Development of a Method for the Rapid Assessment of Flushing Conditions of Coastal Bays

MSC. THESIS

RAPID ASSESSMENT OF FLUSHING OF COASTAL EMBAYMENTS

- WITH APPLICATION TO DOHA BAY -

M.E.A. VAN DER VEN JULY 8, 2014

Milou Elisabeth Anne van der Ven

RAPID ASSESSMENT OF THE FLUSHING OF COASTAL EMBAYMENTS

- With application to Doha Bay -

THESIS

For the degree Master of Science (MSc) in Hydraulic Engineering, Faculty Civil Engineering, Delft University of Technology

AND

For the degree Master of Science (MSc) in Water Resources Management, Faculty of Civil and Environmental Engineering, National University of Singapore

In cooperation with Royal HaskoningDHV (Amersfoort) and Deltares (Delft)

Amsterdam, July 8, 2014

SUPERVISORS

Prof. dr. ir. M.J.F. Stive TU Delft

Dr. ir. R.J. Labeur TU Delft

S. Pande TU Delft

Dr. V. Shua NUS

Dr. L. Y. Min NUS

Ir. J.H. ter Hoeven RoyalHaskoningDHV

Ir. R. Morelissen Deltares

III

ACKNOWLEDGEMENTS With the completion of this Master Thesis, the Double Degree Program has come to an end. These past years in which I have studied both at the TU Delft and at the National University of Singapore were very intensive, yet most rewarding. My semester in Singapore was a challenging and unique experience from which I have learned a lot about science and hydraulic engineering in particular, but also about other cultures and about myself.

I would hereby like to show my gratitude to Paul Visser, who made this incredible journey possible. I sincerely appreciate it that you had faith in me and gave me the chance to participate in this great program.

The past months of which I have spent most of my time at Royal HaskoningDHV have also been an interesting experience in which I have learned a lot about Royal HaskoningDHV and the work of engineering consultants. I would like to thank my supervisor Joost ter Hoeven for giving me the opportunity to graduate within the department of ‘Hydraulica en Morfologie’. Moreover, throughout these months you have given me the guidance and support that were necessary to make this project to a success. You have particularly helped me in distinguishing the relevant matters from those of less importance.

Furthermore, I would like to thank my colleagues at Royal HaskoningDHV, who made feel at home and were always willing to help. Special thanks go to Lars, who helped me a lot with the numerical flow model.

In addition, I would like to thank my supervisor at Deltares: Robin Morelissen. Your enthusiasm for this project was very encouraging and our weekly discussions were often necessary to put me back on track in times of struggle. Being the voice of the future user of my developed method, you have taught me to look at the problem from a more practical or engineering point of view.

I would also like to express my sincere gratitude to my TU Delft mentor Robert Jan Labeur, who pushed me to raise the project to a higher level. Your guidance and extensive feedback were extremely valuable. You have taught me to approach the problem scientifically. Moreover, you have helped me enormously with the structure and contents of the report. I truly appreciate all the time and effort you have spent to help me to bring this report to a higher level.

Furthermore, I would like to thank my professor Marcel Stive, who has had a crucial role particularly in defining the project objective and in making the approach more concrete. I greatly appreciate it that, despite the pressure on your agenda, you really took the time to help me out through this difficult project phase.

I would also like to thank Saket Pande, Dr. Vivien Chua and Dr. Low Ying Min for being part of my graduation committee and making this graduation happen.

Last but certainly not least, I would like to thank Carel, my parents and my sisters. Your endless support, encouragements and expressions of faith were necessary to bring this project to an end. I truly appreciate this.

Milou van der Ven

Amsterdam, July 2014

EXECUTIVE SUMMARY Extensive developments in coastal waters form a potential threat to the water quality worldwide. Most developments result not only in an increase of waste discharges but they might also influence the bay’s flushing conditions. The bay’s flushing can be described as the removal of pollutant concentrations by means of hydrodynamic transport, governed by the exchange of bay water with the external water body (CHOI et al., 2004). Because water is the carrier of (polluted) material, this exchange or flushing is a crucial determinant in the water quality (TAKEOKA, 1984).

To make sure that the developments do not negatively affect the flushing conditions and inherently the water quality, it is therefore important to take the possible consequences of the intended changes to the bay’s flushing into account in the designs. However, no rapid assessment method is available yet, which can be used for engineering practises to obtain quick estimates regarding the flushing conditions in coastal bays under the influence of human developments in the bay. Either the computational time and amount of data required are too large or the level of accuracy is too low. Therefore the objective of this study was to develop a method for the rapid assessment of the influence of intended changes to a coastal bay on its flushing conditions. The study was focused on tidally dominated, small-scale, shallow water coastal bays.

Through an extensive literature review, first the relevant physical processes to flushing were detected. Next, possible indicators were described and analysed in order to find the most suitable way to quantify the flushing conditions. It was found that for applications to environmental problems, it is common and suitable to quantify the exchange process by the use of the relevant transport time scale(s). Furthermore, when the total bay’s flushing conditions are concerned, the average residence time as defined by TAKEOKA (1984) was shown to be the most suitable transport time parameter. The average residence time represents the average time a single particle is situated in the bay.

To define the approach towards the development of the rapid assessment method, the available methods to calculate the average residence time were elaborated. Methods based on a box, one-dimensional or two- dimensional representation of the bay were distinguished. It was decided that the rapid assessment method should be of the box type because this leads to the most rapid calculations. The relevant flushing processes which were not represented in the original box models can be accounted for through basic bay parameters of which the influence on flushing is fundamentally derived by the use of a more detailed, two-dimensional model.

To assess the relation between the chosen relevant basic parameters (e.g. average bay depth, average tidal amplitude) to the average residence time of the bay, numerical calculations are carried out using a schematic 2D model of a coastal bay. Following the systematic approach of the dimensional analysis, the relevant relations between the dimensionless bay parameters and the dimensionless flushing parameter were successfully detected. Based on these relations the following analytical expression of the average residence time was derived:

̅ ( ) ( ) ( ) ( ) ( ) ( ) √ √ √

In which ̅ [days] is the average residence time, [days] the tidal period, [m] the bay’s width, [m] the 2 2 bay’s length, [m] the average bay depth, [m ] the total bay’s surface area, [m ] the cross-sectional 2 inlet area, [m] the Nikuradse roughness number, [m /s] the dispersion coefficient and [m] the tidal amplitude.

Because the value of the dispersion coefficient remains a factor of uncertainty, the most practical approach for the use of this proposed rapid assessment method (RAM) is to estimate the value of the dispersion coefficient and keep this value the same for the different cases of which the flushing conditions are compared. Inherently, this approach leads to the exclusion of the rate of change of internal mixing due to the development (or bay change) of interest.

The use of the derived formulation as a RAM was first validated by application of the RAM to three real cases (Kuwait Bay, Boston Harbour and Venice Lagoon), using the results from detailed flushing studies of the areas in question as a reference. From the resulting values of the dispersion coefficients that were found to reflect the ‘realistic’ average residence times, it was concluded that the RAM performs reasonable in the cases of Kuwait Bay and Boston Harbour. However, the RAM appeared inapplicable to Venice Lagoon, which is attributed to its very shallow water depths.

Finally, the performance of the RAM was further examined by application to Doha Bay. The influence of a number of bay changes on the bay’s flushing conditions is assessed, under which the future planned developments Sharq Crossing and Oryx Island. The predictive value of the rapid assessment method is analysed by comparing the estimates of (the relative change of) the average residence times by the RAM to those values obtained by the use a detailed numerical (depth-averaged) flow model.

From the results of the test cases it was concluded that the RAM overestimates the effect of the tidal pumping mechanism which leads to relatively low absolute values of the calculated average residence times resulting from tidal flushing. This is one of the reasons why one should be very careful in putting any value on the absolute numbers derived by the RAM. However, the RAM has shown to be a useful tool for the estimation of the relative rate of change of the average residence time of the bay as a result of bay developments, under the following assumptions:

 The development is of such a large scale that it has a significant influence on the value one or several of the basic bay parameters in the RAM.  A relatively deep and narrow inlet channel is present.  The bay has one inlet or several inlets of a similar scale (all with deep inlet channels); in the case of more than one inlet, a separated approach is recommended.

Because the presence of Oryx Island leads to a situation in which the above listed assumptions are valid, the influence of Oryx Island on the flushing conditions was successfully predicted by the RAM. Contrary, the estimation of the influence of Sharq Crossing was not possible with the use of RAM because of its relatively small scale.

The poor performance of the RAM to areas that are flushed by a shallow inlet leads to the conclusion that the influence of the inlet configuration on the flushing behaviour is not represented well by the RAM. Further research to the near-inlet processes is therefore highly recommended. It is suggested that when the influence of the inlet width and the inlet depth are considered separately (instead of integrative through the cross- sectional inlet area), the representation of the processes near the inlet by the RAM can largely be improved.

TABLE OF CONTENTS

Acknowledgements ...... IV Executive summary ...... V Table of Contents ...... VII 1 Introduction ...... 1 1.1 Problem description ...... 1 1.2 Objective ...... 3 1.3 Approach ...... 4 2 Theory of flushing of coastal bays ...... 5 2.1 Introduction to Doha Bay ...... 5 2.2 Classification of a coastal bay ...... 6 2.3 Tidal flushing ...... 7 2.4 Transport and mixing processes ...... 8 2.5 Residual currents...... 10 2.6 Mathematical description of the transport processes ...... 13 2.7 Discussion ...... 17 3 Quantification of the flushing process ...... 18 3.1 Transport time scales ...... 18 3.2 Calculation methods ...... 20 3.3 Discussion ...... 24 4 Schematic 2D model tests ...... 25 4.1 Approach ...... 25 4.2 Dimensional analysis ...... 25 4.3 Set up of schematic model ...... 28 4.4 Calculation of the average residence time ...... 32 4.4 Test program ...... 33 4.5 Results ...... 37 4.6 Method performance ...... 44 4.7 Discussion ...... 48 5 Validation ...... 49 5.1 Approach ...... 49 5.2 Kuwait Bay ...... 50 5.3 Boston Harbour ...... 52

VII

5.4 Venice Lagoon ...... 53 5.5 Discussion ...... 54 6 Application to Doha Bay ...... 56 6.1 Case description ...... 56 6.2 Model description ...... 58 6.3 Approach ...... 60 6.4 Test program ...... 61 6.5 Results ...... 68 6.6 Discussion ...... 82 6 Conclusions and recommendations...... 84 7.1 Conclusions ...... 84 7.2 Recommendations ...... 85 References ...... 87 Appendices ...... 89 Appendix A – Complementary data of Doha Bay and the Arabian Gulf ...... 89 Appendix B – Classification of inland coastal waters ...... 94 Appendix C – Hydrodynamic modelling of the Arabian Gulf ...... 95

VIII

1 INTRODUCTION

1.1 PROBLEM DESCRIPTION

1.1.1 WATER QUALITY ISSUES IN DOHA BAY The direct motivation to conduct the present study is the need to understand the consequences of future planned developments (Sharq Crossing and Oryx Island) in Doha Bay on the local water quality. Doha Bay is a relatively small-scale, shallow water bay, located at the southern beach of the Arabian Gulf (Figure 1.1) and surrounded by Qatar’s capital city Doha.

Qatar’s rapid industrialization and related economic growth, has led to increased human activity of all kinds in and near Doha Bay (RICHER, 2009), accompanied by emissions from, amongst others, sewage treatment-, power-, and desalination plants. The result is an increase of waste discharges in Doha Bay, forming a potential threat to nearby ecosystems, human health and the quality of life through the degradation of the water quality (RICHER, 2009). One example is the disposal of brine which is the hyper saline by-product of desalination processes. The presence of brine can have a significant impact on plankton by causing a drop in osmotic pressure. In addition, due to its turbidity, the light filtered through the water column will be reduced which can affect seagrasses and algae (PALOMAR AND LOSADA, 2011).

1.1.2 FLUSHING AS INDICATOR FOR WATER QUALITY The impact of the waste discharges in Doha Bay or any other coastal embayment to the water quality and inherently the environment is determined by a combination of numerous factors related to the type and intensity of the loading, the resistance and vulnerability of the potentially affected environmental component and the flushing characteristics (CHOI et al., 2004). Flushing refers to the removal of (pollutant) concentrations by means of hydrodynamic transport, governed by the exchange with the outer sea (CHOI et al., 2004), such as the Arabian Gulf in the case of Doha Bay. Because water is the carrier of (polluted) material, this exchange or flushing is a crucial determinant in the water quality (TAKEOKA, 1984).

FIGURE 1.1 – LOCATION OF DOHA BAY IN THE ARABIAN GULF AND THE LAY-OUT OF DOHA BAY

1.1.3 IMPACT OF FUTURE DEVELOPMENTS TO THE FLUSHING CHARACTERISTICS OF DOHA BAY To cope with the expected increase of traffic intensity in Doha, plans are on the table for the construction of the bridge-tunnel Sharq Crossing. The design by architect Calatrava comprises three bridges which are interconnected and connected to shore by a total of five submerged tunnels (Figure 1.3). In addition, the reclamation of a new island group with residential and recreation functions has been proposed: Oryx Island (Figure 1.3). In the light of the water quality these developments are concerning in two ways. First, they will attract and accommodate more human activity wich leads to the addition of waste disposal in the bay. From an engineering perspective it is more important how these developments impact the flushing characteristics of the bay. Due to the presence of Sharq Crossing and Oryx Island the bathymetric and/or geometric configuration of the bay will change which will in turn influence the flushing conditions. Given the impact of the flushing process on the water quality, any negative effect on the bay’s flushing is unwanted. It is therefore important to take the possible consequences of the proposed developments to the bay’s flushing characteristics into account in the design.

FIGURE 1.2 – RECENT DEVELOPMENTS IN DOHA BAY. LEFT: PEARL ISLAND; RIGHT: AIRPORT EXTENSION1

FIGURE 1.3 – PLANNED DEVELOPMENTS IN DOHA BAY. LEFT: SHARQ CROSSING; RIGHT: ORYX ISLAND

1 From Google Earth 2

1.1.4 THE NEED FOR A RAPID ASSESSMENT METHOD FOR FLUSHING For the proposed developments in Doha Bay, a flushing assessment will probably come too late: possible negative effects cannot be avoided any more because the designs have already been finalized. To take the impact on flushing into account in the design, a flushing assessment should have been carried out in earlier stages of the design process in which the design can still be adjusted if necessary. However, in practice this appears to be a complicated task.

Doha Bay is just one example of many coastal bays worldwide to which extensive developments form a potential threat to the water quality and where the answers regarding the impact on the bay’s flushing come too late to account for by changing the design. This can be attributed to the fact that the available flushing assessment methods appear unsuitable for application in early stages of the design process.

To keep up with the pace of the design process, it is paramount that the insight in the consequences of a potential design to the flushing conditions is obtained quickly. In addition, in early project stages often minimal data on the bay’s configuration and ambient conditions is available. A flushing assessment should therefore be based only on basic parameters representing the bay and external forcing. Several analytical based methods exist that meet these requirements, however the reliability of these models is subject to discussion as they rely on doubtful assumptions and do not account for the relevant physical processes associated with flushing. The alternative is to use detailed numerical models which are able to represent the relevant processes more accurately, but a rigorous amount of data and time is required to set up such a model and the computations are time consuming.

To conclude, no rapid assessment method is available yet, which can be used for engineering practises to obtain quick estimates regarding the flushing conditions in coastal bays under influence of human developments in the bay. Either the computational time and amount of data required are too large or the level of accuracy is too low. This study therefore aims to develop such a rapid assessment method which is applicable to Doha Bay and similar types of coastal bays. Doha Bay is used as a prototype for a category of bays.

1.2 OBJECTIVE

1.2.1 MAIN OBJECTIVE The main objective of the present study is:

‘To develop a method for the rapid assessment of the relative influence of geometric or bathymetric bay changes on the flushing characteristics of coastal bays.’ , applicable to Doha Bay and any other coastal water body of similar type.’

Typical characteristics for the coastal water bodies considered in this study are listed below. For further explanation of the assumptions refer to section 2.2.

 ‘Bay’ type of coastal water  Tidal dominance  Small topographical scale  Shallow water depth  One inlet  Dispersion dominated

1.2.2 SUB-OBJECTIVES The following sub-objectives are considered:

 To enhance the understanding of the processes of tidal flushing; to detect and describe the relevant physical processes involved.  To choose the most suitable indicator for flushing in the context of the rapid assessment method.  To analyse the available methods to calculate this chosen indicator; to detect and describe the areas of application and the limitations.  To define an approach for the development of the rapid assessment.  To follow the defined approach. This leads to the proposal of a rapid assessment method.  To validate the proposed method; to discover its field of application and its limitations.  To use the developed method to predict the influence of the proposed developments in Doha Bay.

1.3 APPROACH The study comprises roughly three parts: literature review (section 2), development and validation of the rapid assessment method (sections 3, 4 and 5) and the case study Doha Bay (section 6).

First, the mechanism of tidal flushing of coastal bays is thoroughly described based on literature review. This comprises the description of the general flushing mechanisms, the associated physical processes and its mathematical representation.

Next, the possible ways to indicate and quantify the exchange conditions of a coastal bay are explored. After an elaboration of common used indicators, the most relevant indicator in the context of this study is chosen. In addition, the available methods to estimate this indicator are analyzed, based on which a suitable framework for the rapid assessment method is determined and the approach to the development of such a method is defined.

The proposed rapid assessment method is then validated by comparison to three real cases for which highly detailed flushing studies are carried out.

As a final test, the validated method is applied to the situation in Doha Bay, using a detailed numerical model as a comparison. The performance of the method is investigated through several cases in which certain bay features are changed. The final two cases comprise the future developments in Doha Bay: Sharq Crossing and Oryx Island respectively. By comparing the results of the rapid assessment method with those derived through the detailed numerical model, the predictive value of the rapid assessment method is investigated.

2 THEORY OF FLUSHING OF COASTAL BAYS To develop a method for the rapid assessment of flushing, first the process of flushing of a coastal embayment should be fully understood. Therefore in this chapter a theoretical description of the flushing process is given, identifying the driving forces and the physical processes that determine the exchange. Because this study is restricted to the flushing of a category of bays with similar characteristics as Doha Bay, the chapter starts with a brief outline of the situation in Doha Bay, based on which assumptions are made regarding the type of bay considered in this study. When this classification is made, the chapter continues with a qualitative description of the flushing process and the physical transport processes involved, followed by the mathematical representation of the latter processes.

2.1 INTRODUCTION TO DOHA BAY In this paragraph a brief outline is given of the general characteristics of Doha Bay. Unfortunately minimal data is available on the local forcing and the flow conditions in Doha Bay. Therefore the general characteristics are derived from the typical climate and flow conditions of the Arabian Gulf. Estimates of the local conditions in Doha Bay are further obtained from a numerical 2D flow model of Doha Bay, which is developed by Royal HaskoningDHV through nesting procedures in the well-calibrated UAE Regional Model by Deltares (section 6). For more information refer to section 6 in which the case is thoroughly described.

Different from most hydrological studies for engineering purposes, in flushing studies the focus generally does not lie on the extreme conditions. It is not about a structure that should withstand a certain wave height, but the purpose is to represent the system’s flushing behaviour. Because of the typical long-term character of the environmental problems concerned with in the context of flushing of coastal bays, the common or average (e.g. seasonal or yearly) conditions are more relevant.

2.1.1 AREA DESCRIPTION Doha Bay is a roughly crescent shaped, half-open coastal bay (Appendix A) located at the eastern coast of the Qatarian peninsula in the Arabian Gulf. The bay is connected with the Arabian Gulf by two inlets: one relatively wide inlet between the airport and Alsafliya Island which accommodates a deep channel to serve the inner bay harbour, and one narrow and shallow inlet between Alsafliya Island and Pearl Island (Figure 1.1). With a surface area of about 45 km2, its topographical scale is considered to be relatively small (when compared to the length scale of the tidal wave). Further, with an average depth of 5.7 m and local depths of only 2 m, Doha Bay is referred to as a shallow water bay.

2.1.2 HYDRODYNAMIC BEHAVIOUR In the absence of a river inflow, the hydrodynamics in the bay are assumed to be primarily forced by tides and wind. Potential density driven currents induced by differences in salinity as a result of the high evaporation rates in the area are assumed to have a less pronounced effect on the hydrodynamics than the tidal force and wind stress. The tidal signal is of the mixed type as it is mainly composed by the K1 (diurnal) and M2 (semi- diurnal) constituents. The average tidal range is about 1.2 m (Appendix A). Because the wind has a mild character throughout most of the year, the tide is considered to be the dominant forcing type. This might however not be the case during occasional high wind speeds caused by the seasonal event called ‘Shamal’ (section 6).

2.2 CLASSIFICATION OF A COASTAL BAY In principle the primary factors controlling a flushing process are tidal forcing, wind forcing, river inflow, density gradients and the geometry and bathymetry of the embayment (ZHEN-GANG JI, 2008). However, which forces and processes drive the exchange depends on the presence and dominance of the different forcing types, the unique geometric and bathymetric configuration of the considered water body and the prevailing transport processes resulting from these. Therefore it is crucial in the context of present study to clearly define the relevant characteristics of the category of water bodies considered.

Based on the characteristics of Doha Bay (section 2.1) the following assumptions are made regarding the type of coastal water this study is focused on:

 ‘Bay’ type – Coastal waters of the ‘bay’ type are considered, with the definition adopted from KJERFVE AND MAGILL (1989) (Appendix B). Basically, this means that the study is focused on water bodies which cannot be classified as one of the more distinct types such as an estuary, a lagoon, a fjord, a tidal river or a strait.  No fresh water inflow – River inputs are not present or negligible.  Tidal dominance – The dominant forcing type is the tide. Influences of gravitational forces and wind stress are not taken into account.  Shallow water depth – Bays with average depths in the order of 5-10m are considered. The shallow water assumptions lead to the exclusion of the influence of Coriolis, low-frequency water level variations and resonance effects.  Small topographical scale – The spatial scales of the tidal wave are much larger than the spatial scales of the bay which results in a situation in which the bay will be filled and emptied by the tide, rather than having a tidal wave propagating inside the bay causing water level differences within it.  Vertically well-mixed – Given the fact that the water is shallow and the bay experiences negligible fresh water input, the bay is assumed to be well-mixed throughout the depth.  No large scale circulations – Because the typical bay considered is characterized as tidally dominant, the influence of large scale circulations are assumed to be negligible, as these are mainly induced by wind and vertical stratification (GEYER AND SIGNELL, 1992). This leads to the assumption that the transport is dispersion dominated, which will be explained in section 2.6.5.  One inlet: The focus of this study is on bays with one (main) inlet.

