second, the results of model simulations are always associated with uncertainties. These uncertainties are the focus of the following sections.

S0 S1 4.6.5.1 Potential sources of errors S2 ) 1

− In the case of the model study presented, different sources of uncertainties can be distinguished. Some of them were already addressed in TN (mg l TN (mg l Sect. 4.4.4.2 and Sect. 4.4.4.3 but a more systematic overview shall be given hereafter. 1.0 1.5 2.0 2.5 3.0 Model structure In any model, only a subset of the state variables Jan Mar May Jul Sep Nov and interactions existing in nature can be repre- Figure 4.47: Seasonality of the simulated TN concen- sented. Even those most relevant processes which tration at the Ketzin gage for the three management sce- the model takes into account are usually imple- narios S0–S2. mented in a greatly simpliﬁed, conceptual manner. The degree of simpliﬁcation depends on many factors such as the understanding of interactions, the required accuracy, and the solvability of equations. 4.6 Management scenarios 113

It also depends on the spatial and temporal scales ence the model results during the whole simulation which determine data availability and the compu- period. tational effort. Observation data used in calibration In the application-oriented water quality model Apart from being used as initial conditions, obser- presented here, the structural simplification is far- vation data are required for model calibration. reaching. It starts with the use of the reactor con- During calibration, the model is driven by mea- cept (Sect. 3.1.1) which circumvents the need for sured boundary conditions such as discharges and solving the full transport equations. The chosen loads, with the latter being generated from ob- water quality submodels presented in Sect. 4.4 im- served concentrations interpolated in time. In ply even stronger abstractions of reality because rivers with large cross-section areas, flow mea- the complexity of ecosystems is far beyond that of surements are not very accurate but the larger un- purely physical systems. certainty usually results from interpolation of in- It is supposed that structural errors are the pri- frequently measured concentrations. In lowland mary reason for uncertainties in water quality sim- rivers like the Havel, the problem is less severe be- ulations. An analysis of structural uncertainties cause the short-scale variability in flow and con- would require a number of alternative model for- centrations is small. mulations to be compared. With respect to the ap- Furthermore, observed concentrations at moni- plication presented here, this is impractical for two toring stations are used for assessing the goodness reasons: Firstly, the information content of obser- of the simulation. Errors in the data result in dis- vation data is limited insomuch (fortnightly sur- torted optimum values for the calibration param- face water samples, sediment conditions at a sin- eters. In the case of the applied nutrient models, gle date) that the goodness of alternative and more measurement errors may, for example, result from complex model approaches cannot be assessed. spatial heterogeneity at the sampling site (surface Secondly, the effort for calibrating multiple nutri- blooms of algae, drift of organic sediment near the ent turnover models of different structure was too bottom) or incomplete pulping of samples. The lat- high to be undertaken in this work. Consequently, ter is a common problem when analyzing total nu- any simulation results must not be viewed indepen- trient concentrations. dently from the given model structure. Simulated boundary conditions During scenario simulations, TRAM’s boundary Initial values of state variables conditions were adopted from hydrological catch- TRAM requires initial concentrations of the sim- ment models rather than from observation data. ulated components to be specified and, in general, Of course, the simulation results of SWIM and spatially interpolated observation data are used. In ArcEGMO-Urban are uncertain too, e.g. due to the case of mobile components, the errors result- structural errors or inaccurate meteorological in- ing from improper measurement or interpolation put. Finally, the hydrographs of stage and flow fades during the simulation when the initial wa- which drive the mass transport through TRAM’s ter is replaced or mixes with new incoming water. network of reactors are also subject to errors, de- Measurement and interpolation errors may become pending on the quality of the hydrodynamic model severe in systems with long residence times or if, simulations. as was done here, sediment components are sim- A comprehensive analysis of uncertainties ulated. With respect to the phosphorus submodel would need all the errors in the connected models described in Sect. 4.4.3.3, errors in the initial P and to be taken into account. At present, this is im- Fe contents assigned to the lakes’ sediments influ- possible since sufficient information on the uncer- 114 Chapter 4 Model application to the Lower Havel River tainty of catchment modeling results is not avail- 4.6.5.2 Objectives of the analysis able. For a proper view on the results presented in Sect. 4.6.4 it is essential to know how reliable the Parameter values quantitative model output actually is. Because