SALINITY AND TIDES IN ALLUVIAL ESTUARIES By Hubert H.G. Savenije Delft University of Technology Water Resources Section P.O.Box 5048 2600 GA, Delft, The Netherlands Second Completely Revised Edition 2012 Version 2.5 Contents Preface iii Notation v 1 INTRODUCTION: DESCRIPTION AND CLASSIFICATION OF ALLU- VIAL ESTUARIES 1 1.1 What is an estuary? . 1 1.2 Importance of estuaries to mankind . 2 1.3 Classification of estuaries . 4 1.4 Classification by estuary numbers . 7 1.5 Alluvial estuaries and their characteristics . 8 1.5.1 The shape of alluvial estuaries . 9 1.5.2 Dominant mixing processes . 11 1.5.3 How the tide propagates . 12 1.5.4 How the salt intrudes . 16 1.6 What will follow . 19 2 TIDE AND ESTUARY SHAPE 21 2.1 Hydraulic equations . 22 2.1.1 Basic equations . 22 2.1.2 The seventh equation . 25 2.1.3 The 1-dimensional equations for depth and velocity . 28 2.1.4 The effect of density differences and tide . 30 2.2 The shape of alluvial estuaries . 35 2.2.1 Classification of estuary shape . 35 2.2.2 Assumptions about the shape of alluvial estuary in coastal plains . 40 2.2.3 Assumptions about estuary shape in short estuaries . 44 2.3 Relating tide to shape . 45 2.3.1 Why look for relations between tide and shape? . 45 2.3.2 Intuitive derivation . 46 2.3.3 Lagrangean analysis of a water particle . 47 2.3.4 Finding an expression for the phase lag . 51 2.3.5 The Geometry-Tide relation . 52 2.3.6 The Scaling equation . 53 2.3.7 The damping assumption . 54 2.4 Stokes' drift in alluvial estuaries . 54 2.5 The effect of river discharge on the Lagrangean water balance, the phase lag and the Geometry-Tide relation . 57 2.6 Concluding remarks . 60 3 TIDAL DYNAMICS 61 3.1 Tidal movement and amplification . 61 3.1.1 Why is the tidal wave amplified or damped? . 61 3.1.2 Derivation of the tidal damping equation by the envelope method . 61 3.2 Tidal wave propagation . 65 3.2.1 The relation between tidal damping and wave celerity . 66 i ii CONTENTS 3.2.2 Derivation of the celerity equation . 68 3.2.3 Empirical verification in the Schelde and Incomati estuaries . 76 3.2.4 Overview of the set of equations . 76 3.3 Explicit solution of the set of equations . 79 3.3.1 Scaling the equations . 79 3.3.2 Solving the equations explicitly . 80 3.3.3 Specific solutions and asymptotic behaviour of the equations . 83 3.4 Other formulations for the friction term . 87 3.4.1 Different formulations of the damping equation . 87 3.4.2 Performance of the different friction formulations . 88 3.4.3 A combination of the linear and quasi-non-linear equation . 90 3.4.4 Performance of the weighted damping equation . 92 3.4.5 Application to the Schelde estuary . 93 3.4.6 Final words on the friction term . 95 3.5 Effect of river discharge and other higher order effects on tidal damping . 97 3.5.1 Which higher order effects are important . 97 3.5.2 Incorporating river discharge into the derivation of the Celerity equation . 97 3.5.3 Incorporating river discharge into the derivation of the damping equation through the envelope method . 98 3.5.4 Application to the Schelde estuary . 100 3.5.5 Conclusion . 101 3.6 The influence of climate change and human interference on estuaries . 101 4 MIXING IN ALLUVIAL ESTUARIES 105 4.1 What is dispersion and how does it relate to mixing? . 105 4.2 Types of mixing, their relative importance and interaction . 106 4.3 Gravitational circulation . 109 4.4 Mixing by the tide . 110 4.5 Residual circulation through flood and ebb channels . 111 4.6 The decomposition method and why it is not very useful . 115 4.7 Longitudinal effective dispersion . 119 4.8 Van der Burgh's equation . 123 4.8.1 The physical meaning of Van der Burgh's K . 124 4.8.2 Correspondence with other methods . 125 4.9 General equation for longitudinal dispersion . 126 5 SALT INTRUSION IN ALLUVIAL ESTUARIES 127 5.1 Types of salt intrusion and shapes of salt intrusion curves . 127 5.2 Salt balance equation . 129 5.3 Influence of rainfall and evaporation . 132 5.4 Time scales and conditions for steady state . 136 5.5 Predictive model for steady state . 138 5.5.1 Expressions for HWS, LWS and TA . 138 5.5.2 The predictive model compared to other methods . 149 5.6 Unsteady state model . 150 5.6.1 System response time . 150 5.6.2 Unsteady state dispersion . 153 5.6.3 Application of the unsteady state model . 154 5.6.4 Application to the Gambia estuary . 154 5.7 Hypersaline estuaries . 157 5.8 Concluding remarks on the analytical salt intrusion model . 