Based on these assumptions, the following schematic representation of a typical coastal bay is created (Figure 2.1):

FIGURE 2.1 - SCHEMATIC REPRESENTATION OF A TYPICAL COASTAL BAY. WITH INDICATION OF THE BAY’S SURFACE AREA (Ab), THE

AVERAGE BAY DEPTH (db) AND THE TIDAL AMPLITUDE (at) AND TIDAL PERIOD (Tt). 6

2.3 TIDAL FLUSHING The flushing of a coastal water body can be described as the bay’s removal from (pollutant) concentrations by means of hydrodynamic transport, causing the exchange of fluids and constituents (FISCHER et al., 1979; CHOI, 2004; MONSON et al., 2002). This requires an exchange flow which can be driven by the astronomical tide.

2.3.1 TIDAL PRISM Tide-induced exchange is provided by tidal emptying and filling of coastal bay, the so-called vertical tide. The tidal prism, which is defined as the difference between the volume of an embayment during high and low 3 water, denotes the total volume of water [m ] which can be exchanged between bay and sea during one tidal cycle (Equation 2.1). The tidal prism forms the basis of most analytical tidal flushing models (SANFORD et al., 1992).

2.1

2 Where [m ] is the basin surface area, [m] is the tidal range which equals the water level difference between high and low water.

2.3.2 FLUSHING MECHANISMS The tidal current or horizontal tide is the current resulting from the tidal water level fluctuations. Because the horizontal tide in principle just moves the same volume of water (the tidal prism) in and out the basin, additional mixing and transport mechanisms within the bay and near the entrance are necessary for the actual exchange. The following three mechanisms that together determine the effectiveness of the tidal exchange flow can be distinguished:

A. Mixing: Mixing of the incoming sea water with (polluted) bay water. During ebb the diluted water will exit the bay. Each tidal cycle the bay will be further diluted in this way. B. Exchange: Transporting the incoming water along different paths than outflowing water. This will lead to very effective exchange as (hypothetically) ‘pure’ sea water will be exchanged with ‘pure’ (polluted) bay water. C. Prevention of return flow: Transporting the volume of water which has left during ebb away from the inlet, preventing its return during the following flood tidal cycle: decreasing the so-called ‘return flow’.

Mechanism A and B are merely induced by the (tide residual) motions resulting from the interaction of tidal current with bathymetry or geometry and associated ‘physical’ mixing and transport processes, which are described in sections 2.4,2.5 and 2.6. Mechanism C is mainly dependent on the seaward flow conditions, which will not be the focus of this study. Both mechanisms B and C are in addition largely driven by the typical flow patterns that occur near the inlet, which are discussed in section 2.5.3. The mechanisms are schematically represented in Figure 2.2.

FIGURE 2.2 – VISUALISATION OF THE THREE FLUSHING MECHANISMS. (LIGHT) RED IS POLLUTED WATER, (LIGHT) GREEN IS SEA WATER. 7

2.4 TRANSPORT AND MIXING PROCESSES Physical transport and mixing processes can be divided into advection and diffusive processes and advection. Diffusion denotes the spatial spreading of concentration induced by concentration gradients and advection is the transport of dissolved substances by the current. Strictly speaking diffusion occurs only on a molecular scale. Larger scale processes which are denoted as diffusion or similarly dispersion however show close analogy to molecular diffusion. In this paragraph molecular diffusion, the processes of turbulent diffusion and shear flow dispersion are described.

2.4.1 MOLECULAR DIFFUSION Molecular diffusion is the spreading of concentration as a result of the collective activity of many ‘randomly’ moving molecules (FISCHER et al., 1979). Due to the kinetic energy molecules or particles experience in fluids, arising from the fluid’s temperature, the molecules are in constant movement and therefore successively colliding with one another. Collision with other molecules causes the particles to change direction which eventually results in particles traveling by the ‘random walk’, which refers to the randomness of the path the molecule travels (Figure 2.3). Despite this single particle trajectory is observed as totally random and consequently unpredictable, the resulting behaviour of many particles moving randomly together can be effectively predictable (FISCHER et al., 1979).

ADOLPH FICK (1855) described the resulting behaviour as the movement of (dissolved) material from areas of high concentration to areas of low concentration (FISCHER et al., 1979). This leads to the smearing of concentration gradients (or dilution of the concentration), which is generally referred to as diffusion (Figure 2.3). On the scale of molecules we speak of molecular diffusion. Due to the very large time scales and very small spatial scales of this process, the contribution of molecular diffusion to flushing of coastal bays is negligible (FISCHER et al., 1979). However its formulation shows great analogy to the larger scale mixing processes turbulent diffusion and shear flow dispersion (FISCHER et al., 1979). In addition, the concept of the random walk will prove its value when describing transport on the scale of the bay (section 2.6.4).

FIGURE 2.3 –EXAMPLE OF A RANDOM WALK2 (LEFT) THE SPREADING OF DISSOLVED CONCENTRATION BY MOLECULAR DIFFUSION TURBULENT DIFFUSION (RIGHT) 3.

2 From ‘http://butler.cc.tut.fi/~vattula2/teaching-bf/LECTURE03-Nelson-Chapter04-2012.pdf’ 3 From ‘http://www.mhhe.com/biosci/ap/ap_prep/bioC2.html’ 8

2.4.2 TURBULENT DIFFUSION Turbulent diffusion is the result of the combined effect of turbulent flow and molecular diffusion (FISCHER et al., 1979). Turbulence can be characterized by ‘random’ or chaotic velocity and pressure profiles, resulting in the stochastic behaviour of the flow (ROBERTS AND WEBSTER, 2002). Turbulent flow consists of various levels of fluctuating vorticity that can be observed as curved motions in the flow, referred to as eddies or vortices (Figure 2.4). Turbulence can be generated by the tidal current interacting with bottom friction, or if present by wind stress acting on the water surface.

The presence of turbulence enhances the processes of mixing and transport (Figure 2.4): mixing or spreading by turbulent diffusion occurs more rapidly and on larger spatial scales than through by molecular diffusion (FISCHER et al., 1979). This can be attributed to the sharp concentration gradients induced by the smaller scale turbulent eddies which will distort a pollutant plume. These gradients will be smeared by molecular diffusion, causing the volume of the cloud to grow. Turbulent flow thus enlarges the area over which sharp concentration gradients occur, causing the process of molecular diffusion to be much more effective than in laminar flows (FISCHER et al., 1979; ROBERTS AND WEBSTER, 2002).

Vertical diffusion often accounts for the total rate of mixing in vertical direction; however other larger scale processes mostly dominate the spreading horizontally (GEYER AND SIGNELL, 1992). Because the considered bay is assumed to be vertically well-mixed (section 2.2), which means that only mixing in horizontal direction is relevant, in this study the influence of turbulent diffusion is considered negligible compared to the larger scale shear flow dispersion.

FIGURE 2.4 – SCHEMATIC REPRESENTATION OF DIFFUSION OF DYE TRACE IN LAMINAR (LEFT) AND TURBULENT FLOW (RIGHT)4

2.4.3 SHEAR FLOW DISPERSION Shear flow dispersion is the spreading resulting from velocity gradients in the flow profile (either vertical or transverse) which are referred to as shear flows (FISCHER et al., 1979). A typical example of shear flow is the parabolic vertical velocity profile which appears in open channel flows due to the effect of bottom shear stress (Figure 2.5). Transverse velocity profiles result from spatial variations in bathymetry and geometry. This is further explained in section 2.5.

Shear flow dispersion in fact comprises the combined action of advection along the streamlines and turbulent diffusion among them (FISCHER et al., 1979). This results in a similar spreading pattern as through molecular and turbulent diffusion, however acting on much larger spatial scales, up to the order of the entire (vertical or transverse) cross-section of the flow.

4 From ‘http://www.mit.edu/course/1/1.061/OldFiles/www/dream/SEVEN/SEVENTHEORY.PDF’ 9

FIGURE 2.5 – SHEAR FLOW DISPERSION BY VERTICAL (TOP) AND TRANSVERSE (BOTTOM) SHEAR.5

The rates of spreading by transverse shear are generally much higher than those as a result of the vertical velocity profile, because the latter is bounded by the water depth (FISCHER et al., 1979; GEYER AND SIGNELL, 1992; ROBERTS AND WEBSTER, 2002). Both types however are capable of producing rigorous rates of the total mixing within a bay.

A clear example of shear flow dispersion results from flow dispersion around a headland or shoal (Figure 2.6). The bottom of the two concentration patches indicated by squares in Figure 2.6a, is clearly dispersed (locally) when the tidal flow acts along the headland (Figure 2.6b-d).

FIGURE 2.6 – SHEAR FLOW DISPERSION BY FLOW SEPERATION AROUND A HEADLAND6

2.5 RESIDUAL CURRENTS As mentioned in the preceding paragraph, the process of shear flow dispersion is generated by vertical or transverse velocity profiles, which are induced by the interaction of the tidal current with irregular bathymetry and geometry. In addition, advective transport is driven by the current velocity of the flow.

5 From ROBERTS AND WEBSTER (2002) 6 From KASHIWA (1984) 10

Because typical time scales of bay flushing range from weeks to months (GEYER AND SIGNELL, 1992) the tidally averaged flow field is, next to these ‘actual’ water movements, specifically relevant in the context of flushing. The resulting Eulerian residual current pattern shows net flood and ebb-directed currents throughout the bay area.

2.5.1 GENERATION OF RESIDUAL CURRENTS Residual currents result from nonlinear interactions between tidal current7 and bathymetry or geometry (MILLER et al., 1990). Interaction of tidal currents with bathymetric features, generally leads to flood-directed residuals in shallow parts and ebb-directed residual currents in the deeper parts as in Figure 2.7. This spatial separation of the ebb and flood currents lead to the effective exchange mechanism B as described in section 2.3.2. An example of residual currents induced by the tidal current interacting with geometric features is given in Figure 2.7, where residual motions are shown for the tidal current along a headland: flow separation takes place around the headland, which results in a residual pattern.

FIGURE 2.7 – EXAMPLES OF BATHYMETRY (LEFT) AND GEOMETRY (RIGHT) INDUCED RESIDUAL CIRCULATIONS8

2.5.2 INFLUENCE ON MIXING AND TRANSPORT IN THE BAY The Eulerian residual current field in fact represents the long term averaged advective transport. However, the actual transport pattern as a result of advection and diffusive processes (e.g. shear flow dispersion) appears to be more complicated. GEYER AND SIGNELL (1992) argued that the influence of the residual motions as apparent around headlands (Figure 2.7) or shoals to the average residence time of an embayment depends on the relative spacing between geometric and bathymetric variation in relation to the tidal excursion length (GEYER AND SIGNELL, 1992). If this spacing is less or in the order of the tidal excursion length, the generated eddies can interact, having a pronounced effect on the total transport within the bay and the associated total flushing of the embayment (section 2.6.4). However, if this spacing is larger than the tidal excursion length, the effect of (shear) dispersion on the flushing conditions is assumed to remain locally (section 2.6.4). The tidal excursion length equals the horizontal distance over which a water particle is transported during flood:

2.2

In which [s] is the tidal period and [m/s] the velocity amplitude.

7 Similarly for wind-, density- or river induced currents in regions were these are present. 8 From FISCHER et al., 1979 11

2.5.3 RESIDUAL CURRENTS AT THE INLET

TIDAL PUMPING A special type of residual current pattern is induced by the abrupt variation in cross-sectional area at the bays inlet; ‘tidal pumping’. The mechanism of tidal pumping was first described by STOMMEL AND FARMER (1952), who showed the jet-like behaviour of the flood flow entering the bay and a sink type of flow on the sea side of the inlet draining the flood through the inlet. The same flow pattern occurs during ebb, in opposite direction (Figure 2.8). This mechanism ensures that the water that has left the bay during ebb tide will not return during the following flood which corresponds with flushing mechanism C as described in section 2.3.2.

In addition it contributes to mechanism B (section 2.3.2) because the inflow follows different paths than the outflow, which leads to a highly effective exchange of fluids. The more abrupt the transition from the sea to the bay and vice versa, the stronger the ‘jet’ and inherently the influence of tidal pumping will be.

For navigational purposes, many coastal bays have deep inlet channels. These channels accommodate relatively large flow speeds which can as well provide the ‘jet’ flow, resulting in an effective exchange mechanism as tidal pumping.

QUADRUPLET FORMING The flow separation around the inlet can in certain situations lead to the formation of residual eddy pairs, both behind and in front of the inlet. Therefore ZIMMERMAN (1986) speaks of a quadruplet (Figure 2.8). In addition to the regular tidal pumping mechanism, these quadruplets can have a major effect on the tidal exchange as they are directed away from the inlet, removing the exiting water away from the inlet, avoiding its return during the following flood (flushing mechanism C, section 2.3.2).

FIGURE 2.8 – TIDAL PUMPING AT ABRUPT INLET (LEFT)9 AND TIDAL INDUCED QUADRUPLETS (RIGHT)10

The forming and nature of the quadruplets is largely dependent on the bottom friction and the magnitude of the tidal current. In low bottom friction regimes, the eddies can be persistent throughout successive tidal cycles, greatly enhancing the exchange (KASHIWAI, 1984). In the case of moderate friction, the eddies will be formed however they will have a lifetime smaller than the tidal cycle. If the friction is even higher, there will be no residual eddies induced at all (BUTMAN AND SIGNELL, 1992).

9 From BUTMAN AND SIGNELL, 1992 10 From ZIMMERMAN, 1986 12

2.6 MATHEMATICAL DESCRIPTION OF THE TRANSPORT PROCESSES

2.6.1 TRANSPORT BALANCE As explained in the previous paragraphs, the flushing process is determined by processes of transport and mixing which can be divided into advective and diffusive or dispersive processes. Assuming that advection and diffusion are separate, additive processes, the evolution of a mass in a fluid can be described by the well- known advection-diffusion equation, which reads in its general three-dimensional form (FICK, 1855):

( ) ( ) ( )

2.3

In which [kg/m3] is the concentration, [m/s], [m/s], [m/s] the current velocities in , and direction 2 respectively, [m /s] is the coefficient of proportionality or the ‘dispersion coefficient’ representing the rate of spreading in direction and [m3/s] is the source/sink term.

Because the focus of this study is on coastal bays which are vertically well-mixed, the following holds:

and . Therefore it is assumed that the terms and ( ) in equation 2.3 can be neglected, and the resulting two-dimensional diffusion-advection equation is sufficient to describe the exchange in coastal bays with the velocities and , quantifying the advective transport and and the horizontal spreading. For the sake of clarity, when not stated differently, in this chapter the rate of horizontal spreading is assumed uniform in all directions, so: .

2.6.2 DISPERSION COEFFICIENT The nature or meaning of the dispersion coefficient(s) becomes apparent through the solution of the one- dimensional diffusion equation, which describes diffusive spreading (no advection) in direction:

( ) 2.4

2 The fundamental solution to this equation is a Gaussian distribution (FISCHER et al., 1979). The variance [m ] and the standard deviation [m] of this distribution are important properties in this context: indicates the length scale of the diffusing cloud, and its evolution in time is related to the dispersion coefficient:

√ 2.5

This leads to the following relationship between the dispersion coefficient and the variance :

2.6

The dispersion coefficient thus represents the rate of spatial spreading of a diffusing cloud. The value of the dispersion coefficient is closely related to the scale of the transport processes considered. For example the 2 molecular diffusion coefficient [m /s] is much smaller than coefficient representing the rate of shear flow 2 dispersion [m /s] as molecular diffusion is a smaller scale process. In section 2.6.6 the value of the dispersion coefficient is further discussed.

2.6.3 REYNOLDS DECOMPOSITION Although it has been concluded that turbulent diffusion plays no significant role in the total flushing of a coastal bay (section 2.4.2), the description of transport in turbulent flows by Reynolds is explained here because the concept will later show its value for the description of transport on larger scales (section 2.6.4).

Reynolds derived the description of the turbulent diffusion coefficients by decomposition of the turbulent flow into a mean part [m/s] and a fluctuation part [m/s] (ROBERTS AND WEBSTER, 2002):

2.7

FIGURE 2.9 TYPICAL VELOCITY PROFILE OF TURBULENT FLOW11

2 2 Accounting for the spatial variation in turbulent flows, the turbulent diffusion coefficients [m /s], [m /s] 2 and [m /s] appear after Reynolds averaging the advection-diffusion equation, after which the following equation for the diffusive transport can be derived:

( ̅̅ ̅ ̅̅ ) ( ̅̅ ̅ ̅̅ ) ( ̅̅̅ ̅ ̅ ) 2.8

In which ̅̅̅ ̅̅ , ̅ ̅̅ ̅̅ and ̅̅̅ ̅ ̅ are the time-averaged turbulent fluxed in directions , and respectively.

11 From ROBERTS AND WEBSTER, 2002 14

Now the coefficients can be defined as:

̅̅̅ ̅̅ ̅̅ ̅ ̅̅ ̅̅̅ ̅ ̅ 2.9

With equation 2.8 Reynolds shows that advection is driven by the mean part of the flow and turbulent diffusion is the result of the fluctuations on top of the average flow.

This concept can be extended when the processes of transport are averaged over larger spatial and time scales, which is particularly relevant in the context of the flushing of coastal bays. To distinguish between advection and dispersion then becomes a matter of the considered spatial and time scales: the motions that are of a smaller scale than the considered scale can be modelled as a dispersive process, quantified by the dispersion coefficient, even though theoretically these motions drive advective transport. This can be explained by the theory of the tidal random walk.

2.6.4 TIDAL RANDOM WALK The principle of the random walk as explained in the context of molecular diffusion (section 2.4.1), can be applied on larger scales as well. ZIMMERMAN (1986) provides a clear explanation for this concept by his theory of the tidal random walk in which he applies the random walk principle on the scale of the bay, which is the focus of this study. This theory describes the resulting transport from a steady residual current field comprising of eddies with significantly smaller spatial and velocity scales than the tidal flow field (GEYER AND SIGNELL, 1992). In case of a uniform tidal current, ZIMMERMAN (1986) states that the interaction between these two flow fields results in random Lagrangian trajectories. In analogy to the random motion of the molecules (section 2.4.1), the result of these random motions is to spread in a diffusive or dispersive manner, thus according a Fickian law (section 2.4.1) with a constant longitudinal dispersion coefficient. This is applicable to situations in which the space between residual eddies are in the same order as the tidal excursion (AREF, 1984), as explained in section 2.5.2: if the spacing is larger, the effect of the eddies is merely local.

FIGURE 2.10 – RANDOM TIDAL WALK: STEADY RESIDUAL CURRENT PATTERN (LEFT) AND RESULTING LAGRANGIAN PARTICLE TRACKS (RIGHT)12

Particularly relevant is that ZIMMERMAN (1986) describes a way in which also the smaller scale advective motions (residual currents) are represented through a dispersion coefficient. This is analogous to the description of turbulent diffusion, in which the random field of turbulent eddies (which is strictly speaking advection) caused the concentration to spread in a diffusive manner when observed at a larger scale (section 2.4.2). As the tidal random walk is associated with much larger scale processes, the resulting dispersion coefficient will intuitively be several scales larger as well.

12 From GEYER AND SIGNELL, 1992 15

2.6.5 DISPERSION DOMINATED It is explained that the process of exchange or flushing can be described by the advection-diffusion equation in which the advective transport is quantified by the (mean) velocity and the rate of spreading through the dispersion coefficient in the dimension(s) considered. On the scale of the bay, which is particularly relevant in the context of flushing, only the (residual) currents of similar spatial scale as the bay are associated with advective transport. Because the bays of interest are not associated with large scale circulations (section 2.2), the transport in the bay can be classified as ‘dispersion dominated’. This means that on the scale of the bay the total exchange can be described through the diffusion equation and quantified by the horizontal dispersion coefficient.

2.6.6 ESTIMATION OF THE DISPERSION COEFFICIENT Although velocities can be measured relatively easily, to obtain the value of the dispersion coefficient appears a difficult task. Dispersion coefficients can be derived from the field through tracer studies; however such studies are very costly and impractical (RAKHA et al., 2010). Therefore in practise the dispersion coefficient is usually estimated and/or used as a calibration parameter.

VELOCITY AND LENGTH SCALE The dimension of the dispersion coefficient [m2/s] indicates that it can be composed by a velocity scale and a length scale. Because the dispersion coefficient is assumed to represent the spreading by the fluctuating part of the flow, an appropriate velocity scale is the variation on the mean flow velocity, averaged over the considered time scale. The length scale should represent the spatial scale over which the diffusing process acts. This leads to the following rough estimation of the dispersion coefficient:

̅ 2.10

When the dispersive processes deviate in different directions, different values for and can be estimated using equation 2.10 with the velocity and length scale in respectively direction.

ORDER OF MAGNITUDE Dispersion coefficients can be estimated in the field through tracer studies; however such studies are very costly and impractical (RAKHA et al., 2010). Therefore in practice the dispersion coefficient is often used as a calibration parameter.

FISCHER et al. (1979) summarized the values of longitudinal dispersion coefficients which were observed in 2 several inland coastal waters. The values were found to lie in the range of [m /s] = O(10) – O(100). Further, RAKHA et al., (2010) argues that a typical value for the dispersion coefficient of tidally dominated bays could be taken in the ranges between 1 to 20 m2/s.

2 -9 Comparing these values with the order of magnitude of the molecular diffusion coefficient [m /s] = O (10 ) -8 2 -3 -2 – O (10 ) and turbulent diffusion coefficients [m /s] = O(10 ) – O(10 ) immediately shows that these processes are indeed negligible to the total transport in the context of flushing of a bay, which was argued in 2 section 2.4. However, the magnitude of the shear flow dispersion coefficient is in the order of [m /s] = O(10), which means that shear flow dispersion can have a significant influence on the total flushing. The remainder rate of spreading is associated with the bathymetry and geometry induced (residual) eddies (section 2.6.4).

2.7 DISCUSSION It was shown that the process of flushing of a coastal bay can mathematically be described through the diffusion-advection equation. Because shallow, vertically well-mixed bays are considered in this study, the two-dimensional advection-diffusion equation is assumed to be suitable to represent the exchange process.

In order to assess the flushing characteristics and to compare them for different situations, it is necessary to quantify this exchange. Following the two-dimensional advection-diffusion equation the processes of exchange and transport can be quantified through the (mean) velocities and in the two dimensions and the dispersion coefficients and . The values of the dispersion coefficients can then be estimated through Equation 2.10.

However, when the information on the exchange process is applied to environmental assessments, it is common and useful to represent the transport processes in terms of transport time scales (TAKEOKA, 1984). Because the primary reason to assess the flushing characteristics in the context of this study is to indicate on the water quality, the best way to represent the process of exchange is indeed supposed to be by means of (one of) these transport time scales. Yet it is case to determine the most suitable transport time scale(s) in light of the objective of the present study and to develop a method through which quick estimates of the relevant time scale(s) can be obtained.