it The water quality modeling approaches presented is not practically feasible to analyze all the above in Sect. 4.4.3 contain a number of parameters with mentioned possible sources of errors, the following different theoretical foundation. For example, the restrictions apply: dimensionless temperature correction parameters θTN and θprem can only be determined by cali- 1. Only the uncertainty in parameter values and bration since measurement in the field is impos- initial conditions is inspected. The focus is on sible. Similarly, the nitrogen retention rate kTN those parameters which turned out to be par- cannot be directly measured as it lumps the effects ticularly sensitive during model calibration. of different processes. How reliably an optimum value for a calibration parameter can be identified 2. The analysis is limited to the phosphorus sub- depends on many factors such as it’s sensitivity model presented in Sect. 4.4.3.3. as well as the amount and variance of observation 3. The influence of uncertain parameters and ini- data. It also depends on whether a shift in the pa- tial conditions is analyzed for scenario S0 and rameter’s value can be compensated by adjustment S2 only (see Sect. 4.6.3). of another parameter. As exemplified in Fig. 4.34 even sensitive model parameters are not necessar- The following Sect. 4.6.5.3 addresses the specifica- ily well identifiable and different parameter sets tion of reasonable ranges to be considered for each yield a similar goodness-of-fit. input parameter or variable. As described in Sect. 4.4.4.3, other parameters Subsequently, Sect. 4.6.5.4 aims at figuring out of the phosphorus model such as the thickness of how the simulation results for scenario S0 change if parameters and initial values are sampled from the active sediment layer zs or the critical phos- the outer bounds of the conceivable ranges. This is phorus to iron ratio rP :F e were excluded from calibration and estimates were taken from the litera- done by a classical sensitivity analysis with only ture. Even though the adopted values were identi- a single parameter being altered at a time. The fied on similar waters, uncertainty remains whether simulation results obtained with modified param- a transfer from one system to another is actually eter values are compared to those being produced permissible. with the unmodified or ’standard’ values. Finally, Sect. 4.6.5.5 investigates how the simulation results for scenario S0 and S2 would change Numerical accuracy if the combined uncertainties in the model’s input Last but not least, the precision of simulation mod- parameters were taken into account. els is generally limited if numerical methods are involved. In TRAM, numerical imprecisions may appear in numerical integration (Sect. 3.5.2). Since 4.6.5.3 Considered parameters and ranges the system of ODE resulting from the chosen nitrogen and phosphorus turnover models presented Sediment phosphorus and iron content in Sect. 4.4.3.2 and Sect. 4.4.3.3 is simple and the While the iron content of the sediment is a con- processes do not act on very different time scales, stant model parameter, the sediment P concentra- even the Runge-Kutta solver yields a sufficient action is a dynamically simulated variable for which curacy. initial values must be supplied. In Sect. 4.4.4.3 it 4.6 Management scenarios 115 was described how the average Fe and P concen- Table 4.11: Width of the lower and upper 90% con- trations in the upper sediment of the Havel Lakes fidence interval for the mean value of Pas and F eas were derived from core samples. Because only 2– normalized by the corresponding means Pas and F eas (Eq. 4.45–Eq. 4.48). See Fig. 4.5 for lake labels. 4 cores are available for each lake, the uncertainty in the computed averages is high. The impact on P F e modeling results can be assessed if simulations are as as Lake label D D D D carried out with each lake’s P and Fe concentra- L U L U tion in the sediment being set to the upper or lower E 0.17 0.18 0.10 0.11 bounds of the 90% confidence intervals of the re- F 0.13 0.14 0.08 0.09 spective mean values. G 0.22 0.22 0.15 0.20 The application of this straightforward approach H 0.10 0.10 0.10 0.14 is hindered by the small number of 2–4 cores per Average 0.15 0.16 0.11 0.13 lake which does not permit a reliable estimation of confidence intervals. Fortunately, closely corre- Under the assumption that the observed ratios lated sediment parameters such as the acid-soluble DL and DU also hold for the data of interest, i.e. phosphorus and iron concentration (Pas, F eas) the total concentrations of P and Fe in the upper were measured at a greater number of locations (4– 20 cm of sediment, the range for the mean values 14 sites for each lake at 0–3 and 3–6 cm depth). P S and F e at a probability level of 90% can be For these data, lake mean values (Pas, F eas) and estimated from Eq. 4.49–Eq. 4.52. the associated lower and upper bounds of the 90% confidence interval were calculated, using a boot- 23 strap technique (Crawley, 2002) . Then, the L(P S) ≈ P S − P S · DL(Pas) (4.49) width of the lower and upper confidence interval U(P S) ≈ P S + P S · DU (Pas) (4.50) relative to the mean (DL,DU ) was computed according to Eq. 4.45– Eq. 4.48. L(F e) ≈ F e − F e · DL(F eas) (4.51) U(F e) ≈ F e + F e · DU (F eas) (4.52)

