Analytical and numerical analysis of tides and salinities in estuaries; part I: tidal wave propagation in convergent estuaries Leo C. van Rijn Ocean Dynamics Theoretical, Computational and Observational Oceanography ISSN 1616-7341 Volume 61 Number 11 Ocean Dynamics (2011) 61:1719-1741 DOI 10.1007/s10236-011-0453-0 1 23 Your article is protected by copyright and all rights are held exclusively by Springer- Verlag. This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your work, please use the accepted author’s version for posting to your own website or your institution’s repository. You may further deposit the accepted author’s version on a funder’s repository at a funder’s request, provided it is not made publicly available until 12 months after publication. 1 23 Author's personal copy Ocean Dynamics (2011) 61:1719–1741 DOI 10.1007/s10236-011-0453-0 Analytical and numerical analysis of tides and salinities in estuaries; part I: tidal wave propagation in convergent estuaries Leo C. van Rijn Received: 3 February 2011 /Accepted: 9 June 2011 /Published online: 10 July 2011 # Springer-Verlag 2011 Abstract Analytical solutions of the momentum and closed-end channels is important in the most landward 1/3 energy equations for tidal flow are studied. Analytical length of the total channel length. In strongly convergent solutions are well known for prismatic channels but are less channels with a single forward propagating tidal wave, well known for converging channels. As most estuaries there is a phase lead of the horizontal and vertical tide close have a planform with converging channels, the attention in to 90o, mimicking a standing wave system (apparent this paper is fully focused on converging tidal channels. It standing wave). will be shown that the tidal range along converging channels can be described by relatively simple expressions Keywords Tidal propagation . Salinity. Convergent solving the energy and momentum equations (new estuary. Prismatic channel . Analytical model . Numerical approaches). The semi-analytical solution of the energy model equation includes quadratic (nonlinear) bottom friction. The analytical solution of the continuity and momentum equations is only possible for linearized bottom friction. 1 Introduction The linearized analytical solution is presented for sinusoidal tidal waves with and without reflection in strongly The shape of most alluvial estuaries is similar all over the convergent (funnel type) channels. Using these approaches, world, see Dyer (1997), McDowell and O’Connor (1977), simple and powerful tools (spreadsheet models) for tidal Savenije (2005), and Prandle (2009). The width and the analysis of amplified and damped tidal wave propagation in area of the cross-section reduce in upstream (landward) converging estuaries have been developed. The analytical direction with a river outlet at the end of the estuary solutions are compared with the results of numerical resulting in a converging (funnel shape) channel system solutions and with measured data of the Western Scheldt (see Fig. 1). The bottom of the tide-dominated section Estuary in the Netherlands, the Hooghly Estuary in India generally is fairly horizontal. Often, there is a mouth bar at and the Delaware Estuary in the USA. The analytical the entrance of the estuary. Tidal flats or islands may be solutions show surprisingly good agreement with measured present along the estuary (deltas). tidal ranges in these large-scale tidal systems. Convergence The tidal range in estuaries is affected by four dominant is found to be dominant in long and deep-converging processes: (a) inertia related to acceleration an deceleration channels resulting in an amplified tidal range, whereas effects, (b) amplification (or shoaling) due to the decrease bottom friction is generally dominant in shallow converging of the width and depth (convergence) in landward direction channels resulting in a damped tidal range. Reflection in and (c) damping due to bottom friction, and (d) partial reflection at landward end of the estuary, whereas bathy- Responsible Editor: Roger Proctor metric convergence produces continuous reflection along the entire length of the channel (see point b). * L. C. van Rijn ( ) The classical solution of the linearized mass and Deltares and University of Utrecht, Utrecht, The Netherlands momentum balance equations for a prismatic channel of e-mail: [email protected] constant depth and width is well-known (Hunt 1964; Author's personal copy 1720 Ocean Dynamics (2011) 61:1719–1741 Fig. 1 Tidal estuary (planform and longitudinal section) Q-river SEA Bo +x -x B tidal excursion = 10 to 20 km HW High water H MSL mean sea level LW Low water h Mouth bar Dronkers 1964, 2005; Ippen 1966; Le Blond 1978; Verspuy more, they have not given the full solution including the 1985; Prandle 2009). This solution for a prismatic channel precise damping coefficient and wave speed expressions for represents an exponentially damped sinusoidal wave which both amplified