3 QUANTIFICATION OF THE FLUSHING PROCESS After having described the essence of the process of flushing and the underlying relevant physical processes, the current chapter is addressed to finding the most suitable way to determine a bay’s flushing conditions, working towards the development of the rapid assessment method. In order to value the flushing conditions of a coastal bay and, considering the objective of present study perhaps more importantly, to estimate the relative flushing characteristics for different situations, it is essential to define a suitable indicator for flushing.

In the previous chapter it was shown that the relevant transport processes and thus the process of exchange can be quantified in terms of flow velocities and dispersion coefficients (TAKEOKA, 1984). However, it was argued that, for the purpose of this study, concepts of transport time scales are more suitable to indicate the exchange conditions. Therefore this chapter is aimed to find the most relevant transport time scale in this context and to define the approach for the development of the rapid assessment method, striving to include as much of the relevant processes in the smallest time span possible.

3.1 TRANSPORT TIME SCALES

3.1.1 RELEVANCE Flushing is generally indicated by transport time scales which are related to the renewal or the retention time of water masses and dissolved substances within a defined area. These are important measures as, when compared to time scales of the loadings and/or biological and chemical processes, they reveal relevant information on which water quality processes might occur in a specific case or at a specific location and which not (MONSON et al., 2002). Many researchers have emphasized the relevance of transport time scales in water quality studies, supported by studies in which transport time scales were successfully related to biochemical processes and associated concentration distributions of various constituents or living biomasses (TAKEOKA, 1984; BOYTON et al., 1995; MONSON et al., 2002).

3.1.2 DEFINITIONS In literature many concepts of time are found, which are used represent the processes of transport and exchange. The inconsistent use of the definitions is however rather confusing. One reason for this is that the definitions of the concepts of time scales are often based on theoretical cases in which the material transport mechanism is assumed to be steady and well-mixed. This leads to confusion when these definitions are applied to practical situations. For reasons of consistency, in this study the definitions are adopted from TAKEOKA (1984), who in turn uses definitions from ZIMMERMAN (1976) and BOLIN AND RODHE (1973). In addition, the definition of the flushing time is given. This term does not arise in the study of TAKEOKA (1984) but because it is used in many studies, its definition is included for completeness and to avoid confusion.

In the context of this study the time scales representing the system-wide exchange characteristics are particularly interesting. However first the ‘local’ concepts of age, residence time and transit time are defined as these form the basis for the ‘system-wide’ time scales.

LOCAL TRANSPORT TIME SCALES  Age: The time that has elapsed since the water particle has entered the water body through its boundaries (BOLIN AND RODHE, 1973; TAKEOKA, 1984).  Residence time: The time it takes for a particle within a water body to exit the water body (ZIMMERMAN, 1986; TAKEOKA, 1984). This makes residence time the complement of age.  Transit time: The age of a particle leaving the water body or the maximum age (TAKEOKA, 1984).

FIGURE 3.1 - SCHEMATIC REPRESENTATION OF THE TRANSPORT TIME SCALES AGE, RESIDENCE TIME AND TRANSIT TIME FOR THE

PARTICLE ‘P’ AT TIME t=t1. ‘QR’ IS THE RIVER DISCHARGE, ‘S’ THE SOURCE AND ‘t’ INDICATES THE TIME, STARTING AT t=t0 WHICH IS THE MOMENT THE PARTICLE ENTERS THE RIVER.

The above definitions of transport time scales are properties of single particles, which can be visualized using the simple case of a steady state river (Figure 3.1). When the ages or residence times of all particles in a system (e.g. a coastal bay) are known, the exchange conditions can be characterized by the age respectively the residence time distribution function (Figure 3.3). From these the average age and average residence time of the water mass can be derived, which together with the turnover time and average transit time are denoted as ‘system-wide’ transport time scales. TAKEOKA (1984) shows that in steady state conditions the age and residence time distributions are equal (Figure 3.1).

SYSTEM-WIDE TRANSPORT TIME SCALES  Turnover time or flushing time: The time it takes to completely exchange water within a water body, with the external water system.  Average age: The weighted mean of the ages of the individual particles in a water body.  Average residence time: The weighted mean of the residence times of the individual particles in a water body. The average residence time equals the average age in steady state conditions.  Average transit time: The average transit time of all particles leaving the system. The average transit time equals the turnover time in steady state conditions.

In steady state conditions two concepts remain: the flushing time (or turnover time or average transit time) and the average residence time (or average age). TAKEOKA (1984) argues that the average residence time is the most suitable concept to describe the exchange or flushing characteristics of reservoirs when it concerns the total material of the reservoirs. This is illustrated with an example in which two lakes are considered with different distances between the inlet and outlet (Figure 3.2). The residence time and transit time distributions are shown in Figure 3.3. If the two lakes accommodate the same volume of water and the flow rates are equal, in steady state conditions their flushing times will be equal. However, the average residence time of bay (b) will be larger than bay (a), which means that the water in lake (b) stays longer than in lake (a): lake (a) is exchanged more rapidly (Figure 3.2). Therefore this study focuses on the estimation of the average residence time.

Merely because of the oscillating character of the tidal force, coastal bays are not in steady state. However, for the application of the described concepts of time, it is widely accepted to assume steady state conditions. Over larger time scales the criterion inflow equals outflow holds, when the tidal signal is assumed constant. Obviously these are all simplifications of reality, which should be taken into account when applying the time scales to environmental problems.

FIGURE 3.2 - SCHEMATIC REPRESENTATION OF TWO LAKES WITH AN INLET AND AN OUTLET. IN LAKE (A): AVERAGE RESIDENCE TIME < AVERGE TRANSIT TIME. IN LAKE (B): AVERAGE RESIDENCE TIME>AVERAGE TRANSIT TIME. FROM TAKEOKA (1984)

FIGURE 3.3 - RESIDENCE TIME (a) AND TRANSIT TIME (t) DISTRIBUTION FUNCTIONS OF BAY (A) AND BAY (B) IN FIGURE 3.2. FROM TAKEOKA (1984)

3.2 CALCULATION METHODS The next concern is the way the average residence time of a coastal bay can be determined. As mentioned in the introduction (section 1), the available methods appear to be unsuitable for the rapid assessment of flushing. To develop a method which can be used to quickly obtain estimates of the average residence time, first the existing methods are analysed. Methods based on a box, one-dimensional or two-dimensional representation of the coastal bay are distinguished. These will form the starting point of the approach towards the development of the rapid assessment method.

3.2.1 BOX MODEL The simplest and quickest way to estimate the average residence time is through a box model, which uses a zero-dimensional representation of the study area. The tidal prism models (DYER, 1973; FISCHER et al., 1979) are the most relevant to this study as they are based on tide-driven flushing only. In fact these models are based on the definition of the turnover time, or more commonly used the flushing time, rather than on the concept of the average residence time. However, with the assumptions of a steady state and perfectly mixed ( ) system, the average residence time equals the flushing time (TAKOAKA, 1984). The distribution functions of the residence time and the turnover time are in such case identical (Figure 3.4).

FIGURE 3.4 – RESIDENCE TIME AND TRANSIT TIME DISTRIBUTIONS IN THE CASE OF A STEADY STATE AND PERFECTLY MIXED SYSTEM. FROM TAKOAKA (1984) 20

In such a case, the change of concentration [kg/m3] in the bay with volume [m3] and flow rate [m3/s] is described by the following equation:

3.1

The solution is the following exponential function:

3.2

3 Furthermore, the flushing time [s] in this steady state, box system, is described by the flow rate [m /s] and the volume of the water within the system [m3]:

3.3

At time , which equals the flushing time by equation 3.3, the concentration has decreased to or

equivalently to . Most numerical flushing calculations methods are based on this relation between the concentration decrease and the flushing time. This is explained in section 3.2.3.

The classical tidal prism model, which forms the basis of all tidal prism models, uses the definition of the flushing time (equation 3.1) to estimate the average residence time ̅ [s] in coastal reservoirs:

3.4

3 In which [m ] is the volume during high water, which represents the volume in equation 3.5. The flow 3 rate in equation 3.5 is given by the tidal prism volume [m ] divided by the tidal period [s].

Next to the steady state and well-mixed assumptions, this definition implies that the volume which leaves the bay during flood, will not return during the next ebb tidal current. In order words, the return flow is zero. However, in environmental bays there will always be a return current. The effective exchange flow should therefore be smaller than the tidal prism.

STANFORD et al. (1992) therefore modified the model by inclusion of the return flow factor [-], which has a value between 0-1 and represents the fraction of effluent water which returns to the domain during flood (MONSON et al., 2002). The new formula then becomes:

3.5 ̅ ( )

The return flow factor is closely related to flow and mixing conditions near the bay’s inlet (e.g. tidal pumping) and at sea. However, the way this parameter depends on these processes remains unclear (SANFORD et al., 1992). Therefore its value should be determined through calibration.

Although the model has certainly been improved, it is still based on the assumption of complete mixing within the system, which is very unrealistic. This generally results in underestimations of the average residence time (MONSON et al., 2002).

3.2.2 ONE-DIMENSIONAL METHOD TAKEOKA (1984) describes another way to analytically estimate the average residence time. It starts with the one-dimensional representation of the bay with length [m]. Assuming a dispersion dominated environment (section 2.6.5) the problem of flushing can be described by the one-dimensional diffusion equation:

( ) 3.6

With the following boundary condition at the mouth (L=0):

( ) 3.7 In which [s] is the time.

From this TAKEOKA (1984) derives the residence time distribution function of the residence time for the length of the bay which in turn can be used to obtain the average residence time. For a detailed description of the procedure is referred to the article (TAKEOKA, 1984). The following relation between the dispersion coefficient and the average residence time is derived by TAKEOKA (1984):

( )

̅ 3.8

[m] In this equation represents the total distance over which the transport acts: from the inlet towards the end of the bay and back. Instead of assuming a totally mixed bay, like in the box models, in this way the transport processes in the bay seem to be better represented through the inclusion of the dispersion coefficient. However, because only one dimension is taken into account, the bay is assumed to be well-mixed over the width ( ) and only transport over the length of the bay is considered, which does not reflect reality. This makes the estimation of the ‘correct’ value to use for dispersion coefficient also very complicated.

3.2.3 TWO-DIMENSIONAL METHOD

ANALYTICALLY To increase the level of detail, the one-dimensional representation of the bay can be extended with the addition of the second dimension. The transport in the bay can then be described through the two- 2 2 dimensional diffusion equation () with [m /s] to represent the spreading in x-direction and [m /s] in y- direction. The values of and can be estimated through the velocity and length scales in the direction of interest, as explained in section 2.6.6. The problem however becomes too complex now to derive the average residence time analytically, which was done by TAKEOKA (1984) for the one-dimensional representation of the bay. Therefore nowadays most flushing studies make use of advanced numerical modelling techniques.

( ) ( ) 3.9

NUMERICALLY The exchange characteristics of the bay can be represented in detail using detailed flow models, which are capable to solve the advection-diffusion equation numerically over relatively small spatial and time scales. In the context of this study the focus is on the depth-averaged models (2.6.1). A detailed and well-calibrated model is assumed to be able to accurately represent the relevant transport processes and the resulting concentration distributions within the bay. The question is how the average residence time can be derived from this.

This is generally done using the ‘tracer method’. The tracer method is actually addressed to the calculation of the ‘local’ residence times. From these the bay’s average residence time can be derived. For the estimation of the local residence times, the bay area is filled with a homogeneous concentration (the ‘tracer’). The time it takes for the concentration in a computational cell to decrease to about 37% of its initial concentration is taken as the average residence time of the area within this cell. The reasoning behind this can be explained as follows.

Because properties of the water and the flow are uniform within one computational cell as far as the model concerns, the cell is considered as a well-mixed system and the average residence time equals the flushing time which was explained in section 3.2.1.

Under these assumptions, it was shown in section 3.2.1 (equations 3.1 and 3.2) that after a time which is equal to the bay’s flushing time, the concentration in the bay has decreased to 37% of its initial concentration.

When the grid resolution is high enough, the assumption of a totally mixed system within one cell is assumed valid. Contrary, the steady state assumption is questionable. The largest unsteady factor is the tidal force. Because of this the volume within one grid cell is constantly changing, however more or less periodically. In addition, due to the presence of a return flow, the actual concentration decay curve will not be a smooth exponential curve such as shown in Figure 3.4 but it will have an oscillating character. This problem can be dealt with by ‘filtering’ the fluctuations of the concentration curve before the average residence times are subtracted from these.

Up to this point, most flushing studies are in perfect agreement. It becomes confusing when the exchange conditions when the total bay is concerned. The derived ‘local’ residence time represent the average time water particles will be situated within the considered cell. To obtain an estimation of the bay’s average residence time, the weighted mean of the cell averages should be taken.

In most studies the flushing time or turnover time is used to indicate the derived bay’s average residence time. This is incorrect, because the assumption of infinitely large dispersion does surely not hold for the total bay so the average residence time is not the same as the flushing time. Others estimate the cell’s residence times as the time that has passed until the concentration has dropped to 10% or 50% of the initial concentration, instead of the 37%. This can be used to indicate the flushing conditions; however it is a different indicator than the residence time and should therefore not be defined as such. The flushing time or turnover time is in some cases also defined as the time it takes for the average concentration in the bay to decrease to 37%. Again, this is incorrect because the bay is not a totally mixed system.

3.3 DISCUSSION

Several concepts of time were described and discussed. Based on the findings of TAKEOKA (1984) it was concluded that the average residence time is the most suitable concept to be used to characterize the total bay exchange or flushing.

Various calculation methods were elaborated, based on box, one-dimensional or two-dimensional representations of the bay and the associated transport processes. Concerning the rapidness of these different methods, the analytically based box models and one-dimensional model are preferred. However, these models lack a sufficient level of detail concerning the relevant flushing processes as they are based on dubious assumptions. Furthermore, in the light of the primary objective of this study, a higher level of detail is required to be able to account for changes to the bay as a result of bay developments.

To accurately describe the process of flushing the tracer method is the best alternative. By using a detailed depth-averaged representation of the bay and numerically solving the two-dimensional diffusion-advection equation, the relevant flushing mechanisms and the resulting concentration field are assumed to be well represented. Using the numerical approach however is very time consuming and requires a rigorous amount of data, which is generally unavailable in early stage of design in which the rapid assessment method should be used.

Taking all the above into consideration, it is decided that the rapid assessment should be of the box type, using basic bay parameters which account for more relevant processes than resolved for by the existing box models. This higher level of detail can be obtained by fundamentally deriving the influence of certain basic bay parameters from the outcomes of a numerical 2D model. To develop such a rapid assessment method, the following approach is used:

 Schematic 2D model tests are carried out using the numerical software program Delft3D-FLOW, focused on the influence of basic bay parameters.  A dimensional analysis is conducted to systematically detect relevant relations between the basic bay parameters and the estimated average residence time.  On account of the dimensional analysis, the results are interpreted, based on which the rapid assessment method is developed.

4 SCHEMATIC 2D MODEL TESTS Because the final aim of this study is to rapidly obtain an estimation of the average residence time using only basic bay and forcing parameters, schematic model tests are carried out to assess the relation of the relevant basic parameters (e.g. average bay depth, average tidal amplitude) to the average residence time of the bay. In this chapter first the approach is further explained, followed by a description of the software used (Delft3D- FLOW) and the set-up of the schematic model tests following the systematic approach of the dimensional analysis. Finally, the results are given and discussed based on which the rapid assessment method is developed.

4.1 APPROACH To investigate how the average residence time depends on basic bay and tidal characteristic, a dimensional analysis is conducted. A dimensional analysis is a commonly used, physically based method to systematically detect possible relationships between the variables that determine a certain physical problem and to describe the relations mathematically (SONIN, 2001).

The analysis starts with the detection of the parameters that determine the average residence time. The parameters are chosen based on the findings from literature regarding the main drivers of the flushing process (section 2). Following the approach of the ‘Buckingham Pi Theorem’ (SONIN, 2001) the parameters are made dimensionless, which from a physical point of view is the most appropriate form.

The approach of the dimensional analysis is to vary each dimensionless parameter several times and determine their influences on the resulting average residence time. By changing the dimensionless parameters one at a time, their individual influences on the average residence time are isolated. To carry out such an analysis a schematic two-dimensional numerical model is set up using the software program Delft3D-FLOW, based on the bay and forcing parameters. This model is used as a virtual laboratory in which the influence of the dimensionless parameters to the average residence time is assessed using the tracer method (section 3.2.3).

4.2 DIMENSIONAL ANALYSIS A dimensional analysis is a commonly used, physical based method to systematically detect possible relationships between the variables that determine a certain physical problem and to describe the relations mathematically (SONIN, 2001). Two forms of dimensional analysis can be distinguished: ‘blind’ and ‘inspectional’ analysis (SONIN, 2001). An inspectional dimensional analysis can only be conducted if a complete set of equations and boundary conditions that describes the problem is known. In that case, the equations can be normalized resulting in dimensionless equations.

Because the problem of flushing is too complex to be fully described analytically, in this study the blind analysis is used, following the method of the Buckingham Pi-Theorem (SONIN, 2001). A blind analysis is simple to apply and although less powerful than the inspectional form, when carried out carefully it can lead to extremely valuable insights (SONIN, 2001).

4.2.1 RELEVANT PARAMETERS For a correct dimensional analysis following the Buckingham Pi-theorem, it is important to find a complete set of parameters that together define the problem, which is the bay’s average residence time ̅ [s] (section 3.1.2). If the equations and boundary conditions are known, one can be confident that the set is complete. However in this blind analysis, some uncertainty remains. Findings from literature review, showed the importance of the following factors to influence the flushing mechanisms and inherently the average residence time (section 2):

 Tidal forcing (section 2.3)  Bay geometry (section 2.5)  Bay bathymetry (section 2.5)  Inlet characteristics (section 2.5.3)  Bottom roughness (section 2.5.3)  Flow conditions at the sea (section 2.3)

The tidal external forcing can roughly be represented by two parameters: the tidal amplitude [m] and the tidal period [s]. The bay geometry and bathymetry can be described by the basic bay dimensions: the bays length [m], width [m] and depth [m]. The inlet characteristics can be represented by the cross- 2 sectional inlet area [m ]. The influence of irregular bathymetric features, which are not included in the 2 schematic model, are accounted for through the model parameters the horizontal viscosity [m /s] and the 2 horizontal dispersion coefficient [m /s] (section 4.3.3).

Further, to account for the bottom roughness, the geometrical roughness of Nikuradse [m] is considered more suitable than the commonly used Chézy value or Manning coefficient because the dimensions of the latter two lack sufficient physical grounds. Therefore in light of the dimensional analysis, it is more appropriate and meaningful to use the Nikuradse roughness height. The parameters are indicated in Figure 4.1 and listed in Table 4.10.

FIGURE 4.1 - INDICATION OF THE BASIC BAY AND FORCING PARAMETERS IN THE SCHEMATIC COASTAL BAY

Because the main focus is on the relative influence of changes to the bay and to narrow done the scope of the present study, specific seaward conditions such as the presence of a long shore current are left out of the analysis. In the schematic model tests the seaward conditions are kept as constant as possible.

The concentration distribution within the bay and at the sea is not listed in this context because it is considered as output rather than input of the model in this particular case. This is related to the tracer method (section 3.2.3) which is used to determine the average residence time. The tracer concentrations have no physical relation to the flushing problem and are only used to derive the average residence time of the bay. 26

TABLE 4.1 – RELEVANT BAY PARAMETERS

Parameter description Parameter character Parameter dimension Bay length m Bay width m 2 Surface area bay m Bay depth m 2 Cross sectional inlet area m Bottom roughness m 2 Horizontal eddy viscosity m /s Horizontal eddy diffusivity m2/s

Tidal period s Tidal amplitude m

It is important that the parameters are independent, which means that no parameter can be described by a combination of the other parameters (SONIN, 2002). Therefore the bay’s length is not used in the dimensional analysis as it can be described by the length and the surface area of the bay ( ). Assuming that the above parameters form a complete set, the average residence time is a function of these parameters:

4.1 ̅ ( )

4.2.2 DIMENSIONLESS PARAMETERS The next step is to make the parameters dimensionless. In total there are ten parameters and two dimensions. This means that the number of dimensionless parameters to describe the problem can at least be reduced to eight (10-2=8). The next step is to choose three base variables, which represent the two dimensions: [s] and 2 [m ].

The dimensions of the remaining independent parameters are then described by one or both of the base parameters, e.g. the bays width has the dimension ‘m’ and can therefore be described by the base variable

having the same dimension. The last step is to make the parameters dimensionless by dividing them by the (combination of) base variables. This results in the following dimensionless parameters, denoted by the - sign Buckingham (SONIN, 2002):

4.2 √ √ √ √

Using the relation , can be re-written into the following parameter, which is the so-called ‘aspect ratio’ representing the shape of the bay:

4.3

All that remains is to make the dependent variable ̅ dimensionless as well:

̅ 4.4

This leads to the following dimensionless functional representation of the so called ‘dimensionless flushing number’, comprising the remaining eight dimensionless parameters:

̅ 4.5 ( )

√ √ √

Next to the aspect ratio, the parameter ⁄ is physically very interesting. The importance of this ratio to flushing appears in several studies on the behaviour of flushing and is for example discussed in the PIANC Guidelines (2008) which are addressed to flushing of harbours. Intuitively one would expect a larger ratio to be in favour of flushing because in the case of a relatively large cross-sectional inlet area there is more ‘room’ for the exchange flow. However, in several studies the opposite effect is observed (SIGNELL AND BUTMAN, 1992; PIANC Guidelines, 2008). This can be explained by the effect of tidal pumping, which is dependent on abrupt cross-sectional inlet variations, leading to the typical jet type of flow (section 2.5.3). Such a tidal jet occurs due to energy conversion through the inlet resulting in locally increased velocities. This effect will be enhanced when the cross-sectional inlet is taken smaller relative to the basin surface area.

Further, the physical meaning of ̅⁄ seems very relevant in the context of flushing. This so-called ‘mixing number’ can be interpreted to represent the mixing rate during one tidal period within the bay. As previously explained, higher mixing rates are in favour to flushing, so large values of the mixing number are assumed to enhance flushing.