Pas − L(Pas) The obtained values for L(P S), U(P S), L(F e), DL(Pas)= (4.45) Pas and U(F e) represent the lower and upper estimates of P S and F e at p= 0.9 which can be used in the U(Pas) − Pas DU (Pas)= (4.46) uncertainty analysis. Combining the upper esti- P as mate of P S with the lower estimate of F e (or vice F eas − L(F eas) DL(F eas)= (4.47) versa) assumes that the errors in both P and Fe data F eas are independent. Since P and Fe are in fact corre- U(F eas) − F eas lated (see Fig. 4.49), the above mentioned combi- DU (F eas)= (4.48) F eas nations represent extreme cases with probabilities of about 1/4%. where L( ) and U( ) represent the lower and upper limits of the 90% conﬁdence intervals of its Thickness of the ’active’ sediment layer arguments. The obtained values are shown in Ta- The thickness of the active sediment layer con- ble 4.11. tributing to P export cannot be estimated from

23 available data on the investigated system and a Non-parametric estimates of the conﬁdence limits were vague literature value of 0.2 m had to be adopted. determined as the empirical 0.05 and 0.95 quantiles of the distribution of mean values obtained from 10000 bootstrap repli- In order to ﬁgure out the sensitivity of simulation cates. results, zs was altered by ±10 cm, i.e. values 116 Chapter 4 Model application to the Lower Havel River )

1 2 not tested because the model performance during − y = 0.284*x − 3.375, r = 0.72 the calibration period deteriorated significantly. Phosphorus release parameters As shown in Fig. 4.33, a number of parameter sets (kprem, θprem) fitted the observed P concentrations in the calibration period equally well. The GLUE philosophy (Beven, 1993, 2000) suggests carrying out a simulation with each parameter set unless it is 0 2 4 6 8 10 12 acid soluble P (mg (g TM) classified as ’unbehavioral’25. By interpreting the 0 10 20 30 40 50 goodness-of-fit for each parameter set as a measure acid soluble Fe (mg Fe (g TM)−1) of its likelihood, a quasi-distribution function can be obtained from which – similar to confidence in- Figure 4.49: Correlation of acid soluble iron (F eas) tervals – uncertainty estimates can be derived. Al- and phosphorus (Pas) in the Havel sediments. The 90% though the GLUE method does not yield probabili- confidence interval of the regression is shown as dotted ties in a strict sense, it provides a simple and trans- line. parent framework for (non-parametric) uncertainty estimation. of 0.1 and 0.3 m were tested. The choice of a In order to make the uncertainty in the values of much larger value has little effect as the phospho- kprem and θprem accessible with minimum com- rus excess in the sediment declines with increasing putational effort, simulations were carried out with depth (Fig. 4.32). A much smaller value seems un- four selected parameter sets only. These sets were likely as numerical experiments with a 1D diffu- selected based on Fig. 4.50. The four combinations 24 sion model have shown. of kprem and θprem represent the intersections of the border of ’zone 1’ with the major axes of an Threshold P:Fe ratio imagined ellipse. That is, one set combines low The threshold phosphorus to iron ratio is another values of kprem and θprem, a second set assumes highly uncertain parameter. The sensitivity of the large values for both parameters and the two other model to altered values of rp:fe directly follows sets combine small and large values. All four sets from its structure (Sect. 4.4.4.3). The assumed are equally likely with respect to their associated standard value of 0.025 (mass ratio) was varied by goodness-of-fit. ±0.015. The lower estimate of 0.01 is near to the A summary of the test values for the parameters value found by Maassen et al. (2005) and much zs, rp:fe, and upset is given in Table 4.12. higher values seem unlikely according to Fig. 4.31.