158 Preface This is the open access version of the fully revised second edition of \Salinity and Tides in Alluvial Estuaries", first published by Elsevier Publications in 2005. This open access edition is fully updated, uses the latest insights in the field of estuary hydrodynamics and includes references to the most recent publications. It also links to a wide range of applications and examples and includes valuable data and information that other researches may want to use. As it is, this remains the only book on alluvial estuaries that presents an integrated theory on estuary shape, estuary hydrodynamics, mixing and salt intrusion. Traditional publications focus on sub-disciplines: the hydrodynamics, the mixing (e.g. Fischer et al., 1979), the salinity, or the morphology, but seldom are these fields considered as interdependent. Although the literature on tidal hydrodynamics is vast, most concentrates on flow in prismatic (i.e. constant cross-section) channels, or, if a variable topography is used, then authors make use of 2-D or 3-D simulations using an imposed bathymetry. By this fragmented approach we miss opportunities. An estuary system and its component parts work as a single physical entity, where potential and kinetic energy is dissipated through the interaction between water, sediment and salinity. This interaction leads to clear patterns in the morphology, the hydrodynamics and the salinity, many of which can be surprisingly simple and even predictable. It was this realisation that inspired me to write this book. An estuary is the transition between a river and a sea. There are two main drivers: the river that discharges fresh water into the estuary and the sea that fills the estuary with salty water, on the rhythm of the tide. The salinity of the estuary water is the result of the balance between two opposing fluxes: a tide-driven saltwater flux that penetrates the estuary through mixing, and a freshwater flux that flushes the saltwater back. Both fluxes strongly depend on the topography: the salt water flux because the amount of water entering the estuary depends on the surface area of the estuary; and the fresh water flux, because the cross-sectional area of the estuary determines the efficiency of the fresh water flow to push back the salt. So, the topography is crucial. It provides the most important boundary condition for tidal hydraulics, mixing and salt intrusion. One of the innovations of this book is that, throughout, it works with the natural topography of alluvial estuaries. This natural topography is one with converging banks following an exponential function. Both the width and the cross-sectional area obey exponential functions. Moreover, in coastal plain estuaries, the depth is constant and there is no bottom slope. Estuaries in coastal areas with a strong relief are generally too short for this type of estuary to develop. They form a special category of alluvial estuaries where standing waves occur and where the depth decreases in the upstream direction. These estuaries have been described by others (e.g. by Wright et al., 1973; Prandle, 2003). The topography of alluvial estuaries, whether short or long, can be characterised by a single parameter: the convergence length: the length scale of the exponential function. This length scale is a key parameter for understanding tidal and mixing processes. In classical literature on tidal hydraulics, tidal mixing and salt intrusion, this parameter has been overlooked more often than not, leading to incom- plete, or even flawed, dimensional analysis. The reason probably lies in the fact that the early literature was based on laboratory experiments in prismatic channels. This book is unique in that it systematically integrates these natural topographies with tidal movement, mixing and salt intrusion. Mixing in estuaries is driven by both the tide and the density gradient. The density gradient induces vertical mixing, while the tide mainly causes horizontal mixing through tidal trapping and residual circulation (and to a minor extent turbulent mixing). It is recognized that residual iii iv PREFACE circulation is a dominant mechanism, particularly near the mouth of the estuary, but it is poorly understood. Several mixing mechanisms have been well documented in the literature, such as the vertical density-driven circulation and turbulence-driven mixing, but, until now, no theory exists for residual circulation. This book demonstrates that residual circulation is strongly related to an estuary's topography, and particularly to its width, which in alluvial estuaries is predictable. The book therefore provides an integrated mixing theory and a practical computational approach for the prediction of salt intrusion and tidal propagation in alluvial estuaries. The result is a book that tries to build a bridge between science and engineering.