The combination of the parameters ⁄√ and ⁄√ is also interesting. Dividing these two parameters by each other results in , which is closely related to the tidal prism model (equation 3.4). When the parameters of the tidal prism model are described through the bay parameters of equation 4.5

( ( ) and ), equation 3.4 can be written as:

̅ 4.6

4.3 SET UP OF SCHEMATIC MODEL Now the relevant parameters have been detected and made dimensionless, the schematic model can be set up using these basic parameters. The model is set up using the numerical software program Delft3D-FLOW. For information on this software program refer to the Delft3D-FLOW Manual by Deltares. It is chosen to use a schematic bay representation instead of a more detailed and realistic one, because this is the easiest way to isolate the individual effects of the basic parameters. In addition, the large computational times associated with highly detailed models are very inconvenient considering the large amount of calculations required to carry out the dimensional analysis. The disadvantage however is that the representation of the bay is less detailed which can have its consequences to the accuracy of the results. Therefore validation of the results is important (section 5Error! Reference source not found.). This paragraph describes the set-up of the reference case model. The values of the input parameters are based on the situation in Doha Bay (section 2.1). 28

4.3.1 GRID AND BATHYMETRY

The representation of the bay’s geometry through the parameters and and leads to a rectangular basin with a uniform depth. Based on the dimensions of Doha Bay which are approximately 9x9 km (Figure 1.1), dimensions of 10x10km are chosen (which leads to a larger surface area than Doha Bay). Important is that the assumption of a small-scale bay holds. The bay is given a uniform depth of 5 m (section 2.1), which is of the same order as the average depth of Doha Bay (section 2.1). The cross-sectional inlet area is based on the initial estimate of the inlet area of Doha Bay being 15,000 m2, which, with an average depth of 5 m results in an inlet width of 3 km. For symmetry reasons a width of 2.8 km is used. The inlet is constructed through dry cells at both sides.

To avoid additional effects due to the abrupt bed level variation at the inlet, the depth at the inlet is the same as the depth in the bay. For the conditions offshore, a uniform slope is used, ranging from 50m at the offshore boundary to the depth of the basin at the inlet (in the reference case 5m). This represents the transition from the deep sea to the shallower coastal shelf.

The ocean area is represented by 100x150 grid cells, covering an area of 20x30 km, which is assumed to be large enough to avoid boundary effects from propagating into the bay area. In total the grid comprises 150x150 grid cells (excluding dummy values). A rectangular grid is used with uniform grid cell sizes of 200x200 m for both the sea as the bay area (Figure 4.2).

Bed level (m) 35 -5

-10 30 -15 25

 -20

20 -25

15 -30

y coordinate y (km) coordinate -35 10 -40 5 -45

0 0 5 10 15 20 25 30 x coordinate (km) 

FIGURE 4.2 – MODEL GEOMETRIC AND BATHYMETRIC CONFIGURATION OF THE REFERENCE CASE

4.3.2 INITIAL AND BOUNDARY CONDITIONS Although the wave signal in Doha Bay is mixed with diurnal and semi-diurnal tidal components (section 2.1), for reasons of simplicity the reference case will be forced by a single wave signal. The tidal wave with a period of 12.5 hours (equivalent to the M2 astronomical tide) and amplitude of 0.5 m is imposed at the offshore open boundary, using a water level type boundary. Both cross-shore boundaries are closed. The initial water level is zero for the total domain. As mentioned in section 4.2.1, the focus is on the processes at the bay and no long shore current is imposed.

4.3.3 MODEL INPUT

BOTTOM ROUGHNESS In Delft3D-FLOW the bottom roughness can be added to the model through a Chézy or Manning coefficient (Delft3D-FLOW Manual, 2009). Both are commonly used in numerical modelling. It is chosen to use Chézy as this parameter is easier to relate to the Nikuradse roughness height. The Chézy coefficient [m/s] is in fact a smoothness coefficient, which means that a larger value represents less bottom friction. A reference value of =65m/s is used. This is a commonly used value for coastal bays.

Through the formula of White–Colebrook the coefficient of Nikuradse can be calculated for the total water level [m]:

( ) 4.7

Where = + , with [m] is the water depth and [m] the surface elevation.

This leads to the following description of [m]:

4.8

To estimate , the average water depth in the bay is taken for in the White-Colebrook formula (Equation 4.8).

BACKGROUND VISCOSITY AND DIFFUSIVITY The representation of dispersion in the numerical model is less straightforward. As emphasized before, horizontal dispersion represents the rate of horizontal mixing. Gradient transport is accounted for by Delft3D- FLOW through the transport equation (Delft3D-FLOW Manual). However, horizontal turbulent motions and forcing on scales smaller than the scale of the grid cannot be resolved. In Delft3D-FLOW these influences are parameterized by horizontal viscosity and diffusivity coefficients. In the depth averaged two-dimensional Delft3D-FLOW model, the horizontal eddy viscosity is calculated as follow (Delft3D-FLOW Manual):

4.9

is the contribution of larger sub grid scale motions which is calculated by Delft3D-FLOW through the

Horizontal Large Eddy Simulation (HLES). In this study this is not used. Is the user defined value and should account for the internal shear forces resulting from transfer of momentum between regions of flow with varying velocities by means of turbulent mixing (Delft3D-FLOW Manual; ZHEN-GANG JI, 2008). Because the momentum and transport equations are depth-averaged the shear flow dispersion due to vertical flow velocity differences will have to be added as well. This can be well estimated by the Elder’s formula (Error! Reference source not found.). Therefore will represent the two dimensional turbulence and the dispersion coefficient.

The horizontal eddy diffusivity is used in the transport equation and accounts for all unresolved mixing (Delft3D-FLOW Manual). In the depth averaged two-dimensional Delft3D-FLOW model, the horizontal eddy diffusivity is calculated as follows:

4.10

The eddy diffusivity depends on the constituent. Both the viscosity and diffusivity depend on the flow, the size of the grid and the numerical method which is used to solve the transport equation. Therefore their values can neither be directly measured nor observed but should be calibrated using (flow) data (ZHEN-GANG JI, 2008).

Common values for the viscosity parameter lie between 1 and 10 m2/s (Delft3D-FLOW Manual) For the 2 reference case values of =5 m /s is chosen. In section 2.6.6 it was shown that the dispersion coefficient can be estimated by a length scale and velocity scale of the motions associated with mixing. Based on information obtained from the Hydrographical Impact Study by Royal HaskoningDHV the average velocity fluctuations in the bay were estimated to be around 0.05m/s. With a length scale of approximately half the 2 grid cell size, a dispersion coefficient of = 5 m /s is obtained which is used as input for the reference case.

TRACER CONCENTRATION Related to the average residence time calculation using the tracer method (section 4.3.1), in the tracer run (section 4.4.2) tracer concentrations have to be added to the domain. This will be discussed in section 4.4 in which also the tracer concentrations are given.

TIME STEP AND SIMULATION TIME Test simulations of the flow model with the set up as described in this paragraph shows that a time step of one minute gives accurate and stable results. The simulation time of the ‘restart run’ (section 4.4.1) has been set to two weeks. The simulation time for the following or ‘tracer run’ (section 4.4.2) will vary per case, as it depends on the (estimated) residence times. For the reference case the average residence time lies around one month. Therefore a standard simulation time of 50 days is used. Obviously, if the local residence times exceed this value, a longer simulation time is necessary. To summarize, the following input values can be listed, representing the model input of the reference case (Table 4.2):

TABLE 4.2 – MODEL INPUT REFERENCE CASE

Parameter description Parameter Dimension Value Restart run Time step min 1

Simulation time days 14 Bottom roughness m/s2 65 2 Horizontal eddy viscosity m /s 5 2 Horizontal eddy diffusivity m /s 5 Tracer run Time step min 1

Simulation time days 50 Bottom roughness m/s2 65 2 Horizontal eddy viscosity m /s 5 2 Horizontal eddy diffusivity m /s 5 3 Concentration in the bay kg/m 2 3 Concentration at the sea kg/m 1 31

4.3.4 MODEL OUTPUT

STORAGE INTERVALS Because the average residence time calculations are based on data saved in the map file (Delft3D-FLOW Manual), it is paramount that the map intervals are short enough to include the influence of the tide-induced concentration oscillations. Map intervals of one hour are assumed to be suitable to correctly represent the tidal motion. In the restart run, the restart interval is one day. Further, a history interval of 10 minutes is used. The output values are listed in Table 4.3.

TABLE 4.3 – MODEL OUTPUT REFERENCE CASE

Parameter description Parameter Dimension Value Restart run

Map interval min 60 Restart interval min 1440 History interval min 10 Restart run

Map interval min 60 Restart interval min 1440 History interval min 10

4.4 CALCULATION OF THE AVERAGE RESIDENCE TIME The average residence times are calculated by the tracer method which is explained in section 3.2.3. This requires a restart run.

4.4.1 RESTART RUN When starting a simulation in Delft3D-FLOW, some time is necessary for spin-up effects to die out resulting in the ‘actual’ flow field. This time is referred to as the spin-up time, resulting of a mismatch between the initial and the boundary conditions (Delft3D-FLOW Manual). Only after the spin-up time the model results are assumed to be reliable. For a correct calculation of the average residence time the tracer mass which is necessary for the calculation via the tracer method has to be inserted after the spin-up time. This can be easily done by the use of a restart file. First the simulation will be conducted without the tracer, which is called the ‘restart run’ and a restart file will be created at a time after all initializing erroneous effects have died out.

4.4.2 TRACER METHOD The calculation of the average residence time by the tracer method comprises the following five steps (section 3.2.3):

1. Restart run simulation – Flow simulations of the reference case have shown that the spin up time is between 5 to 7 days. Therefore a restart file after two weeks is assumed to be sufficient, also for the test cases in which the boundary and initial conditions will deviate from the reference case.

2. Tracer injection – The restart file which is obtained through the simulation of the restart run is then modified by adding the tracer concentrations. The basin is filled with the uniform concentration of 3 3 tracer mass ( =2 kg/m ) and the sea with a smaller concentration ( =1 kg/m ), which is illustrated inFigure 4.3. Finally an initial conditions file is created (Delft3D-FLOW Manual).

3. Tracer run simulation – The tracer run is carried out with the obtained initial conditions file which contains the tracer concentrations.

4. Calculation of (average) residence times per cell or local residence times – First, the cell’s concentrations in time, which are stored in the output trim-file, are filtered to account for the unsteadiness of the tidal wave. Subsequently the average residence times of the cells are taken as the time after which the concentration has decayed to 37% of its initial concentration. This is the case when the concentration in a cell reaches 1.37 kg/m3.

5. Calculation of the bay’s average residence time – Finally, the bay’s average residence time is obtained by taking the weighted mean of the calculated (average) residence times of the total cells. For this purpose first the average residence times of the cells are multiplied with the cells’ volumes. Then the sum of all the resulting values is taken and divided by the total volume of the bay. The result is an estimation of the bay’s average residence time.

Initial tracer concentrations (kg/m3) 2

30 1.8

25  1.6 20

15 1.4

y coordinate y (km) coordinate 10 1.2 5

1 0 5 10 15 20 25 30 x coordinate (km) 

FIGURE 4.3 – INITIAL TRACER CONCENTRATION DISTRIBUTION

4.4 TEST PROGRAM To detect relations between the dimensionless flushing parameter and each of the dimensionless parameters, a test program is set up for the average residence time calculations with varying dimensionless parameters.

The program starts with the reference case which is described in section 4.3. Further, each dimensionless parameter is varied four times, so in total five calculations (including the reference case) per parameter are carried out to test its influence to the dimensionless flushing parameter. To isolate the parameter influence, it is important that only one dimensionless parameter is varied, without changing the value of the other dimensionless parameters.

First, the aspect ratio ⁄ is varied by changing and in such a way that the surface area remains constant. In this way the other dimensionless parameters are not affected. These tests are followed by tests with varying values of the inlet width representing the influence of ⁄ . Further, the roughness 1/2 parameter ⁄√ is varied by changing the Chézy value. Values of in the range of 55-75 m /s are chosen

as these are common used values in practise. The influence of the viscosity and mixing numbers ⁄ , respectively ⁄ , are tested by changing values for the horizontal eddy viscosity and the horizontal eddy diffusivity . In addition several tests were set up with varying values of the tidal amplitude in order to detect the influence of the dimensionless parameter ⁄√ . Realistic values for the tidal amplitude are chosen ranging from 0.3 to 1.5m.

To test the parameter influence of ⁄√ , is more complicated as both the bay’s depth as the bay’s surface area appear in other dimensionless parameters as well. However, once the relations of ⁄√ respectively ⁄ with the flushing parameter ̅⁄ are known, the influence of ⁄√ can also be detected through tests with varying values of : the influences of changing values of and can then be extracted from the outcomes. For values between 3 and 7 m are chosen which results in situations that can still be classified as shallow water depth (section 2.2).

A summary of the test program is given in Table 4.4. The associated values for the dimensionless parameters are shown Table 4.5.

TABLE 4.4 – INPUT PARAMETER VALUES FOR THE DIFFERENT SCENARIO'S

Test Lb Wb db Wi C DH VH at Tt Ai ai ks 1/2 2 2 2 2 [km] [km] [m] [km] [m / [m /s [m /s [m] [hr] [km ] [km ] [m] s] ] ] 1 10 10 5 2.8 65 5 5 0.5 12.5 100 1.4 1.4E-2 2 12.5 8 5 2.8 65 5 5 0.5 12.5 100 1.4 1.4E-2 3 10.8 9.2 5 2.8 65 5 5 0.5 12.5 100 1.4 1.4E-2 4 9.2 10.8 5 2.8 65 5 5 0.5 12.5 100 1.4 1.4E-2 5 8 12.5 5 2.8 65 5 5 0.5 12.5 100 1.4 1.4E-2 6 10 10 5 2.0 65 5 5 0.5 12.5 100 1.4 1.4E-2 7 10 10 5 4.0 65 5 5 0.5 12.5 100 1.4 1.4E-2 8 10 10 5 6.0 65 5 5 0.5 12.5 100 1.4 1.4E-2 9 10 10 5 7.0 65 5 5 0.5 12.5 100 1.4 1.4E-2 10 10 10 5 2.8 55 5 5 0.5 12.5 100 1.4 5.3E-2 11 10 10 5 2.8 60 5 5 0.5 12.5 100 1.4 2.8E-2 12 10 10 5 2.8 70 5 5 0.5 12.5 100 1.4 7.7E-3 13 10 10 5 2.8 75 5 5 0.5 12.5 100 1.4 1.7E-3 14 10 10 5 2.8 65 3 5 0.5 12.5 100 1.4 1.4E-2 15 10 10 5 2.8 65 7 5 0.5 12.5 100 1.4 1.4E-2 16 10 10 5 2.8 65 10 5 0.5 12.5 100 1.4 1.4E-2 17 10 10 5 2.8 65 20 5 0.5 12.5 100 1.4 1.4E-2 18 10 10 5 2.8 65 5 3 0.5 12.5 100 1.4 1.4E-2 19 10 10 5 2.8 65 5 7 0.5 12.5 100 1.4 1.4E-2 20 10 10 5 2.8 65 5 10 0.5 12.5 100 1.4 1.4E-2 21 10 10 5 2.8 65 5 20 0.5 12.5 100 1.4 1.4E-2 22 10 10 5 2.8 65 5 5 0.3 12.5 100 1.4 1.4E-2 23 10 10 5 2.8 65 5 5 0.7 12.5 100 1.4 1.4E-2 24 10 10 5 2.8 65 5 5 1.0 12.5 100 1.4 1.4E-2 25 10 10 5 2.8 65 5 5 1.5 12.5 100 1.4 1.4E-2 26 10 10 3 2.8 65 5 5 0.5 12.5 100 1.4 1.4E-2 27 10 10 4 2.8 65 5 5 0.5 12.5 100 1.4 1.4E-2 28 10 10 6 2.8 65 5 5 0.5 12.5 100 1.4 1.4E-2 29 10 10 7 2.8 65 5 5 0.5 12.5 100 1.4 1.4E-2

TABLE 4.5 – VALUES OF DIMENSIONLESS PARAMETERS FOR THE DIFFERENT SCENARIO'S

Test 1 2 3 4 5 6 7 2 2 Wb/ Lb db /Ab ai/Ab Ks/Ab DTt/Lb vTt/ Lb At /Ab [-] [-] [-] [-] [-] [-] [-]

1 1 5E-4 0.00014 1.47E-06 0.002235 0.002235 5E-5 2 0.64 5E-4 0.00014 1.47E-06 0.002235 0.002235 5E-5 3 0.85185185 5E-4 0.00014 1.47E-06 0.002235 0.002235 5E-5 4 1.17391304 5E-4 0.00014 1.47E-06 0.002235 0.002235 5E-5 5 1.5625 5E-4 0.00014 1.47E-06 0.002235 0.002235 5E-5 6 1 5E-4 0.0001 1.47E-06 0.002235 0.002235 5E-5 7 1 5E-4 0.0002 1.47E-06 0.002235 0.002235 5E-5 8 1 5E-4 0.0003 1.47E-06 0.002235 0.002235 5E-5 9 1 5E-4 0.00035 1.47E-06 0.002235 0.002235 5E-5 10 1 5E-4 0.00014 5.28E-06 0.002235 0.002235 5E-5 11 1 5E-4 0.00014 2.78E-06 0.002235 0.002235 5E-5 12 1 5E-4 0.00014 7.75E-07 0.002235 0.002235 5E-5 13 1 5E-4 0.00014 4.09E-07 0.002235 0.002235 5E-5 14 1 5E-4 0.00014 1.47E-06 0.00135 0.002235 5E-5 15 1 5E-4 0.00014 1.47E-06 0.00315 0.002235 5E-5 16 1 5E-4 0.00014 1.47E-06 0.0045 0.002235 5E-5 17 1 5E-4 0.00014 1.47E-06 0.009 0.002235 5E-5 18 1 5E-4 0.00014 1.47E-06 0.002235 0.00135 5E-5 19 1 5E-4 0.00014 1.47E-06 0.002235 0.00315 5E-5 20 1 5E-4 0.00014 1.47E-06 0.002235 0.0045 5E-5 21 1 5E-4 0.00014 1.47E-06 0.002235 0.009 5E-5 22 1 5E-4 0.00014 1.47E-06 0.002235 0.002235 3E-5 23 1 5E-4 0.00014 1.47E-06 0.002235 0.002235 7E-5 24 1 5E-4 0.00014 1.47E-06 0.002235 0.002235 1E-4 25 1 5E-4 0.00014 1.47E-06 0.002235 0.002235 1.5E-4 26 1 3E-4 0.00014 1.47E-06 0.002235 0.002235 5E-5 27 1 4E-4 0.00014 1.47E-06 0.002235 0.002235 5E-5 28 1 6E-4 0.00014 1.47E-06 0.002235 0.002235 5E-5 29 1 7E-4 0.00014 1.47E-06 0.002235 0.002235 5E-5

4.5 RESULTS The results of the tracer runs are the spatial distributions of the average residence times of the cells, which are regarded as estimates of the local residence times. From these spatial distributions, the overall bay’s average residence times are derived, following the procedure as explained in section 4.4. First, the general flushing behaviour of the schematic bay is shortly analysed based on the results regarding the local residence times of the reference case.

Further, the focus is on the resulting average residence times of the tests which are given in section 4.5.2. Based on the results, analytical relations between the dimensionless parameters and the dimensionless flushing parameters are derived and discussed. Finally, this leads to the development of an analytical description of the average residence time: the proposed rapid assessment method.

4.5.1 RESIDENCE TIME DISTRIBUTION To obtain more insight in the behaviour of the ‘system’ (although artificial), three plots of the reference case are briefly discussed, regarding the residence time distribution (Figure 4.4) and the depth averaged flow velocities (Figure 4.4; Figure 4.5).

In the residence time plot (Figure 4.4), a clear pattern is shown: the local residence times increase from the inlet towards the bay. This means that the area close to the inlet is better flushed, which is not a surprise. Further, maximum values of about 40 days are observed.

The velocity profile (Figure 4.4) which indicates the situation at flood tide, clearly shows the velocity increase through the inlet: the tidal jet flow as described in section 2.5.3 which is a typical feature of the tidal pumping mechanism. In the velocity vectors show that two eddies are formed behind the inlet. This is a result of shear around the obstruction, as described in section 2.5.3. The velocities in the rest of the bay, further away from the inlet, are less than 0.05 m/s. These results suggest that the mechanisms at the inlet account for a substantial part of the total exchange.

depth averaged velocity (m/s) Indicative residence time (days) 0.5 08-Jan-2014 05:00:00

50 25 20 0.45 45 0.4 40 18 20 35 0.35

30 0.3 16 25 15 0.25

14 20 0.2

Y coordinate Y(km) coordinate Y coordinate Y (km) coordinate 15 10 0.15 12 10 0.1 5 10 5 0.05 0 22 24 26 28 30 15 20 25 30 X coordinate (km) X coordinate (km) 0

FIGURE 4.4 – RESIDENCE TIMES OF THE REFERENCE CASE (TEST 1) (LEFT) AND THE DEPTH AVERAGED VELCOTIES OF THE REFERENCE CASE DURING FLOOD (RIGHT)

depth averaged velocity (m/s) 08-Jan-2014 05:00:00 19 0.5

0.45 18 0.4 17 0.35

 16 0.3

15 0.25

0.2

14 y coordinate y (km) coordinate 0.15 13 0.1 12 0.05

11 0 18 19 20 21 22 23 24 x coordinate (km) 

FIGURE 4.5 - DEPTH AVERAGE VELOCITY (MAGNITUDE AND VECTORS) OF THE REFERENCE CASE, ZOOMED IN AT THE AREA NEAR THE INLET.

4.5.2 AVERAGE RESIDENCE TIMES The resulting average residence times, as calculated through the tracer method (section 4.4.2) are listed in Table 4.6 together with the corresponding dimensionless flushing parameter.

TABLE 4.6 - RESULTS SCHEMATIC MODEL TESTS

Test Tr [days] Tr /Tt [-] Test Tr [days] Tr /Tt [-]

1 21.78 46.16 16 17.47 33.54 2 42.62 88.16 17 13.42 25.77 3 27.61 57.92 18 21.94 42.12 4 17.60 37.84 19 21.64 41.55 5 13.30 29.60 20 21.44 41.17 6 14.81 28.43 21 20.92 40.16 7 29.25 56.15 22 42.16 80.94 8 49.99 95.98 23 13.63 26.18 9 55.94 107.40 24 8.44 16.20 10 33.56 64.43 25 5.27 10.12 11 27.13 52.09 26 21.33 40.96 12 17.35 33.31 27 20.79 39.91 13 14.24 27.33 28 24.91 47.82 14 25.09 48.18 29 27.52 52.83 15 19.65 37.73

4.5.3 ANALYTICAL RELATIONS To assess the influence of the dimensionless parameters on the dimensionless flushing parameter and to detect relevant relations, for each parameter, the logarithmic values are plotted against the logarithmic values of the corresponding flushing parameter. The graphs are shown in Figure 4.8. Starting with the results of the first 25 runs, so excluding the runs with varying depth parameter values, very high correlations are found through linear regression analysis. This indicates that the relations for each of the dimensionless parameters (excluding ⁄√ ) to the flushing parameter can be described through 2 a power law (PARSA AND SHAHIDI, 2010). A correlation coefficient of R =0.96 is found for the analysis of 6 2 ( ⁄ ) and a value of R =0.99 for the dimensionless parameters 1, 3, 4, 5, and 7. The parameters and their correlations to the dimensionless flushing number are listed in Table 4.7.