Effective settling velocity 4.6.5.4 Sensitivity of individual parameters For the scenario simulations, a standard value of Fig. 4.51 displays the proportional change in the −1 upset= 0.04 m d was used (Sect. 4.4.4.3). For as- simulation results for scenario S0 that results from sessing the impact of increased or lowered settling modifying the parameters upset, zs, rP :F e and the of phosphorus, simulations with upset= 0.02 and initial sediment concentrations of P and Fe individ- −1 0.1 m d were carried out. Higher values were ually. For each model conﬁguration with altered

24A simple explicit finite-difference solution of Fick’s sec- 25The term ’unbehavioral model’ is used for parameter ond law was implemented in a spreadsheet program. Only combinations that yield a low goodness-of-fit, e.g. efficien- molecular diffusion was considered. cies below zero or some positive threshold. 4.6 Management scenarios 117

As Fig. 4.51 shows, the sensitivity of results is Zone 2 Zone 4 not equal for all years of the simulation period because of the variability in ﬂow dynamics. Further- more, the deviation between the simulated concen- prem

θ trations obtained with standard and modiﬁed val- Zone 1 Zone 3 ues for zs, rP :F e, and the initial P and Fe levels increases with time. This is because these parameters basically control the quantity of the sediment 1.0 1.4 1.8 phosphorus excess. If the P excess is assumed to 0.00004 0.00008 0.00012 be larger (high zs, low rP :F e, high initial P:Fe ra-

kprem tio), its reduction takes more time and the rate of P remobilization declines more slowly. In contrast, Figure 4.50: Reclassified response surface of the phos- if the sediment phosphorus excess is assumed to phorus model with respect to the parameters kprem be lower, the simulated P concentrations decline and . Zone 1: & θprem E > 0.95 · max(E) M < faster. Therefore, the deviations in the statistics 1.05 · min(M), Zone 2: E > 0.9 · max(E) & M < 1.1 · min(M), Zone 3: E > 0.75 · max(E) turn more and more negative for a lowered value & M < 1.25 · min(M), Zone 4: remaining sets. In this of zs, an increased rP :F e, or a low initial P:Fe ra- definition, E is the Nash-Sutcliffe efficiency, M is the tio. mean absolute percentage error, and min & max are the lowest & highest observed values for the two goodness- Fig. 4.52 illustrates how the predicted P concen- of-fit measures. The locations of the four selected pa- trations under scenario S0 would change, if the rameter sets are marked by circles. simulations were carried out with near-optimum sets of kprem and θprem (see Fig. 4.50) instead of the true best-fit parameter set. Like zs and rP :F e parameters or initial values, the two main calibra- in Fig. 4.51, the parameters kprem and θprem act on the magnitude of phosphorus release in sum- tion parameters kprem and θprem were fitted anew. mer. Hence, a modification of the values primar- Since only a single parameter was altered at a ily causes a shift in the simulated 90% percentile time, the sensitivity of results with respect to that while the 0.1 quantile is much less affected. particular parameter can be assessed. By compar- While the sensitivity of simulation results ing the individual sensitivities, it can be figured out against altered values of zs and rP :F e or initial what additional information is required in order to P and Fe concentrations increased over the simu- reduce the uncertainties most effectively. lation period (Fig. 4.51) an opposite trend is ob- According to Fig. 4.51, the simulation results served if kprem or θprem are modified (Fig. 4.52). are most sensitive to an increase in the phospho- This can be explained by the significant decline in rus settling velocity (upset, uppermost panel) and the sediment’s P excess over time, causing the im- altered values of the sediment’s phosphorus bind- pact of P release on the concentrations to fade more ing capacity related to iron (rP :F e, third panel from and more. Consequently, the sensitivity of the con- the top). In the case of modified values for upset, trolling parameters kprem and θprem must decrease the 10% percentile is most affected (low concentra- too. tions in winter and spring). In contrast, changes in According to Fig. 4.52, the change in the sim- those parameters and variables which control the P ulation results is largest if the values for kprem remobilization from sediments primarily affect the and θprem are both increased or lowered simulta- 90% percentile (late summer concentrations). neously. This is a result of parameter compensa- 118 Chapter 4 Model application to the Lower Havel River