and damped converging channels. Hunt dies out gradually in a channel with an open end or is (1964) briefly presents his solution for a converging reflected in a channel with a closed landward end. In a channel and focusses on an application for the Thames frictionless system with depth ho, both the incoming and Estuary in England. The analytical model is found to give 0.5 reflected wave have a phase speed of co=(gho) and have very reasonable results fitting the friction coefficient. Hunt equal amplitudes resulting in a standing wave with a virtual shows that strongly convergent channels can produce a wave speed equal to infinity due to superposition of the single forward-propagating tidal wave with a phase lead of incoming and reflected wave. Including (linear) friction, the the horizontal and vertical tide close to 90°, mimicking a wave speed of each wave is smaller than the classical value standing wave system (apparent standing wave). A basic co (damped co-oscillation). According to Le Blond (1978), feature of this system is that the wave speed is much larger 0.5 frictional forces dominate accelerations over most of the than the classical value co=(gho) , in line with observa- tidal cycle and hence a diffusive type of solution (neglect- tions. For example, the observed speed of the tidal wave in ing accelerations) provides accurate results for most of the the amplified Western Scheldt Estuary in The Netherlands tydal cycle in a prismatic channel. Using the classical is between 11 and 15 m/s, whereas the classical value is of approach, the tidal wave propagation in a funnel-type about 10 m/s. estuary can only be considered by schematizing the channel Parker (1984) has given a particular solution for a into a series of prismatic subsections, each with its own converging tidal channel with a closed-end focusing on the constant width and depth, following Dronkers (1964) and tidal characteristics (only sinusoidal (M2) tide) of the many others. Unfortunately, this approach eliminates to a Delaware Estuary (USA). He shows that the solution based large extent the effects of convergence in width and depth on linear friction and exponential decreasing width yields on the complex wave number and thus on the wave speed very reasonable results for the Delaware Estuary fitting the (Jay 1991). A better approach is to represent the planform friction coefficient. of the estuary by a geometric function. When an exponen- Harleman (1966) also included the effect of width tial function with a single-length scale parameter (Lb)is convergence by combining Greens’ law and the expressions used, the linearized equations can still be solved analyti- for a prismatic channel. Predictive expressions for the cally and are of an elegant simplicity. friction coefficient and wave speed were not given. Instead, The analytical solution for a funnel-type channel with he used measured tidal data to determine the friction exponential width and constant depth is, however, not very coefficient and wave length. well known. Hunt was one of the first to explore analytical Godin (1988) and Prandle and Rahman (1980) have solutions for converging channels using exponential and addressed a channel with both converging width and depth. power functies to represent the width variations. Both Le They show that that the analytical solution can be Floch (1961) and Hunt (1964) have given solutions for formulated in terms of Bessel functions for tidal elevations exponentially converging channels with constant depth. and tidal velocities in open and closed channels. However, However, their equations are not very transparent. Further- the complex Bessel functions involved obscure any Author's personal copy Ocean Dynamics (2011) 61:1719–1741 1721 immediate physical interpretation. Therefore, their results horizontal and vertical tide. Only bulk parameters are were illustrated in diagrammatic form (contours of ampli- considered; hence, the time effect is not resolved. Since tude and phase) for a high- and low-friction coefficient. reflection is not considered, their equations cannot deal Like Hunt, Jay (1991),basedonananalyticalpertur- with closed-end channels. Various approaches have been bation model of the momentum equation for convergent used to arrive at their four equations. According to the channels (including river flow and tidal flats), has shown authors, the combination of different approaches may that a single, incident tidal wave may mimic a standing introduce inconsistencies, which may limit the applicabil- wave by having an approximately 90° phase difference ity of the equations. This may not be a real problem as between the tidal velocities and tidal surface elevations long as measured data sets are available for calibration of and a very large wave speed without the presence of a the tidal parameters. reflected wave. The tidal wave behavior to lowest order is In this paper, analytical solutions of the momentum and dominated by friction and the rate channel convergence.