TABLE 4.7 - CORRELATIONS OF THE DIMENSIONLESS PARAMETERS TO THE DIMENSIONLESS FLUSHING PARAMETER

2  Parameter Exponent R

1 ⁄ -1.32 0.99

2 ⁄√ -0.77 0.95

3 ⁄ +1.07 0.99

4 ⁄√ +0.34 0.99

5 ⁄ -0.33 0.99 6 ⁄ -0.03 0.96

7 ⁄√ -1.31 0.99

The relation between the aspect ratio ( ⁄ ) and the flushing parameter ( ̅⁄ ) can be described through the following power law:

̅ ( ) 4.11

Similarly, the relations between each of the other dimensionless parameters and the flushing parameter can be represented by power functions:

̅ ( ) 4.12

̅ ( ) 4.13 √

̅ ( ) 4.14

̅ ( ) 4.15

̅ ( ) 4.16 √

The effect of ⁄√ on the tidal flushing number can now be examined through the results of test cases 24 to 29 in which the depth was varied. Assuming a correctly carried out dimensionless analysis and a relation between ⁄√ and ̅⁄ which can also be described by a power function, the flushing parameter can be calculated through the following equation:

̅ ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4.17 √ √ √

In which , and are unknown constants.

This means that the relation of this depth parameter to the dimensionless flushing parameter can be detected by dividing the resulted values of ̅⁄ (of tests 25 to 29) by the known dimensionless parameter values as presented in the above equations 4.11-4.16 and carry out the same procedure which was used for the other parameters (Figure 4.6). Through this analysis the following relation between ⁄√ and ̅⁄ was found to be described by a power function (R2=0.98):

̅ ( ) 4.18 √

FIGURE 4.6 - CORRELATION PLOT OF THE DIMENSIONLESS DEPTH NUMBER WITH THE DIMENSIONLESS FLUSHING NUMBER

This means that the values for , and can be derived and the problem of flushing (regarding the simplified 2D model) can be described by the following exponential function, with a correlation coefficient of R2=0.99 when applied to all the test cases (Figure 4.7):

̅ ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4.19 √ √ √

The results are very interesting in two ways. Firstly, conclusions can be drawn on the relative influence of the main factors which describe the flushing problem, which will be discussed in the following paragraph. Second, the results indicate that the flushing as computed by the simplified numerical model, comprising the relevant basic parameters of a coastal bay, can also be calculated through the above analytical equation. This reveals the possibility of a rapid assessment method in analytical form, which is the most preferred as it is very time efficient.

FIGURE 4.7 – RESULTING DIMENSIONLESS RESIDENCE TIMES FOR TESTS 1 TO 29: COMPARISON OF THE RESULTS OBTAINED WITH THE SCHEMATIC 2D MODEL AND WITH THE DERIVED RAPID ASSESSMENT EQUATION13

13 These are based on the results of EQ 36 instead of Equation 4.19 because in section 4.5.4 will be concluded that ( ) will be left out from Equation 4.19 due to its low influence.

FIGURE 4.8 – REGRESSION PLOTS FOR THE LOGARITHMIC VALUES OF THE DIMENSIONLESS PARAMETERS AGAINST THE LOGARITHMIC VALUES OF THE RESULTING DIMENSIONLESS FLUSHING PARAMETER.

4.5.4 PARAMETER INFLUENCE Before investigating the possibility of the use of the derived formula (equation 4.19) as a rapid assessment method further, first the behaviour of the dimensionless parameters are discussed based on relational functions indicated by equations 4.11-4.16.

Firstly, these functions indicate the nature of the influence of the dimensionless parameters to the dimensionless flushing parameter: the larger the power, the larger its influence. The very low value of the exponent of the viscosity parameter ⁄ indicates that the flushing parameter is hardly influenced by this parameter. This is in accordance with the expectations, as the problem is merely described by the transport equation which does not include the viscosity. The problem of flushing is driven by the tidal emptying and filling and therefore not associated with dynamic flow behaviour, or in other words the transfer of momentum (related to the viscosity parameter) plays in principle no role.

The marginal influence can be explained by tidal pumping effects near the abrupt inlet, introducing some dynamics. Because the influence is so little, the influence of this parameter is not taken into account in further analysis. Consequently this parameter can be deleted from Equation 4.19, resulting in the following equation:

̅ ( ) ( ) ( ) ( ) ( ) ( ) 4.20 √ √ √

High powers were found for parameters ⁄ , ⁄ and ⁄√ indicating the importance of these parameters. The large influence of ⁄√ is no surprise, as the (relative) tidal amplitude is closely related to the volume of water which is available for the exchange (section 2.3).

Further, the high influence of the cross-sectional inlet area to bay surface area ⁄ is not remarkable either and corresponds with findings from literature in which the importance of this parameter to flushing conditions was emphasized. The high influence associated with the aspect ratio ⁄ is understandable as well: intuitively one would expect that the shape of the basin plays a large role in the redistribution (flood) and extraction (ebb) of water and associated pollutants within respectively from the bay.

Smaller but still significant influences of the mixing parameter ⁄ , roughness parameter ⁄√ and the depth parameter ⁄√ were found. The mixing number was expected to have a relatively higher influence, because flushing conditions are assumed to highly depend on the mixing processes. The reason for this could be the dominance of the tidal pumping and quadruplet forming mechanisms near the inlet, as the cross- sectional variation is very abrupt.

In addition to the relative influence of the parameters, the functions also indicate the nature of the relation.

The negative sign of the power shows that parameters ⁄ , ⁄√ , ⁄ and ⁄√ have a negative relation with the flushing parameters: an increasing value of these dimensionless parameters results in a decrease of the flushing parameter. Because the flushing parameter represents the average residence time relatively to the tidal period, this means that higher values of the parameters corresponds with relatively faster or better flushing, or better flushing conditions. These findings are mostly in accordance with the expectations: higher rates of mixing ⁄ and higher relative tidal amplitude ⁄√ are in favour to flushing. For ⁄√ the relation is somewhat more complicated as on the one hand it was expected that relatively larger depth enhance flushing as the tidal currents will be higher; contrary, it also represents a relative volume increase which would suggest the opposite effect. The first effect is obviously dominant.

To continue, the negative power associated with the aspect ratio ⁄ indicates that larger widths compared to the length of the basin have a positive effect on the flushing conditions. This is remarkable as intuitively one would expect that a ratio of 1 would be most favourable to flushing. This can be explained through the tidal pumping effects, which are less when the width compared to the length is smaller: there is less ‘space’ for the ebb ‘sink flow’ as explained in section 2.5.3, which results in a returning flow consisting for a great part of incoming water. This would mean that this effect is closely related to the near inlet width.

Contrary, the parameters ⁄ and ⁄√ are positively related to the dimensionless flushing parameter. Larger (relative) roughness means a decrease of current velocities, which indeed has a negative effect on the flushing conditions. The relation inlet cross-sectional area to bay surface is however more complicated. Intuitively one would expect that a smaller value of this ratio would leads to lower flushing rates as the connection with the external ‘fresh’ water source (the sea) is relatively smaller. However, the tests show the opposite. This can be explained by the tidal pumping and quadruplet forming effects which are emphasized when the cross-sectional inlet is smaller: by decreasing the inlet width, the cross-sectional variation is more abrupt and a decrease of cross-sectional flow area also increases the tidal jet velocities enhancing tidal pumping and thus flushing (section 2.5).

4.5.5 RETURN FLOW FACTOR

In section 4.2.2, it was shown that the classical tidal prism model can be re-written using the average depth , tidal amplitude and tidal period (equation 4.6). To recall, this tidal prism model was improved by the addition of the return flow factor , which has a value between 0 and 1. When the constant factor (+0.32) in equation 4.20 is neglected, which is assumed to be appropriate because its value is very small compared to the typical order of magnitudes of the dimensionless flushing parameters of coastal bays, a description of the return flow factor can be derived, based on the basic parameters. Similarly to equation 4.6, the tidal prism model with the return factor (equation 3.5) can be written using the basic bay parameters of the present study:

( )

Re-writing equation 4.20 in the same form, leads to the determination of the return flow factor:

̅ ( ( ) ( ) ( ) ( ) ( ) ( ) )

√ √ √

When the term between brackets in equation is denoted as , the return flow factor . For the reference case this leads to a return factor of is found. This high value illustrates once again that the tidal prism model produces overestimates of the average residence times.

4.6 METHOD PERFORMANCE Before applying the derived formula (equation 4.20) to realistic cases, several additional tests are carried out to further investigate the performance of the method and to obtain better insight into the way the model should be used in practice, for example when the conditions are not so ‘perfect’ as in the schematic models. This chapter is concerned with the influence of the surface area, the tidal wave and the inlet channel respectively.

It should be noted that for the purpose of this paragraph, the emphasis lies on the estimation of the absolute average residence times, although the primary goal is to be able to use the RAM to determine the relative change of the averages residence time considering different situations. Equation 4.20 is from now on referred to as the (proposed) rapid assessment method, or RAM.

4.6.1 SURFACE AREA In each of the 29 tests (Table 4.4) only one dimensionless parameter was varied (with the exception of the last four tests). If the dimensional analysis was carried out correctly, the equation should also be valid when more dimensionless parameter values are changed at the same time. This can perfectly be verified by changing the value of the bay’s surface area . This parameter remained constant through the analysis as it appears in more than one dimensionless parameter.

In order to check whether the derived equation (4.20) is still valid for varying bay’s surface area, the same set up as in the reference case (run 1, Table 4.4) are used, but with changing values for . The set up for the three test cases are shown in Table 4.8, together with the resulting values for the average residence time, calculated from the 2D model and from the rapid assessment method (RAM) in the form of 4.20. The minor differences between the flushing times calculated by the schematic 2D model and those computed through 4.20 indicate on a correctly derived equation to predict the outcomes of the 2D model.

TABLE 4.8 – RESULTS TESTS WITH VARYING BAY'S SURFACE AREA

Test Bay width Bay length Bay surface Tr Tr [km] [km] area 2D model RAM [km2] [Days] [Days] 30 5 5 25 19.2 18.5 31 8 8 64 20.1 20.8 32 12 12 144 23.0 23.0

4.6.2 TIDAL FORCING

TIDAL PERIOD Similarly to the surface area, for the tidal period only one value was used in the test cases (Table 4.4), namely the M2 tide with an approximate period of 12.5 hours (or 45000 seconds). However, as mentioned before, other tidal components and associated periods can dominate in certain areas. It has been mentioned that in

Doha Bay next to the M2 tide also the K1 tidal constituent is important. Therefore a test is carried out with the

K1 tidal period, which is approximately 24 hours (diurnal). All other parameters remain the same as in the reference case (run 1, Table 4.4). The average residence time obtained from the 2D model is ̅=19.9 days.

Compared with the average residence time ̅=18.5 days as calculated by 4.20, there is a deviation of less than 10%. This is assumed to be small enough to suggest that the derivation of 4.20 was successful and that the equation is also valid for diurnal tidal periods.

The analytical method is based on one single sinusoidal signal with constant period and amplitude. However, in reality the tidal wave consists of many constituents, all with different periods and amplitudes. Most environments can be categorized as diurnal, semi-diurnal or mixed tidal regimes. In the first two cases either diurnal or semi-diurnal constituents dominate, which intuitively leads to the assumption that in such cases the tidal period should be diurnal respectively semi-diurnal. The latter case is more complicated.

TIDAL AMPLITUDE For the tidal amplitude the average tidal amplitude should be taken. Care should be taken with the use of the tracer method in the case of distinct spring and neap tidal cycles. The start time of the simulation influences in such cases the resulting average residence time. To illustrate this effect and to investigate in which order of magnitude this can influence the results, several tests were conducted with two tidal components: M2 with a period of 12.5 hours (as in the previous test cases) combined with the S2 tidal constituent which has a period of 12 hours. All the other settings remain the same as in the reference case.

The following results are obtained regarding the average residence times:

TABLE 4.9 – RESULTS OF TESTS WITH COMBINED M2 AND S2 TIDAL COMPONENTS

Test M2-tide S2-tide Tr Tr 2D model RAM [Days] [Days]

34 at=0.5m; ϕ=0° at=0.5m; ϕ=0° 12.9 14.8

35 at=0.5m; ϕ=0° at=0.5m; ϕ=90° 13.4 14.8

36 at=0.5m; ϕ=0° at=0.5m; ϕ=120° 14.0 14.8

37 at=0.5m; ϕ=0° at=0.5m; ϕ=150° 14.2 14.8

38 at=0.5m; ϕ=0° at=0.5m; ϕ=180° 15.3 14.8

Test 34 started at spring tide and test 38 at neap tide. The average tidal amplitude is approximately the maximum tidal amplitude ( =0.5+0.5=1m) divided by two squared (20.7), which results in an averaged amplitude of =0.7m. Using the amplitude of the S2 period to calculate the average residence time through

4.20 results in ̅=15.5 days. First of all this shows that the equation including only the average tidal amplitude is capable of being used for the combination of two tidal constituents, as the derived average residence time lies within the range as computed by test 34 to 38 (Table 4.9). Further, it can be concluded that, due to the differences between spring and neap tides, ‘actual’ average residence times can vary for about 10% of the average residence times as calculated by 4.20 due to spring and neap tidal differences which are not accounted for by the equation.

4.6.3 INLET CHANNEL In the schematic 2D model, the cross-sectional inlet depth was similar to the average depth of the bay. In reality, the inlets often accommodate channels which are much deeper than the average bay width, which is also the case in Doha Bay. What the effect of this channel is and whether the RAM can be used to predict the average residence time when a channel is present, is tested by additional 2D model tests with a modified bottom profile (Figure 4.9). Channels with maximum depths of 15m, 16m, 17m and 20m were constructed. The transitions between the channel bottom and the depth besides the channel were smoothened as shown in Figure 4.9. The results are shown in Table 4.10.

TABLE 4.10 – RESULTS TESTS WITH DEEP INLET CHANNELS

Test Maximum Cross-sectional Average bay Tr Tr channel depth inlet area Depth 2D model RAM [m] [m2] [m] [Days] [Days] 39 15 34800 6.2 64.5 55.0 40 16 36850 6.3 69.1 58.1 41 17 38900 6.5 73.5 60.7 42 20 45000 6.8 82.0 69.5

The resulting average residence times calculated by RAM, using the cross-sectional inlet area and average bay depth as in Table 4.10 and all other settings equal to the reference case (Table 4.4) are significantly lower (approximately 20%) than those by the 2D model. However, the relative deviation from the 2D model results is constant. This means that the use of the formula with respect to the relative changes of the flushing characteristics due to bay developments seems to be appropriate even when an inlet tidal channel is present.

However, when the results are compared to the reference case, in which no inlet channel was present, the results suggest that the presence of a channel decreases the exchange. This is remarkable, because the influence of a deep channel is assumed to be in favour of flushing because they accommodate high current velocities through the inlet. That the tidal channel in the schematic model has the opposite effect, might be related to the relative scale of the channel which is large compared the relative scale of inlet channels in real cases. To conclude, the performance of the RAM when a tidal inlet is present is rather doubtful, especially when the absolute values are concerned.

TABLE 4.11 - RESULTS TESTS WITH DEEP INLET CHANNELS FOR THE ADJUSTED NIKURADSE ROUGHNESS HEIGHT

Test Maximum Cross- Average Total water Nikuradse Tr Tr channel sectional bay depth [m] roughness 2D RAM depth [m] inlet area Depth (H in eq.4.8) height [m] model [Days] [m2] [m] [Days] 39 15 34800 6.2 12.4 0.036433 64.5 69.3 40 16 36850 6.3 13.2 0.038666 69.1 74.2 41 17 38900 6.5 13.9 0.041 73.5 78.2 42 20 45000 6.8 16.7 0.047 82.0 92.6

Bed level (m) 35 -5

-10 30

-15 25 -20

20 -25

15 -30 Y coordinate Y(km) coordinate -35 10

-40

5 -45

0 0 5 10 15 20 25 30 X coordinate (km)

FIGURE 4.9 - BATHYMETRY OF SCHEMATIC MODEL WITH AN INLET CHANNEL WITH A DEPTH OF 15 M

4.7 DISCUSSION Through the systematic approach of a dimensional analysis, the relations between basic bay and tidal parameters to the overall flushing conditions of the bay were detected. Using these relations, the following analytical box-type of formula was derived to estimate the average residence time:

̅ ( ( ) ( ) ( ) ( ) ( ) ( ) ) 4.21 √ √ √

Through several test cases using the schematic 2D model set up, it was shown that this formula is also capable of producing relatively accurate estimates of the average residence times of the schematic bay model in the following situations which were not accounted for through the initial test cases:

 Varying bay’s surface area  Diurnal tidal period  More than one tidal constituent

In the case of more than one tidal constituent, the average tidal amplitude and the smallest tidal period should be chosen to represent the tidal force through the parameters in equation 4.21.

When a deep tidal channel is present, the RAM was found to underestimate the values of the average residence time. The results for the inlet channel raise doubts on the correct representation of the near inlet processes by the RAM. It remains uncertain whether all types of modification to the inlet can be accounted for through the RAM. This should be taken into account when situations with different inlet configurations (compared to the schematic models) are considered.

Further, it was shown that spring and neap tidal cycle can influence the absolute results in the order of 10%. In theory, this should however not be a problem when one is concerned with the relative flushing conditions.

To conclude, equation 4.21 can be used to estimate the outcomes of the schematic 2D model, even when more realistic configurations and tidal forcing are concerned. This suggests that the box-type formula can be used as a rapid assessment method for the flushing of coastal bays. However, because the schematic model is, like the name reveals, only a schematic representation of reality, for the application of this proposed rapid assessment method (RAM) to practical cases, validation of the formula is paramount.

5 VALIDATION Validation of the derived analytical equation to be used for the rapid assessment of the flushing conditions is crucial to indicate its predictive value in practice. This is done by comparing the results concerning the average residence times calculated by the RAM with those computed with the use of a detailed numerical model. Three cases are considered: Kuwait Bay, Boston Harbour and Venice Lagoon. Kuwait Bay is chosen because it fits in the category of bays considered and shows similarities with Doha Bay regarding the bathymetric and geometric irregularities. Further, Boston Harbour is much more irregular. Finally, as the name reveals, Venice Lagoon is used to assess the possibility of applying the method to lagoon types of coastal inlet waters, with more than one inlet and very small water depths (<2m).

5.1 APPROACH The validation to practical cases is not so straightforward. The first problem is that the actual average residence times cannot be measured. Therefore the answers of the RAM are compared to the answers obtained by the use of a detailed numerical model which is assumed to be the best possible alternative. The problem which then arises is that two of the three available validation cases are concerned with the definition of the turnover time (section 3.1.2) instead of the average residence time. Nevertheless, because the order of magnitude of both parameters is assumed to be similar14, the validation by comparison with these cases can still give valuable insights on the behaviour and field of applications of the rapid assessment method.

Another difficulty is the role of the dispersion coefficient. In the RAM the dispersion coefficient should account not only for the sub-grid scale dispersion which was not included in the schematic bay model, but also for the spreading as a result of motions induced by irregular bathymetry and geometry: in the schematic bay model these were not accounted for because the bottom and geometry were unrealistically regular shaped. On the other hand, the dispersion coefficient in the detailed numerical model is supposed to represent the rate of spreading which is not resolved for by the model. Its value is thus closely related to the model features.

It is therefore difficult to compare these values and to choose the ‘correct’ value to be used in the rapid assessment method. Therefore the dispersion coefficient is in fact used as a calibration parameter. To summarize, the approach is as follows:

1. An estimation of the average residence time is derived from a scientific paper which concerns the flushing assessment using a detailed numerical model of the bay of interest.

2. The values for the parameters of the RAM are obtained from the information given in the paper.

3. The RAM is conducted several times, using different values of the dispersion coefficient.

4. The answers of the RAM are compared to the average residence times from the paper: the dispersion coefficient, for which the average residence time calculated by the RAM matches the average residence time computed by the detailed model, is detected.

5. The value of the dispersion coefficient is evaluated, assisted by the given ranges of the value of the dispersion coefficient of 1 to 20 m2/s and O(1) to O(100) m2/s (section 2.6.6).

14 For the schematic model tests the flushing times were also derived, next to the average residence times. The difference in absolute value was about 10%. For the later test cases using the detailed model of Doha Bay (section 6), a similar deviation between the two parameters was found. 49

5.2 KUWAIT BAY The first case is Kuwait Bay, an elliptical shaped bay protruding from the Arabian Gulf (Figure 5.1). Due to increased bay activity (e.g. ports, power plants), concerns have risen regarding the quality of the water. Therefore several flushing studies have been conducted (POKAVANICH AND ALOSAIRI, 2014; RAKHA et al., 2010). For validation purposes the study of RAKHA et al., (2010) is the most suitable as it accounts for tidal forcing only and the residence times are computed by the tracer method. This was done twice, using different values for 2 2 the constant dispersion coefficient: =2 m /s and =5 m /s. The resulting turnover times were =400 days 2 2 when =2 m /s was used and =200 days for =5 m /s.

These values have to be compared to the average residence times calculated by 4.20. The relevant parameters are estimated from the data in the article RAKHA et al., (2010) and listed in Table 5.1. For the width to length ratio (aspect ratio), the part of the bay which has its depth below sea water level is considered (Figure 18). This results in a width to length ratio of about 1/1.5 (Figure 5.2). With a surface area of 720 km2 this leads to values for and of approximately 33km and 22km respectively.

FIGURE 5.1 – LOCATION OF KUWAIT BAY IN ARABIAN GULF. FROM RAKHA ET AL. (2010)

TABLE 5.1 – PARAMETER VALUES REPRESENTING KUWAIT BAY

Parameter description Parameter Dimension Value

Average bay length Lb [km] 33

Average bay width Wb [km] 22 2 Bay surface area Ab [km ] 720 2 Cross-sectional inlet area ai [m ] 240000

Average bay depth db [m] 5.2 Chezy coefficient C [m/s] 65

Nikuradse height Ks [m] 1.5E-2

Tidal period Tt [s] 45000

Average tidal amplitude at [m] 1.5

For the width to length ratio (aspect ratio), the part of the bay which has its depth below sea water level is considered (Figure 18). This results in a width to length ratio of about 1/1.5 (Figure 18). With a surface area of 2 720 km this leads to values for and of approximately 33km and 22km respectively.

Calculations of the average residence time through 4.20 with the parameter values of Table 5.1 and varying dispersion coefficient results in the graph shown in Figure 19. Figure 19 shows that the value of the dispersion 2 coefficient which leads to an average residence time of ̅=400 days is approximately 1 m /s. Similarly, a value 2 2 of 7 m /s corresponds with the results of RAKHA et al. (2010) using a dispersion coefficient of 5 m /s in the detailed model.

Considering the processes which the different dispersion coefficients represent, it is remarkable that similar values of the dispersion coefficient in the RAM and the detailed numerical model respectively are used to obtain the same results concerning the average residence time. The dispersion coefficient in the RAM should account for the additional rate of spreading by the motions which are not represented in the schematic model, for example those that are induced by the irregular bathymetry of Kuwait Bay. Higher values of the dispersion coefficient are thus expected to represent this additional dispersion. The RAM therefore seems to slightly underestimate the average residence time. However, the values lie within the common range (section 2.6.6).

FIGURE 5.2 – BATHYMETRY KUWAIT BAY. FROM RAKHA ET AL. (2010). INDICATION OF LENGTH TO WIDTH RATIO

FIGURE 5.3 – CALCULATED AVERAGE RESIDENCE TIMES BY THE RAPID ASSESSMENT EQUATION FOR VARYING VALUES OF THE DISPERSION COEFFICIENT

5.3 BOSTON HARBOUR The second validation case is Boston Harbour which is a shallow water, tidally dominated embayment in Massachusetts Bay (Figure 5.4). This bay is used for validation purposes, as its appearance differs greatly to that of Kuwait Bay and a detailed description of a flushing study of this region is available (SIGNELL AND BUTMAN, 1992). The embayment is characterized by its irregular geometry and bathymetry, comprising many flats and shoals, channels and side embayments. The need to obtain more information on the flushing conditions of Boston Harbour stems from the environmental problems the region is facing, due to waste discharges from the Boston metropolitan area in combination with other polluting activities such as dredging. SIGNELL AND BUTMAN (1992) have performed a detailed flushing study using a numerical depth-averaged flow model and tide as only forcing. Futher, the tracer method was used to compute a turnover time of 17 days.

FIGURE 5.4 – BOSTON HARBOUR IN MASSACHUSSETS BAY WITH TWO INLET CHANNELS: PRESIDENT ROADS AND NANTASKET ROADS

The relevant parameters to use the analytical equation (4.20) are subtracted from the information given in the article (SIGNELL AND BUTMAN, 1992) and listed in Table 5.2 (case A). For the length to width ratio roughly 1/2 is 2 taken. With a surface area of approximately 125 km this leads to a length of 7.9km and width 15.9km. However, because the inlet accommodates two deep channels (Figure 5.4), it seems more appropriate to consider two parts separately (case B, Table 5.2).

TABLE 5.2 PARAMETER VALUES REPRESENTING BOSTON HARBOUR

Parameter Parameter Dimension Value Value description (Case A) (Case B) Parts 1 and 2

Average bay length Lb [km] 7.9 7.9

Average bay width Wb [km] 15.9 7.9 2 Bay surface area Ab [km ] 125 62.5 2 Cross-sectional inlet ai [m ] 60000 30000 area

Average bay depth db [m] 4.9 4.9 Chezy coefficient C [m/s] 50 50

Nikuradse height Ks [m] 9.8E-2 9.8E-2

Tidal period Tt [s] 45000 45000

Average tidal at [m] 1.35 1.35 amplitude

With a value of 12 m2/s, the derived average residence time for case A equals the calculated turnover time of 17 days. A slightly larger value was found for the average residence times of separate parts (case B): 16 m2/s. The differences between case A and case B is the result of the differences in the aspect ratios used in the RAM. It suggests that such a separated approach can indeed be valuable.

As expected, the values for the dispersion coefficients are higher than those which were derived for Kuwait Bay. The irregular configuration of Boston Harbour is assumed to be responsible for the inducement of shear and (residual) current motions which leads to enhanced dispersion which should be represented by the dispersion coefficient. This suggests that the relative influence of the remaining input parameters in the RAM on the flushing conditions is well represented.

The absolute value of the derived dispersion coefficient seems to be somewhat on the low side for such an irregular environment.

5.4 VENICE LAGOON Venice Lagoon is a lagoon in the Adriatic Sea (Figure 5.5). In many ways, Venice Lagoon is similar to the type of bay considered in this study (section 2.2): its topographical scale is relatively small, the tidal force is dominant and the influence of fresh water is negligible. However, its depth is characterized as very shallow with an average depth of =1.5m, wind forcing is assumed to play a major role next to the forcing by tides, the lagoon has three distinct inlets and the length to width ratio is very large ( / 5). Therefore this case is used to investigate the limits of the application area of the rapid assessment method.

For this purpose, the outcomes of the study of GUCCO AND UMGIESSER (2006) are used to make comparisons. For the tide only condition (also cases with wind were considered), an average residence time of ̅44.5 days was calculated by the tracer method.

FIGURE 5.5 – VENICE LAGOON IN ADRIATIC SEA (LEFT) AND LOCATION OF THREE PARTS (RIGHT). FROM GUCCO AND UMGIESSER (2006)

Similarly to Boston Harbour, for the calculation of the average residence times by the RAM, two cases are considered: case A in which the lagoon is schematized as one water body with one inlet and case B in which the area of the lagoon is separated into three parts, one per inlet. The three sub-areas are shown in Figure 5.5 (right). The parts 1, 2 and 3 correspond with the section SB, CB and NB as indicated in Figure 5.5 (left). The parameter values are listed in Table 5.3.

For case A, a dispersion coefficient of 0.05 m2/s leads to the same average residence time for the entire bay as derived by GUCCO AND UMGIESSER (2006).

The average residence times of the three parts were also considered in the study of GUCCO AND UMGIESSER

(2006), which resulted in the following average residence times for parts 1, 2 and 3 respectively: ̅30 days,

̅42 days and ̅50 days. To obtain similar results through the RAM, the following dispersion coefficient are found for 1, 2 and 3 respectively: 0.05 m2/s, 0.08 m2/s and 0.15m2/s.

These values are very low, which is unrealistic. Venice Lagoon is a very irregular environment with comparable conditions as Boston Harbour concerning the rate of spreading. For Boston Harbour however the values that were found for the dispersion coefficients were orders of magnitudes larger. These results give the impression that the RAM is not applicable to lagoons or bays with very small water depths (<2 m), at least not to estimate the absolute value of the average residence time.

TABLE 5.3 – PARAMETER VALUES REPRESENTING VENICE LAGOON

Parameter Parameter Dimension Value Value Value Value description (Case A) (Case B) (Case B) (Case B) Part 1 Part 2 Part 1

Average bay length Lb [km] 9 9.0 11 9

Average bay width Wb [km] 46 9.0 15.5 18 2 Bay surface area Ab [km ] 415 80 170 160 2 Cross-sectional inlet ai [m ] 24500 4000 6500 14000 area

Average bay depth db [m] 1.5 1.5 1.5 1.5 Chezy coefficient C [m/s] 65 65 65 65

Nikuradse height Ks [m] 9.8E-2 9.8E-2 9.8E-2 9.8E-2

Tidal period Tt [s] 45000 45000 45000 45000

Average tidal at [m] 0.5 0.5 0.5 0.5 amplitude

5.5 DISCUSSION The results of the three validation cases are summarized in Table 5.4. The results of the cases of Kuwait Bay and Boston Harbour are first of all promising, regarding the predicted value of the rapid assessment method: the values for the dispersion coefficients found lie within the common range, which suggests that the RAM produces indeed respectable estimates of the average residence time. Further, for Boston Harbour, which has more irregular features than Kuwait Bay, higher values were found than those for Kuwait Bay. This corresponds to the expectations based on the theory of the flushing process (section 2.3.2).

The results of Kuwait Bay and Boston Harbour also suggest that the RAM overestimates the flushing conditions as relatively low dispersion coefficients were needed to obtain the same results as in the papers. There are several explanations to this. First, this can be related to the tidal pumping effect which is assumed to be overestimated by the schematic 2D models due to the schematic representation of the inlet. It can also be attributed to the influence of the inlet channel. Both Kuwait Bay as Boston Harbour have relatively deep inlet channels. In section 4.6.3 it was shown that the RAM in such cases indeed produces relatively low average residence times.

The results of Venice Lagoon reveal that the RAM is not applicable for lagoon type of environments, with very shallow water depths. However, when not the absolute but the relative flushing conditions are concerned, the RAM might be applicable to these types of bays too. This should be further investigated.

Using Boston Harbour as an example, the RAM seems to be applicable to bays with more than one inlet. In that case it is recommended to divide the area into parts and calculate the average residence times per part.

To conclude, the rapid assessment method seems to be valid for the application to environmental bays with one or more inlets. In principle the rate of spreading as a result of irregular bay configurations can be accounted for through the dispersion coefficient. It is recommended to use the RAM for the relative comparison of different cases; the absolute numbers should be handled with great care.

TABLE 5.4 – OVERVIEW OF RESULTS VALIDATION CASES

2 Case Detailed study by Tt or Tr [days] D [m /s] RAM Kuwait Bay 2 Case A - D=2 m /s RAKHA et al. (2010) 400 1 Case B - D=5 m2/s 200 7 Boston Harbour Case A SIGNELL AND BUTMAN (1992) 17 12 Case B - part 1,2 - 15 Venice Lagoon Case A GUCCO AND UMGIESSER 45 0.05 (2006) Case B - part 1 30 0.05 Case B - part 2 42 0.08 Case B - part 3 50 0.15

6 APPLICATION TO DOHA BAY In the current chapter the performance of the rapid assessment method is tested by application to Doha Bay. The influence of a number of bay changes on the bay’s flushing conditions is assessed, under which the future planned developments Sharq Crossing and Oryx Island (section 1.1). The predictive value of the rapid assessment method is analysed by comparing the estimates of (the relative change of) average residence times to the values obtained by using a detailed numerical depth-averaged flow model.

First a description of the cases of Doha Bay is given, which is followed by an explanation of the set-up of the numerical model of the area which will be used for the comparison of the results of the RAM. After this, the focus is on the test cases, which involves the approach, the test program and the results respectively.

6.1 CASE DESCRIPTION Doha Bay has been introduced briefly in section 2.1.1. For completeness, some more background on the area, climate and hydrodynamic behaviour are given in this paragraph. In addition, the two proposed future developments Sharq Crossing and Oryx Island, which were mentioned in the introduction (1.1), are further described. Inevitably some fragments are repeated.

6.1.1 LOCATION AND AREA Doha Bay is located at the eastern coast of the Qatarian peninsula in the Arabian Gulf, an inner sea which is connected to the Gulf of Oman by the Strait of Hormuz (Figure 6.1). With maximum depths of around 100 m, it is a shallow sea. Doha Bay lies in the southern part of the Gulf where water depths range from 10 to 30 m (Figure 6.1).

FIGURE 6.1 – BATHYMETRY OF THE ARABIAN GULF AND THE LOCATION OF QATAR (LEFT). DOHA BAY (RIGHT)

Doha Bay is a half-open basin which comprises an area of about 45 km2 (Figure 6.1). In the north the artificial island group Pearl Island is located. The bay has one main connection to the Gulf with a width of approximately 4.5 km which is bounded by the airport extension in the south and Al Safliya Island to the north (Figure 6.1). The shallow flats at the eastern side of the bay reduce the opening’s effective width. In addition, the bay is connected through a smaller inlet at the north, with a width of no more than 1.0km (Figure 6.1). With an average depth of 5-6m and local depths of only 2 m, Doha Bay is considered a shallow bay. For navigational purposes a channel has been dredged from offshore approaching Doha Bay from the south-east. The bed of the navigational channel is maintained at a level between 12 to 14 m below Chart Datum (Figure 6.4). 56

3.1.1 CLIMATE The Arabian Gulf is located between latitudes 24°-30° N, a desert region that can be characterized by high temperatures, low rainfall, high humidity and high evaporation (REYNOLDS, 1993). Due to the extra tropical weather climate, evaporation and salinity rates are high. Also, seasonal variation in temperature and inherently pressure results in about 0.2 m higher mean water levels in the summer than in the winter. Further, there is no significant source of fresh water discharging into the bay (e.g. river).

The wind climate in the Gulf is largely determined by the seasonal event called ‘Shamal’, meaning ‘north’ in Arabic. A distinction is made between summer and winter Shamal, of which the latter is the most extreme in its occurrence (REYNOLDS, 1993). Winter Shamal is recognized by its abrupt appearance and characteristic duration between 24-36 hours, occurring several times per month during winter time (PERRONE, 1979). In off- Shamal periods, which covers most of the year, the wind climate in Doha Bay is assumed mild (Appendix A): offshore conditions are moderate throughout most of the year and with the dominant wind direction is offshore (Appendix A), the wind speeds at Doha Bay are assumed to be much smaller.

3.1.2 TIDES, WAVES AND CURRENTS The dimensions and the bathymetry of the Arabian Gulf are such that a system of ‘standing waves’ is generated. The combination of the two main components K1 and M2 result in a mixed tidal regime as both dominant diurnal and semi-diurnal constituents are present, with an average tidal range of 1.2 m (Appendix A).

Due to the narrow and curved Strait of Hormuz swell waves are not able to penetrate in the Gulf. As a result the wave climate within the Gulf and similarly in Doha Bay is a result of local wind-generated waves. During extreme Shamals, waves will be depth-limited in the relatively shallow southern part of the Gulf.

Outcomes of the numerical model of Royal HaskoningDHV, which is described in section 0, suggest that currents in the bay are moderate with maximum tidal current speeds less than 0.5 m/s (Appendix A).

3.1.3 FUTURE PLANNED DEVELOPMENTS After the construction of Pearls Island in 2013 (Figure 6.1) and the extension of the airport, the development of Doha Bay has not finished. Two major projects are currently going on which involve changes to the bay: Sharq Crossing and Oryx Island.

SHARQ CROSSING

FIGURE 6.2 – LOCATION OF SHARQ CROSSING IN DOHA BAY (LEFT) AND A VISUALISATION OF ONE OF THE DESIGN OF ONE OF THE THREE BRIDGES: THE CULTURAL CITY BRIDGE (RIGHT).

To cope with the increased traffic intensity near the bay as the result of Qatar’s rapid industrialization and economic growth (RICHER, 2009), plans are on the table for the construction of the bridge-tunnel Sharq Crossing. The current design by Calatrava comprises three bridges which are interconnected through two submerged tunnels (Figure 6.2). In addition three short tunnel segments connect the three bridges to shore. One of the three bridges is shown in Figure 6.2. At the transition of the bridges to the tunnels artificial Island are located with small marinas. These Island, together with the bridge pillars will influence the current pattern within the bay. This could in turn have an impact on the tidal exchange or flushing.

ORYX ISLAND As mentioned before, plans are on the table for a second artificial island in Doha Bay: Oryx Island. The main purpose of the Island is to attract more tourism in the area. The current design shows an area of approximately 200 km2 covered with hotels, restaurants, shops, residential towers, resorts, private beaches, water parks etc. (Figure 6.3).

FIGURE 6.3 – ORYX ISLAND. LOCATION IN THE DOHA BAY (LEFT) AND A VISUALISATION OF THE DESIGN

6.2 MODEL DESCRIPTION For the preparation of Sharq Crossing, Royhal HaskoningDHV conducted a hydrographical impact study, using a detailed 2D depth-averaged flow model in Delft3D-FLOW (Delft3D-FLOW Manual, Deltares). This model is used to relate to the outcomes of the rapid assessment method. Relevant information of the model is listed in this paragraph comprising the grid configuration, depth profile and the boundary conditions.

6.2.1 GRID AND BATHYMETRY For the numerical flow model computations, a curvilinear grid with 310x200 grid cells and a spatial resolution of 30-80m was constructed (Figure 6.4). For the bathymetry of the bay survey data was used compiled from a bathymetric survey of Qatar coastal waters in 2002 (Hydrographical Impact Study Sharq Crossing, Royal HaskoningDHV). For the offshore conditions the CMAP digitalized nautical maps15 were used. The bathymetry is shown in Figure 6.4.

15 Admirality Chart Data made available by Jeppesen Marine, 2012 58

FIGURE 6.4 – GRID (LEFT) AND BATHYMETRY (RIGHT) OF NUMERICAL FLOW MODEL DOHA BAY. FROM HYDROGRAPHICAL IMPACT STUDY SHARQ CROSSING16

6.2.2 BOUNDARY CONDITIONS The flow conditions for the Doha Bay model were derived from the calibrated Arabian Gulf Model, which is a large scale flow model of the entire Arabian Gulf developed and owned by Deltares (Appendix C). The Arabian Gulf model includes 30 years of 3 hourly wind, wave, currents and water level data and 10 years of 1 hourly data (Hydrographical Impact Study Sharq Crossing, Royal HaskoningDHV). To obtain the required level of detail three subsequent nesting procedures were conducted. Incomplete flow conditions of the largest scale model (Level A, Figure 6.5) were established from nesting of the Arabian Gulf model. The necessary information for the level B model was in turn derived from the level A model. Finally the boundary conditions of the level C model were obtained through nesting of the B model (Figure 6.5).

The Arabian Gulf model, the level A model and the level B model are all calculated on rectangular grids. For the detailed model of Doha Bay (level C) however a curvilinear grid is used. The model used for the present study was forced by the astronomical tide only describing water levels at the eastern boundary and current velocities at the eastern and northern boundaries.

FIGURE 6.5 – OVERVIEW OF THREE LEVELS OF FLOW MODELS USED FOR NESTING; LEVEL A (RED), LEVEL B (GREEN) AND LEVEL C (YELLOW).14

16 From the Hydrographical Impact Study Sharq Crossing by Royal HaskoningDHV 59

6.2.3 AVERAGE RESIDENCE TIME CALCULATION The bay’s average residence time is calculated with the tracer method, following the same procedure as carried out for the schematic model tests. The different steps which are explained in section 4.4 comprise the 3 3 simulation of a restart run; the ‘injection’ of a uniform concentration of =2kg/m in the bay and =1kg/m in the sea; the calculation of the average residence times per cell which is estimated by the time it takes for the concentration within the cell to decrease to 1.37kg/m3; the calculation of the total average residence time which is the weighted mean of the local residence times. It should be emphasized that contrary to the schematic models in which the cell volumes are equal, the scaling of the derived ‘local’ residence times (the average residence time of one computational cell) to the volume of the cells is crucial.

Based on the model performance, the simulation time of the restart run is set to seven days, starting from April 10, 2006. The simulation time of the tracer run is 2.5 months. This is assumed to be large enough to reach the local residence times. These and the other input parameters are listed in Table 6.1.

TABLE 6.1 – VALUES OF INPUT PARAMETER FOR THE DETAILED DOHA BAY MODEL, REFERENCE CASE

Parameter description Parameter Dimension Value Restart run Time step min 0.1 Simulation start time DD-MM-YYYY-HH-MM-SS 10-04-2006-00-00-00 Simulation stop time DD-MM-YYYY-HH-MM-SS 17-04-2006-00-00-00 Bottom roughness m/s2 65 2 Horizontal eddy viscosity m /s 1 2 Horizontal eddy diffusivity m /s 1 Tracer run Time step min 0.1 Simulation start time DD-MM-YYYY-HH-MM-SS 17-04-2006-00-00-00 Simulation stop time DD-MM-YYYY-HH-MM-SS 30-06-2006-00-00-00 Bottom roughness m/s2 65 2 Horizontal eddy viscosity m /s 1 2 Horizontal eddy diffusivity m /s 1

6.3 APPROACH

6.3.1 RELATIVE CHANGES TO THE FLUSHING The primary concern of the rapid assessment method is the estimation of the relative influence of bay changes on the flushing conditions, which are indicated by the average residence time. Therefore several cases are set up in which the configuration of Doha Bay is adapted. Among the adaptations are both hypothetical bay changes (e.g. deepening of the channel) and actual proposed developments (e.g. Sharq Crossing).

First the value of the dispersion coefficient to be used in the RAM is determined through calibration. For this purpose, the average residence time of the reference case, which reflects the current situation in Doha Bay, is estimated by the RAM for varying values of the dispersion coefficient.

Subsequently, the value of the dispersion coefficient that results in a similar average residence time as calculated by the detailed model is determined. This value is then used to represent the rate of spreading using the RAM in the further cases.

When the value of the dispersion coefficient is obtained, the average residence time is computed for varying bay configurations, each of which is addressed to a specific modification of the current situation. The computations are carried out both with the RAM and the detailed numerical model. Because the relative changes of the flushing conditions are the primary concern, the focus is on the increase or decrease of the average residence times computed for these cases relatively to the reference case. The RAM is considered to be successful when it is capable of approximating the order of the relative change of the average residence time derived by the detailed model.

6.3.2 DISPERSION COEFFICIENT Because the value of the dispersion coefficient to be used in the RAM and its relation to the dispersion coefficient used in the numerical model remain highly uncertain, in practise the dispersion coefficient will be used as a calibration parameter: one value of the dispersion coefficient is estimated and this value is further kept constant when the relative influence of changes to the bay are being assessed. This is the same approach which is explained in section 6.3.1. This approach works in principle quite well: the relative change of the flushing conditions is in this way independent on its precise value.

The problem of this approach however is the fact that the increase or decrease of the overall spreading as a result of the considered bay change(s) is not accounted for: to include the influence of such changes, the value of the dispersion coefficient should be adjusted. Although it is impossible to determine the exact rate of change of the dispersion coefficient that is associated with the considered change in the overall bay dispersion resulting from certain bay developments, it is valuable to have an idea of the order of magnitude one must consider in this context.

For this purpose, two additional cases are conducted. The first one is focused on the extra rate of spreading resulting from the irregular bathymetry of Doha Bay, which is determined by the comparison of the reference case to the case in which the bottom depth is made uniform. Similarly, to get a feeling of the dispersion as a result of irregular geometric features, one case is considered in which Doha Bay is shaped into a rectangular shaped basin. It is expected that these changes lead to a decrease of dispersion: the relative rate of decrease is estimated by comparison of the results to the reference case.

6.4 TEST PROGRAM The test program comprises 12 cases in total, which are listed below. Case A is the reference case. The cases B reflect hypothetical bay changes and the real cases, Oryx Island and Sharq Crossing, are denoted with a C. Finally the cases D are focused on the value of the dispersion coefficient.

In this paragraph each case is introduced and the associated values of the input parameters for the RAM are given. The input parameter values are listed in Table 6.1. Section 6.5 is concerned with the results.

Test cases:

 A – Reference case: the current situation  B1 – No inlet channel  B2 – No Alsafliya tail  B3 – One inlet (closure of the northern inlet)  B4 – Inlet channel 2 m deeper  B5 – Inlet channel 4 m deeper  B6 – Bed level 2 m deeper (excluding the inlet channel)  B7 – Bed level 2 m shallower (excluding the inlet channel)  C1 – Reclamation of Oryx Island  C2 – Construction of Sharq Crossing  D1 – Uniform bed level  D2 – Rectangular shaped basin

6.4.1 CASE A – REFERENCE CASE For the reference case the model set up which is described in section 6.2 is used. The depth profile is shown in Figure 6.7. Because Doha Bay has two inlets, for this analysis the bay is divided into two separate parts (section 5). Part 1 comprises the part which is flushed by the inlet channel and part 2 (Figure 6.6), which contains the smaller and shallower area north of Alsaflyia ‘tail’ (Figure 1.1).

Most of the values of the input parameters for the rapid assessment, such as the surface area, are derived from the numerical model. To represent the tidal conditions, the average amplitude and the M2 tidal period are used, which corresponds with the findings in section 4.6. The values of the input parameters are listed in Table 6.2. For the aspect ratio of the total bay, a value of 1 is chosen because this is assumed to reflect the near inlet conditions the best (section 5). In Table 6.2 the input parameters are shown.

FIGURE 6.6 - INDICATION OF PART 1 (BLACK) AND PART 2 (WHITE) IN DOHA BAY

TABLE 6.2 – INPUT PARAMETER VALUES OF THE TEST CASES FOR THE RAM

Test case Modification Wb /Lb Ab ai db C Tt at [-] [km2] [m2] [m] [m/s] [s] [m] Case A Reference case Total 1.0 43.6 20000 5.7 65 45000 0.6 Part 1 1.0 30.0 18000 6.4 65 45000 0.6 Part 2 0.5 13.0 2000 4.3 65 45000 0.6 Case B1 No channel 1.0 43.6 19000 5.5 65 45000 0.6 Case B2 No tail 1.0 43.6 20000 5.7 65 45000 0.6 Case B3 One inlet 1.0 43.6 18000 5.7 65 45000 0.6 Case B4 Channel -2m 1.0 43.6 20900 5.8 65 45000 0.6 Case B5 Channel -4m 1.0 43.6 21700 5.9 65 45000 0.6 Case B6 Bed level -2m 1.0 43.6 25200 7.5 65 45000 0.6 Case B7 Bed level +2m 1.0 43.6 15000 4.6 65 45000 0.6 Case C1 Oryx Island 0.7 41.1 18500 5.5 65 45000 0.6 Case C2 Sharq Crossing 1.0 43.6 20000 5.7 65 45000 0.6 Case D1 Uniform depth 1.0 43.6 20000 5.7 65 45000 0.6 Case D2 Rectangular shaped 1.0 38.7 20000 6.2 65 45000 0.6

6.4.2 CASES B – HYPOTHETICAL CHANGES As listed above, in total the effect of seven different, hypothetical modifications to Doha Bay are investigated. For this purpose the initial depth profile (Figure 6.4) to be used for the numerical model simulations is modified. The resulting configurations are shown in this paragraph. Furthermore, in the light of the RAM computations, the associated changes to the input parameters are given.

BAY CONFIGURATIONS The bathymetry of the reference case is modified using Delft3D-QUICKIN. To remove the inlet channel (case B1), the depth in the channel domain is changed to a similar level as the surrounding bed, creating a smooth transition. In this way unwanted side effects are avoided. Similarly, the ‘tail’ of Alsaflyia Island (case B2) is removed from the bay by assigning depth values to its domain which are similar to those of the connecting area. For case B3, the northern, smallest inlet is closed with the use of dry cells (Figure 6.7).

Further, the deepening of the inlet channel is conducted by adding 2 m (case B3) or 4 m (case B4) to all the depth values within the domain of the channel. The same procedure is carried out for the entire bay area, with the exclusion of the channel (case B6; case B7). The bed level of the channel is in these cases not modified, because the focus lies on the effect of a deeper respectively shallower bay and not on the impact of adjustments to the channel.

The depth profiles of cases B1-B7 are shown in Figure 6.7 – Bathymetries of the cases A and B1-B3Figure 6.7 and Figure 6.8.

INPUT PARAMETER VALUES The modifications can be accounted for through the input parameters for RAM. For example deepening of the channel has its consequences for the average bay depth and the cross-sectional inlet area. The relevant parameters are obtained from the modified bay configurations. The input values for each case are listed in Table 6.2.

Case A – Reference case Case B1 – No channel

Case B2 – No tail Case B3 – One inlet

FIGURE 6.7 – BATHYMETRIES OF THE CASES A AND B1-B3

Case B4 – Channel -2 m Case B5 – Channel -4 m

Case B6 – Bed level -2m Case B7 – Bed level +2 m

FIGURE 6.8 – BATHYMETRIES OF THE CASES B4-B7

6.4.3 CASES C – PROPOSED DEVELOPMENTS

BAY CONFIGURATIONS The influences of the actual proposed developments in Doha Bay, namely Sharq Crossing and Oryx Island, are also investigated by the modification of the initial model configuration. To represent the presence of Sharq Crossing, the pillars and ‘Island’ that are part of this construction, which was described in section 3.1.3, are added to the model through dry cells. Oryx Island are represented by depth values of -2 m, or in other words the Island rise 2 m above the water level. The resulting bathymetries are shown in Figure 6.9.

INPUT PARAMETER VALUES The presence of Sharq Crossing (case C1) has no significant influence on any of the basic bay parameters which are included in the RAM, other than the dispersion coefficient: the area which is covered by Sharq Crossing is negligible relatively to the total bay area. The presence of the pillars and Island are expected to increase the rate of spreading, at least locally. However, it is difficult to estimate the associated increase of the dispersion coefficient as a result of this. Cases D1 and D2 are focused on this issue.

Contrary to Sharq Crossing, the presence of Oryx Island (case C2) is associated with changes to the input parameters of the RAM. Both the cross-sectional inlet area and the total bay surface area are decreased as a result of Oryx Island. Furthermore, a lower aspect ratio is assumed to represent the bay conditions in the presence of Oryx Island. This is because Oryx Island significantly decreases the width near the inlet (section 5). The values for the input parameters are listed in Table 6.2.

6.4.4 CASES D – DISPERSION COEFFICIENT These last two cases are concerned with the relative rate of change of the dispersion coefficient as a result of certain bay changes.

BAY CONFIGURATIONS For the investigation of the influence of the irregular bathymetry to the rate of internal mixing, in case D1 the bed level of the bay is made uniform (Figure 6.9). To isolate the effect of the bathymetric features, the inlet channel remains present. Similarly, in case D2 the large scale irregular features are for a great part erased by making the shape of the bay more or less rectangular. The initial bathymetry remains the same (Figure 6.9).

INPUT PARAMETER VALUES Because the value of the depth in case D1 is equal to the average bay depth in the reference case (5.7 m), all parameter values (when the dispersion coefficient is left out of account) remain the same as in the reference case. This is favourable considering the purpose of this case, because in this way the influence of the changes on the rate of spreading is isolated. This does however not hold for case D2: the modification in this case results in an increase of the total bay’s surface area. Therefore the analysis with respect to the influence of the changes on the rate of internal mixing is less straightforward.

Case C1 – Oryx Island Case C2 – Sharq Crossing

Case D1 – Uniform bed Case D2 – Rectangular shaped

FIGURE 6.9 - BATHYMETRIES OF THE CASES C1, C2, D1 AND D2

6.5 RESULTS An overview of the calculated average residence times for all the test cases is given in Table 6.3. The spatial distributions of the calculated residence times are shown in Figure 6.16, Figure 6.17, Figure 6.19 and Figure 6.22.

First the results of the reference case are discussed. As explained in section 6.3 based on these results the dispersion coefficient to be used in the rapid assessment method is determined. The obtained value of the dispersion coefficient is subsequently used in the test cases. The results are discussed per cluster of tests (starting with test A then B etc.).

6.5.1 CASE A – REFERENCE CASE First the system behaviour is briefly discussed, based on the residence time distribution and depth averaged velocity plots (Figure 6.17; Figure 6.10).

RESIDENCE TIME DISTRIBUTION The residence time distribution plot (Figure 6.17) clearly shows that the area near the inlet channel is flushed most rapidly. The further away from this inlet, the larger the residence times are, with maxima around 45 days. This is the expected pattern which is commonly found near tidal inlets. It is remarkable that the area close to the northern inlet does not show this typical pattern. In fact, the average residence times in this area are relatively large (about 40-45 days). This suggests that the flushing through the northern inlet is relatively slow.

This flushing behaviour is further illustrated by Figure 6.10 showing the depth averaged velocities. The deep channel clearly accommodates relatively high current velocities. This leads to a similar jet and sink pattern which also occurred in the schematic models (section 2.5.3). Contrary, very low velocities are observed at the northern inlet and the tidal pumping mechanism does not occur. This explains why the average residence times are relatively low in the area north of the Alsaflyia tail.

FIGURE 6.10 - DEPTH AVERAGE VELOCITIES IN DOHA BAY (CASE A) DURING HIGH TIDE

AVERAGE RESIDENCE TIMES Using both the RAM as the detailed model, three values for the average residence time are computed, namely for part 1, part 2 and the total bay respectively (Table 6.3). For each of the three areas, the average residence times are estimated by the RAM for varying values of the dispersion coefficient. The matching values of the dispersion coefficient can be derived from the graphs which are shown in Figure 6.13, Figure 6.11 and Figure 6.12.

Starting with part 1, an average residence time of 26.8 days is derived from the numerical model calculation. Approximately the same value is found through RAM with a dispersion coefficient of 1.2 m2/s (Figure 6.13). This value is of similar order as the value which was found for Kuwait Bay (section 5). Although on the low side, it is considered to be of a reasonable order of magnitude (section 2.6.6).

As expected from the residence time distribution plot (Figure 6.16), the average residence time of part 2 is much higher. A value of 39.3 days is the result obtained by the numerical model. The results of the RAM for this part are remarkable. Using the dispersion coefficient which was found for part 1 ( =1.2 m2/s) leads to an average residence time of approximately 7 days, which is significantly lower than the average residence time of part 1 which is illustrated by the graph in Figure 6.12. Clearly, the results of the RAM for part 2 are unrealistic.

An explanation for this is found in the behaviour of the flow near the inlets. The RAM is based on the schematic models (section 4) in which the tidal pumping mechanism occurs as a result of the abrupt variation in the cross-sectional inlet area (section Figure 2.8). This highly effective flushing mechanism is also present at the southern inlet, where the deep inlet channel accommodates high flow velocities. This leads to the rapid flushing of part 1. However, in part 2 no such a mechanism is observed. Due to the shallow water depth at the inlet, the velocities are too low to produce the typical jet flow (Figure 6.10). The RAM therefore strongly overestimates the flushing through this inlet. Apparently, the tidal pumping effect in the schematic model, in which also no deep channel is present, is overestimated because of the schematic inlet representation. In reality the transition at the inlet is not as abrupt but more smoothly.

In that case, when the current velocities are not high enough, the tidal pumping mechanism does not occur. This leads to the conclusion that the RAM is not valid in areas with shallow inlets.

Because the RAM cannot be used to estimate the flushing of part 2, considering the bay as one total area seems to be the best approach in this case, concerning the total bay’s average residence time.

When the total area is concerned, an average residence time of 29.9 days is derived from the numerical model. This corresponds to a dispersion coefficient of 1.9 m2/s (Figure 6.10). Intuitively this value seems to be on the low side, similarly as what was found for Kuwait Bay (section 5). Again, the order of magnitude is considered to be reasonable (section 2.6.6). This value is used as input for the further test cases.

FIGURE 6.11 – RESULTING AVERAGE RESIDENCE TIMES FOR CASE A (PART 1) FOR DIFFERENT VALUES OF THE DISPERSION COEFFICIENT COMPUTED BY THE RAM, COMPARED TO THE AVERAGE RESIDENCE TIME CALCULATED BY THE NUMERICAL MODEL

FIGURE 6.12 - RESULTING AVERAGE RESIDENCE TIMES FOR CASE A (PART 2) FOR DIFFERENT VALUES OF THE DISPERSION COEFFICIENT COMPUTED BY THE RAM, COMPARED TO THE AVERAGE RESIDENCE TIME CALCULATED BY THE NUMERICAL MODEL

FIGURE 6.13 - RESULTING AVERAGE RESIDENCE TIMES FOR CASE A (TOTAL BAY) FOR DIFFERENT VALUES OF THE DISPERSION COEFFICIENT COMPUTED BY THE RAM, COMPARED TO THE AVERAGE RESIDENCE TIME CALCULATED BY THE NUMERICAL MODEL

TABLE 6.3 - RESULTS OF THE TEST CASES SHOWING THE AVERAGE RESIDENCE TIMES DERIVED BY BOTH THE NUMERICAL MODEL AND THE RAM AND THE RELATIVE DIFFERENCES TO THE REFERENCE CASE

Test case Modification Average residence time [days] Relative difference to reference case [%] By Delft3D By RAM By Delft3D By RAM (D=1.9m2/s) Case A Reference case Total 29.9 29.9 - - Part 1 26.8 22.8 - - Part 2 39.3 5.9 - - Case B1 No channel 33.0 28.7 +10.4 -3.9 Case B2 No tail 32.8 29.9 +9.6 +0.0 Case B3 One inlet 37.6 26.7 +25.8 -11.1 Case B4 Channel -2m 30.0 31.1 +0.50 +4.0 Case B5 Channel -4m 31.6 32.1 +5.48 +7.5 Case B6 Bed level -2m 40.8 34.0 +36.4 +13.8 Case B7 Bed level +2m 24.1 24.1 -19.4 -19.4 Case C1 Oryx Island 35.4 35.2 +18.5 +17.1 Case C2 Sharq Crossing 25.017 25.518 -2.019 +0.0 Case D1 Uniform depth 34.6 29.9 - - Case D2 Rectangular shaped 29.0 28.4 - -

17 This value is derived with different model settings as for the reference case and other cases: the simulation starts two days earlier (at 15-04-2014). 18 This value is obtained with a dispersion coefficient of 3.0 m2/s to match the situation with the different model settings as described above. 19 This represent the difference to a different reference case, namely with a simulation start time at 15-04- 2014. The average residence time for this reference case is 25.0 days. 71

6.5.2 CASES B – HYPOTHETICAL CHANGES For the cases B the computed values of the average residence times by the numerical model and the RAM respectively are listed in Table 6.3 and shown in Figure 6.14. For the RAM a dispersion coefficient of 1.9 m2/s is used (section 6.5.1). Further, the relative differences of the average residence times compared to the average residence time which was estimated for the initial situation (case A) are shown in Figure 6.15, both for the numerical model calculations and the RAM.

ABSOLUTE VALUES As far as the absolute values are concerned (Figure 6.14), the average residence times computed by the RAM for cases B4, B5 and B7 are similar to the corresponding values which are derived through numerical modelling (Figure 6.15). This suggests that the RAM works well for these cases. However, for the remaining cases large deviations are observed between the residences times resulting from the numerical model and the RAM respectively.

RELATIVE DIFFERENCES Particularly interesting in the context of this study is the relative change of the average residence times of the cases B compared to the reference case. The RAM is assumed to be successful if the relative change of the flushing conditions can be predicted. The graph in Figure 6.15 shows that for the cases B4 to B7 the direction of the change in average residence time (positive or negative) is correctly predicted through the RAM.

However, this is not the case for the cases B1 and B3. When the numerical model shows an increase of the average residence time, the RAM predicts a decrease. Also for case B2 the results are not very promising. The RAM predicts no significant change in the flushing conditions but the numerical model calculations result in a 9% increase of the average residence time. Let’s take a closer look to the behaviour of the separate cases, starting with case B1 in which the tidal channel was ‘removed’ from the bay.

CASE B1 – NO CHANNEL Calculations with the numerical model indicate that this modification causes an increase of the residence time of approximately 10%. This can be attributed to the tidal pumping mechanism which is either decreased, or does not occur at all anymore, when there is no tidal channel anymore. The flow velocities drastically decrease through the inlet because of the smaller water depths (Figure 6.10). As argued before, the RAM is based on this highly effective flushing mechanism. The consequence is that the average residence times are underestimated when the tidal pumping effect is absent. Because the dimensionless ratio decreases, the RAM predicts a decrease of the average residence time. Clearly, the influence of the inlet characteristics is not well represented by the RAM. One solution might be to take the inlet depth and inlet width into account separately in the RAM instead of through the cross-sectional inlet area.

CASE B2 – NO TAIL The removal of the ‘tail’ from Alsaflyia Island does not significantly change any of the input parameters of the RAM. Therefore the prediction of the RAM reads that this procedure does not change the average residence time. However, the results obtained by the numerical model show that the removal of the tail does lead to an increase of the average residence time. This is a remarkable result as intuitively one would consider the tail is an obstruction that ‘blocks’ the northern part of the bay. However, apparently the pier causes a certain current pattern (or circulation) which is in favour to the exchange. Obviously, (large scale) changes in the current pattern cannot be accounted for through the RAM.

FIGURE 6.14 - ABOSULUTE VALUES OF THE COMPUTED AVERAGE RESIDENCE TIMES IN DAYS FOR CASES B1-B7: COMPARISON OF THE RESULTS OF THE NUMERICAL MODEL AND THE RAM

FIGURE 6.15 – RELATIVE DIFFERENCES OF THE COMPUTED AVERAGE RESIDENCE TIMES IN DAYS FOR CASES B1-B7 COMPARED TO THE AVERAGE RESIDENCE TIME OF THE REFERENCE CASE (CASE A) IN PERCENTAGES. NEGATIVE VALUES INDICATE A DECREASE OF THE AVERAGE RESIDENCE TIME (BETTER FLUSHING) AND VICE VERSA.

CASE B3 – ONE INLET The results of the RAM for case B3, in which the northern inlet is removed, do not correspond to those produced through the numerical model either. Closing the inlet has a significant effect on the average residence time when the numerical model calculations are concerned. This is not a surprise as in this way the total area has to be flushed through the southern inlet. However, in the RAM the decrease of total cross- sectional inlet area is associated with a decrease of the average residence time, which is opposing the results of the numerical model. This raises doubts on the integrative approach when more than one inlet is present.

CASES B4 AND B5 – CHANNEL LEVEL Changing the level of the inlet channel is relatively well predicted by the RAM. Both the numerical model as the RAM show increasing average residence times when the channel is deepened by 2 m respectively 4 m. The RAM predicts a slightly higher relative change in both cases. However, when looking at the residence time distributions the decrease in residence time is remarkable in the most areas and exceeds the simulation time of 74 days in the remote areas. This means that the actual average residence times will be slightly higher. To conclude, the RAM performs well when changes to the inlet channel are concerned.

CASES B6 AND B7 – BED LEVEL The influence of raising the bed level of the entire bay (case B7) is very well predicted by the RAM. When the bed level is raised by 2 m, both through the RAM and the numerical model a decrease of the average residence time of about 19% is estimated. This is remarkable because in section 5 it was argued that the RAM does not perform well in very shallow areas. Apparently the shallowness was not the (only) reason why the predictions of the RAM for Venice Lagoon (section5) were unrealistic. When the bay is deepened, the RAM ‘correctly’ predicts the direction of the change to the average residence time. An increase of the average residence time was found, which corresponds to the findings of the numerical model. However, the relative rate of change is significantly larger when the results of the numerical model are concerned. This can be related to the tidal pumping mechanism which becomes less pronounced when the bed level approaches the level of the tidal channel.

Case A – Reference case Case B1 – No channel

Case B2 – No tail Case B3 – One inlet

FIGURE 6.16 - RESIDENCE TIME DISTRIBUTIONS OF THE CASES A AND B1-B3; COMPUTED WITH THE USE OF THE NUMERICAL MODEL

Case B4 – Channel -2 m Case B5 – Channel -4 m

Case B6 – Bed level -2 m Case B7 – Bed level +2 m

FIGURE 6.17 - RESIDENCE TIME DISTRIBUTIONS OF THE CASES B4-B7; COMPUTED WITH THE USE OF THE NUMERICAL MODEL

6.5.3 CASES C – PROPOSED DEVELOPMENTS For the cases C the computed values of the average residence times by the numerical model and the RAM respectively are listed in Table 6.3 and shown in Figure 6.18. Similarly to the cases B, for the RAM a dispersion coefficient of 1.9 m2/s is used (section 6.5.1). In addition, the relative differences of the average residence times compared to the average residence time which was estimated for the initial situation (case A) are shown in Figure 6.15, both for the numerical model calculations and the RAM.

CASE C2 – ORYX ISLAND The results of this case are promising. The RAM predicts an average residence time of 35.0 days which is approximately the same value as that is found for the numerical model computations: 35.4 days. The increase of the average residence time is merely due to the change of the aspect ratio of the area near the inlet. Due to Oryx Island, the open space next to the inlet channel is ‘blocked’. This decreases the effect of the tidal pumping mechanism, which was explained in section 5.

The only thing that is not accounted for through the RAM is the rate of increase of internal mixing due to Oryx Island. The average residence time distribution plot (Figure 6.19) clearly shows that in the area around the Island the residence times are relatively low. Because the area which is influenced by Oryx Island is relatively large compared to Doha Bay and because it is located close to the inlet, it is assumed to have at least some influence to the total average residence time. When this increased rate of internal mixing would be accounted for by the RAM, the resulting average residence time would be somewhat lower. This suggests that the average residence time of the RAM is on the low side; however this does not change the conclusion that the RAM has proven its capability of estimating the influence of Oryx Island.

CASE C1 – SHARQ CROSSING Due to complications with the numerical model, the influence of Sharq Crossing is tested with a model with a simulation start date of two days earlier than the reference case: 15-04-2006. The results of the case with Sharq Crossing are therefore compared to a different reference case as case A1, with the start date as only difference. This leads to a different average residence time, namely 25.5 days when computed by the numerical model. The deviation is related to the differences in the tidal forcing, which is explained in section 5. It underlines the fact that the absolute values should be handled with care and not taken too explicitly. The results are presented in Table 6.3.

As mentioned before, the input parameters of the RAM have not changed as a result of Sharq Crossing. This is, when the dispersion coefficient is kept constant. Higher rates of internal mixing are assumed in the near presence of the ‘obstructions’. From literature its was expected that these merely effect the local conditions rather than the flushing of the total bay, unless the obstructions are located close enough for the induced motions to interact (section 2.5). In the case of Sharq Crossing, the obstructions can clearly be divided into three groups in which interaction will occur. However the distances among the groups are assumed to be large. Therefore it is expected that the dispersion coefficient should only be raised marginally to account for the slight increase of mixing predicted.

The results of the numerical model reveal that the presence of Sharq Crossing indeed only has a small influence on the total bay’s average residence time (Table 6.3). An average residence time of 25.0 days is found for the new situation. This leads to a relative difference of less than 2% when compared to the reference case. To conclude, the local influence of Sharq Crossing can only be accounted for through the dispersion coefficient. In this case the dispersion coefficient should be raised from 3.0 m2/s (which reflects the value of the dispersion coefficient for the ‘new’ reference case) to 3.2 m2/s.

FIGURE 6.18 - RELATIVE DIFFERENCES OF THE COMPUTED AVERAGE RESIDENCE TIMES IN DAYS FOR CASES B1-B7 COMPARED TO THE AVERAGE RESIDENCE TIME OF THE REFERENCE CASE IN PERCENTAGES. NEGATIVE VALUES INDICATE A DECREASE OF THE AVERAGE RESIDENCE TIME (BETTER FLUSHING) AND VICE VERSA.

Reference case for Oryx Island (Case A) Case C1 – Oryx Island

Reference case for Sharq Crossing Case C2 – Sharq Crossing

FIGURE 6.19 – RESIDENCE TIME DISTRIBUTIONS OF THE CASES C1 (ORYX ISLAND) AND C2 (SHARQ CROSSING); COMPUTED WITH THE USE OF THE NUMERICAL MODEL

6.5.4 CASES D – DISPERSION COEFFICIENT As mentioned before, the aim of the cases D is to get a feeling of the rate of internal mixing that is associated with certain irregular bay features. For this purpose the entire bathymetry was made uniform (case D1), with the exception of the inlet channel, and the geometry was modified to a rectangle (case D2).

CASE D1 – UNIFORM DEPTH Without the irregularities in the bathymetry, the numerical model calculations show a significant decrease in the average residence time. A value of 32.8 days was found. This is purely attributed to the decrease of internal mixing.

Because none of the input parameters of the RAM have changed, the same graph can be constructed as for the reference case, showing the results of the RAM for varying rates of dispersion (Figure 6.20). From this graph the value of the dispersion coefficient which leads to an average residence time of 32.8 days can be derived.

The graph (Figure 6.20) shows that a dispersion coefficient of approximately 1.5 m2/s reflects the situation with the uniform bed level. Comparing this value to the value which was used for the reference case (1.9 m2/s) it can be concluded that about 20% (=0.4 m2/s) of the total rate of spreading (=1.9 m2/s) can be attributed to the rate of internal mixing that is induced by the irregularities in the bed of Doha Bay. The remainder part is related to the tidal pumping effects near the inlet (channel), the irregular geometric features and the smaller scale motions which are accounted for in the detailed numerical model through the background diffusivity (=1 m2/s in the reference case).

FIGURE 6.20 – RESULTING AVERAGE RESIDENCE TIMES FOR CASE A (PART 1) FOR DIFFERENT VALUES OF THE DISPERSION COEFFICIENT COMPUTED BY THE RAM, COMPARED TO THE AVERAGE RESIDENCE TIME CALCULATED BY THE NUMERICAL MODEL

CASE D2 – RECTANGULAR SHAPED For case D2 the analysis of the results is not so straightforward because the modification to the geometry also resulted in a change of the total bay area and the average bay depth. Therefore the effect of the change to the bay’s shape is not isolated. To exclude the ‘additional effects’, the computed value of the average residence time calculated by the numerical model is compared to the values found through the RAM for this case.

For a dispersion coefficient of 1.9 m2/s, the RAM predicts an average residence time of 28.4 days for case D2. When this is compared to the value found by the numerical calculations (29.0 days), there is only a slight increase of average residence found due to this ‘smoothing’ of its shape. This can be in a similar way as for case D1 related to a decrease of internal mixing which is represented by the dispersion coefficient. The associated value for the dispersion coefficient in this modified case is 1.8 m2/s. When the difference is only attributed to the decrease in dispersion, this means that the geometric features of the bay are responsible for only 5% of the total rate of mixing, which suggests that the influence of geometric irregularities to dispersion is less pronounced than that of the bathymetry.

However, when looking at the residence time distribution plot (Figure 6.22), the area where the difference is the largest, is near the inlet. This is far away from the shore and is therefore not assumed to be attributed to the decrease of shear from irregularities in the geometry. Furthermore, the parts that are erased from the bay are side embayments with low residence times. Excluding these parts is therefore associated with a decrease of the average residence time, rather than an increase. To conclude, in this case it is much more difficult to relate the change directly to the rate of change of the internal mixing.

To summarize, the relative increase or decrease of the dispersion coefficient associated with certain bay changes are shown in Figure 6.21. In this overview, the influence of Sharq Crossing (section 6.5.3) to the total bay’s internal mixing is also shown. This figure can be used to estimate the order to which the dispersion coefficient should be increased or decreased to account for changes to the internal mixing of the bay of interest as a result of certain bay changes.

FIGURE 6.21 - RELATIVE CHANGE OF THE DISPERSION COEFFICIENT RESULTING FROM CERTAIN MODIFICATIONS: THE PRESENCE OF SHARQ CROSSING, A UNIFORM DEPTH PROFILE AND A RECTANGULAR SHAPE RESPECTIVELY.

Case D1 – Uniform bed Case D2 – Rectangular shaped

FIGURE 6.22 – RESIDENCE TIME DISTRIBUTIONS OF THE CASES D1 (UNIFORM BED) AND D2 (RECTANGULAR SHAPED); COMPUTED WITH THE USE OF THE NUMERICAL MODEL

6.6 DISCUSSION Using Doha Bay as a case study, the influence of several changes to the bay were analysed with respect to the bay’s flushing conditions. This resulted in relevant findings on the performance and behaviour of the rapid assessment method. One crucial finding was that the RAM largely underestimates the average residence time when the water depths at the inlet are shallow and no deep channel is present. In such a situation the effective tidal pumping mechanism, which is accounted for in the RAM, does not occur.

The behaviour of the RAM for bays with more than one inlet appears unreliable. Dividing the total are into separate parts is still assumed to be the best approach, however one should be careful when the dimensions of the inlets deviate significantly. In that case their individual influence is not clear.

When a deep inlet channel is present, the RAM seems to be capable of predicting the relative change of large scale changes to the bathymetry, such as deepening of the inlet channel or the presence of Oryx Island, quite accurate. However, smaller scale changes like Sharq Crossing can only be accounted for through the dispersion coefficient, which value remains uncertain and is in principle used as a calibration parameter. The dispersion coefficient is estimated and kept constant in order to assess the relative influence of bay changes.

However, this approach should be followed with caution when the developments in question are expected to influence the (total) rate of internal mixing as this should be accounted for through the dispersion coefficient. To obtained estimates of the rate of change of the value of the dispersion coefficient as a result of certain bay changes should give an idea of the order of magnitude one should consider.

Overall, the RAM seems to predict relatively low values of the average residence time. This can be attributed to the unrealistic representation of the inlet in the schematic models on which the RAM was based. On the other hand, the exclusion of all other forcing types than the tide, such as wind, is assumed to lead to conservative answers regarding the flushing conditions.

In the case of Doha Bay, especially the Shamal winds could have a reasonable influence on the bay’s flushing, which will vary greatly per season. As a final remark, it should be noted that the results of the numerical model do not reflect the actual situation in Doha Bay either. This model has its own limitations and inaccuracies.

This summation of factors of uncertainty regarding the value of the average residence time as derived by the RAM illustrates that the RAM should merely be used to assess the relative change of the flushing conditions, rather than considering the absolute values of the average residence time.

6 CONCLUSIONS AND RECOMMENDATIONS

7.1 CONCLUSIONS The present study was focused on the development of a rapid assessment method to analyse the influence of intended changes to a coastal bay to the bay’s flushing conditions. The approach was to derive a model of the box type based on basic bay parameters of which the influence to flushing was detected by schematic 2D numerical model tests using the systematic approach of a dimensional analysis. The following conclusions can be drawn from the study:

 The material exchange (or flushing) between the bay and the sea by advection and dispersion is mainly driven by the following mechanisms:  Shear flow dispersion  Residual currents  Tidal pumping and quadruplet pair forming

 The occurrence of these processes and mechanisms is dependent on the following factors:  The characteristics of the tidal wave in general (e.g. the tidal period, the tidal amplitude and the tidal prism)  The bathymetric and geometric configuration (e.g. the presence of channels, shoals and headlands)  The inlet characteristics (e.g. presence of a tidal channel, cross-sectional inlet area)

 The average residence time by the definition of TAKEOKA, 1984 is the most suitable transport time parameter to indicate the overall exchange conditions of a coastal bay.

 The relevant relations between the basic bay (and forcing) parameters and the average residence time were successfully derived through the systematic approach of the dimensional analysis.

 Based on these relevant relations an analytical expression of the average residence time was successfully derived and the use of this equation as a RAM for the flushing of coastal bays was validated by comparison to real cases.

 The most practical approach for the use of the RAM is to estimate the value of the dispersion coefficient and to keep this value the same for each of the different cases that are compared. Following this approach means that the rate of change of internal mixing due to the development (or bay change) of interest cannot be accounted for by the RAM.

 The RAM overestimates the effect of the tidal pumping mechanism which leads to relatively low absolute values of the calculated average residence times for tidal flushing.

 In general, the exclusion of all other external influences than the tide generally leads to results that are on the conservative side.

 The developed RAM can be used to estimate the relative rate of change of the average residence time of the bay as a result of bay developments under the following assumptions20:  The development is of such a large scale that it has a significant influence on the value one or several of the basic bay parameters in the RAM.  A relatively deep and narrow inlet channel is present.  The bay has one inlet or several inlets of a similar scale (all with deep inlet channels); in the case of more than one inlet, a separated approach is recommended.

 The developed RAM is assumed to be inapplicable to the following situations:  One (or more) of the assumptions listed above is (are) not valid.  One (or more) of the assumptions of a typical coastal bay is (are) not valid.  The scale of the developments of interest is relatively small.  The inlet depth is shallow.  The water depths of the bay are very shallow (<2 m).  Drastic changes to the inlet representation are concerned.  The developments of interest result in the re-direction of large scale (circulation) currents.  The absolute values of the average residence time are concerned.  The local differences are concerned.

 The influence of Oryx Island on the flushing conditions was successfully predicted by the RAM. However, the estimation of the influence of Sharq Crossing was not possible with the use of RAM because of its relatively small scale.

7.2 RECOMMENDATIONS To improve the predictive value of the derived rapid assessment method as well as its areas of application, the following is recommended:

 Further research is particularly recommended on the way the influence of the inlet configuration on flushing is represented by the RAM. The dimensional analysis should be repeated using the inlet depth and inlet width to represent the inlet characteristics instead of the cross-sectional area.

 To eliminate the effects of the unrealistic tidal pumping mechanism as included in the method in its present form, the schematic 2D model tests should be repeated for a more natural configuration. This would require a higher grid resolution which means that the simulations will become more time consuming.

 The influence of the seaward conditions was not investigated in this study. The current velocity and the offshore depth are however assumed to have an effect on the average residence time and should therefore be investigated.

 Although some valuable estimates were derived from this study, the prediction of the value of the dispersion coefficient remains a difficult task. Further research is required to enhance the understanding of the value of this parameter.

20 In addition to the assumptions for the category of bays concerned in this study. These were given in section 2.2 85

 The possibility of including the influence of other forces into the rapid assessment method should be investigated. Wind and gravitational forces are the most relevant in this context.

 This study was focused only on the average bay flushing conditions. However, in many practical cases one is (also) interested in the spatial distribution of the residence time. Therefore it is highly recommended to investigate the possibilities of extending the rapid assessment method to predict not only the bay’s average residence time but also the local residence times. In the cases considered typical patterns of the residence time distribution were observed. The local residence times appear closely related to its distance to the inlet and the local water depth. This suggests that the local residence times might be estimated from the bay’s average residence time, the local water depth and the distance to the inlet.

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APPENDICES

APPENDIX A – COMPLEMENTARY DATA OF DOHA BAY AND THE ARABIAN GULF

A.1 TIDAL CONSTITUENTS IN ARABIAN GULF

FIGURE A 1 – THE M2 AND K1 TIDAL CONSTITUENTS IN THE ARABIAN GULF21. TIDE HEIGHTS ARE SHOWN IN METERS.

A.2 TIDAL ELEVATION DOHA BAY Table A 1 gives the tide levels for Doha Bay according to UDPA (2005). It should be noted that this table indicates tidal levels (astronomical) only, so wind and pressure influences are not included. Due to the nature of the tide in this part of the Arabian Gulf (high variation between winter and summer, neap and spring), the values for Mean High Water Spring (MHWS) and Mean Low Water Spring (MLWS) are not considered (Hydrographical Impact Assessment Doha Bay by Royal HaskoningDHV).

TABLE A 1 – TIDAL LEVELS DOHA BAY

WATER LEVEL

[m +CD] [m +QNHD]

HAT Highest Astronomical Tide +2.08 +1.20 MHHW Mean Higher High Water +1.53 +0.65 MLHW Mean Lower High Water +0.91 +0.02 MSL Mean Sea Level +0.88 +0.00 QNHD Qatar National Height Datum +0.88 +0.00 MHLW Mean Higher Low Water +0.68 -0.20 MLLW Mean Lower Low Water +0.32 -0.56 CD Chart Datum +0.00 -0.88 LAT Lowest Astronomical Tide -0.06 -0.94

The values in the table for MHHW, MLHW, MHLW and MLLW are taken from the geographic section of the UDPA (2005). These values are common used abbreviations in tide tables to explain the characteristics of the astronomical tide. These values occur on a daily to 7 daily (neap-spring cycle) base. From the table, it can be concluded that the difference between HAT/LAT and MHHW/MLLW is quite significant. This is due to the differences between summer and winter water levels caused by seasonal changes in barometric pressure and the fact that Doha Bay is located close to the transition zone between a mixed and a diurnal tide (see Figure A 2).

21 From Lardner et. al, 1982 89

FIGURE A 2 – TIDAL ELEVATION THROUGH A YEAR INCLUDING AVERAGE, MAXIMUM AND MINIMUM TIDAL ELEVATION PER MONTH (RED LINES) OVER 30 YEARS

A.3 Wind conditions 25km offshore Doha Bay The wind conditions based on NCEP-CFSR wind for a location 25 km offshore of Doha are presented in Figure A 3. The table shows the probability of occurrence for each class of wind directions and for each class of wind speeds. Especially the high occurrence (34% of total time) of wind from the direction 315°-345°N is remarkable. Furthermore the small increase in occurrence of wind from around 120° shows that this direction should not be neglected.

Seasonal variations of the wind speed and the wind direction are visualized in Figure A 4. Figure A 4 presents the wind speeds which are exceeded 1%, 10% and 50% of the time for every month. The wind speeds exceeded 1%, 10% and 50% of the time are highest in February and are lowest in September.

Figure A 5 depicts the monthly and annual operational wind conditions by means of wind roses. For each bin of wind direction the distribution of the wind speeds are given by a colour scale. For example in the wind rose representing the annual wind data, it can be seen that wind will blow from the north approximately 5% of the time, about 2% of the time the wind from this direction is smaller than 4 m/s.

The wind roses show in the months August to October that the occurring wind directions are more evenly distributed over all directions compared to for example the month January when the prevailing wind direction is northwest.

Wind direction [degrees N] 345 15 45 75 105 135 165 195 225 255 285 315 15 45 75 105 135 165 195 225 255 285 315 345 Sum Cumulative sum 0.0 1.0 0.18 0.28 0.29 0.30 0.30 0.26 0.24 0.22 0.21 0.20 0.23 0.36 3.07 3.07 1.0 2.0 0.63 0.94 1.01 0.97 0.82 0.61 0.46 0.45 0.42 0.52 0.64 1.07 8.55 11.62 2.0 3.0 1.12 1.59 1.63 1.74 1.32 0.95 0.62 0.53 0.50 0.65 1.04 1.93 13.63 25.25 3.0 4.0 1.42 1.80 1.69 1.97 1.75 1.27 0.67 0.41 0.43 0.72 1.43 3.13 16.70 41.95 4.0 5.0 1.48 1.27 1.24 1.57 1.63 1.20 0.60 0.29 0.26 0.57 1.83 4.19 16.12 58.07 5.0 6.0 1.12 0.64 0.55 0.96 1.15 0.93 0.43 0.20 0.13 0.40 1.87 4.71 13.09 71.16 6.0 7.0 0.78 0.23 0.17 0.46 0.71 0.60 0.29 0.09 0.05 0.19 1.49 5.07 10.12 81.28 7.0 8.0 0.36 0.09 0.07 0.21 0.39 0.29 0.19 0.03 0.02 0.07 1.08 4.74 7.55 88.82 8.0 9.0 0.16 0.04 0.04 0.09 0.19 0.12 0.07 0.02 0.01 0.02 0.60 3.82 5.18 94.01 9.0 10.0 0.08 0.02 0.02 0.06 0.08 0.05 0.04 0.01 0.00 0.01 0.31 2.60 3.28 97.28 10.0 11.0 0.02 0.01 0.01 0.03 0.05 0.03 0.01 0.00 0.00 0.13 1.36 1.65 98.93 11.0 12.0 0.01 0.00 0.00 0.01 0.01 0.01 0.00 0.00 0.04 0.64 0.73 99.67 12.0 13.0 0.00 0.00 0.01 0.00 0.01 0.00 0.01 0.20 0.23 99.90

13.0 14.0 0.00 0.00 0.00 0.00 0.01 0.05 0.06 99.96 14.0 15.0 0.00 0.00 0.03 0.03 99.99 15.0 16.0 0.00 0.00 100.00 16.0 17.0 0.00 0.00 100.00 17.0 18.0 0.00 0.00 100.00

18.0 > 0.00 100.00 wind speed [m/s] speed wind sum 7.37 6.92 6.72 8.4 0 8.39 6.33 3.64 2.24 2.03 3.35 10.71 33.91 100.00

FIGURE A 3 - PROBABILITY OF OCCURRENCE [%] AS FUNCTION OF WIND DIRECTION AND WIND SPEED AT LOCATION 25 KM OFFSHORE OF DOHA BAY [BASED ON NCEP-CSFR WIND DATA, SEE SAHA ET AL (2010)]

Deltares model data 3hr average 12

8 Exceeded 1% of time Exceeded 10% of time

7 Exceeded 50% of time wind velocitywind m/s 6

3 Jan Feb Mrt Apr Mai Jun Jul Aug Sep Oct Nov Dec

FIGURE A 4 - MONTHLY VARIATIONS IN WIND SPEED 25 KM OFFSHORE OF DOHA (BASED ON NCEP-CSFR WIND DATA, SEE SAHA ET AL (2010)).

FIGURE A 5 MONTHLY AND ANNUAL WIND ROSES AT A LOCATION OF 25KM OFFSHORE DOHA BAY (BASED ON NCEP-CSFR WIND DATA, SEE SAHA ET AL (2010)).

A4 CURRENT SPEEDS DOHA BAY shows the resulting extreme maximum depth-average flow velocities for different locations, these can be applied for 10 year return period design purposes. Table A 2 shows extreme maximum depth-average flow velocities for a 120 year return period.

Due to model uncertainties in this phase of the project, an additional margin for model uncertainties of 0.1 m/s was included in the results.

The resulting flow velocities in Table A 2 consist of:

 Maximum tide-induced flow velocity during one of the 3 design situations (as described in paragraph Table A 2).  Wind-induced flow velocity for 10 year and 120 year return period.  A margin for model uncertainties of 0.1 m/s.

The local flow velocities near the bridge piers and artificial reefs could be higher than the values presented below. The presence of the artificial reefs under the bridge causes complex (3D) flow features which may have negative impact on the local erosion and create unfavourable conditions for navigation of small craft. It is advised to study this impact in detail in a 3D model, and possibly to optimise the design.

TABLE A 2 - ESTIMATED MAXIMUM DEPTH-AVERAGE FLOW VELOCITIES [M/S] FOR 10 YEAR RETURN PERIOD

MAXIMUM WIND –DRIVEN TOTAL TIDE- AND WIND TIDE-DRIVEN CURRENT [m/s] INDUCED MAXIMUM FLOW NUMBER DESCRIPTION CURRENT FOR 10 YEARS VELOCITIES INCL. MARGIN [m/s] RETURN PERIOD FOR MODEL UNCERTAINTIES [m/s] 1 Approach Sharq 0.4 0.1 0.6 Transition Sharq Bridge – 0.1 0.1 0.3 2 Immersed tunnel Sharq 3 Immersed tunnel Sharq 0.2 0.2 0.5 4 Marine interchange 0.2 0.1 0.4 5 Approach West Bay 0.2 0.1 0.4 6 Immersed tunnel Cultural City 0.4 0.2 0.7 Transition Cultural city bridge – 0.3 0.2 0.6 7 Immersed tunnel Cultural City 8 Approach Cultural City <0.1 0.1 0.3

APPENDIX B – CLASSIFICATION OF INLAND COASTAL WATERS Coastal waters can be classified into different types. Because the nature of the flushing processes is often closely related to the type of coastal water considered, it is important to be aware of the existing types and the associated characteristics in the context of flushing. In this way the type of bay of interest can be easily identified. Moreover, in the light of present study notification of the different types is particularly relevant, because this study only focuses on the ‘bay type’.

B1 TYPES OF INLAND COASTAL WATERS Inland coastal water bodies range from completely natural to artificial or man-made such as harbours. Focusing on the natural waters, there are many ways to classify these environments, e.g. based on their geographical location, geomorphological processes and physical processes. Some useful attempts have been made (PRITCHARD, 1952; FISCHER et al., 1979) of which the categories of KJERFVE AND MAGILL (1989) seem to provide the most complete picture. Their classification is based on both geomorphological as well as physical processes. The following six categories are described (KJERFVE AND MAGILL, 1989):

 Estuary: A drowned inland river valley or section of the coastal plain, affected by tides and with significant river discharge. An estuary is typically horn-shaped, from wide at the coast to narrow at the river side and depths are relatively shallow (<20m). Estuaries are associated with relatively good flushing conditions, especially when the typical estuarine circulation is present.  (Coastal) lagoon: A long and narrow, very shallow (<2m) inland water body, parallel orientated to the coast and separated by the ocean through one or several barriers with one or more narrow inlets. In coastal lagoons the wind generally plays a large role to the flushing conditions, due to the shallow water depths. Generally, coastal lagoons are associated with relatively poor exchange.  Fjord: A glacially scoured, very deep (>100m) and highly stratified (vertically) inland marine area.  Bay: A coastal indentation, affected by tides with or without river discharge. Depths range from shallow to deep.  Tidal river: An inland river valley, containing only fresh water, affected by tidal water level oscillations.  Strait: An inland marine waterway, connecting two oceans/seas.

B2 ‘BAY’ TYPE Doha Bay belongs to the bay type. This type comprises in fact all coastal basins (the first four categories listed above) which cannot be classified as an estuary, lagoon or fjord. Those three are associated with more distinct features, such as the shape of the embayment, common water depths and dominant forcing types. Sub- classification based on specific characteristics (2.2) is thus necessary in order to further define the category of bays which present study is focused on.

APPENDIX C – HYDRODYNAMIC MODELLING OF THE ARABIAN GULF