Relationships between Tidal Dynamics and Bathymetry in Strongly Convergent Estuaries

D. Prandle Proudman Oceanographic Laboratory, Bidston Observatory, Merseyside, United Kingdom

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Abstract

Localized analytical solutions are derived for the propagation of a single (predominant) tidal constituent in estuaries with strongly convergent triangular cross sections. The advective term is neglected, and the friction term is linearized. The solutions apply at any locality where the cross-sectional shape remains sensibly congruent and the spatial gradient of tidal elevation amplitude is small (i.e., “synchronous”). Then, for specified tidal elevation amplitude ζ̂ and water depth D, these solutions indicate values of tidal current amplitude û, ratio of friction to inertial terms, slope of the estuarine bed, rate of energy dissipation, and phase difference θ between ζ̂ and û. Illustrations of these results over a range of ζ̂ and D provide reference frameworks to assess observed conditions at any position in any estuary satisfying the criteria cited. They may also indicate the range of parameters likely to be consistent with bathymetric equilibrium in dynamic sedimentary regimes. The ratio of friction to inertial terms is shown to be represented by tanθ. Moreover, this ratio is further shown to be approximated by 10ζ̂/D, the latter demarking the classification of shallow and deep estuaries. For shallow estuaries, by specifying a constant value of ζ̂, an expression for estuarine length is determined approximating 2350D5/4/ζ̂1/2 (m). This expression is consistent with observed values from some 50 estuaries spanning, without selection, both the entire U.K. coastline and that of the U.S. eastern seaboard. (The U.S. data are extracted from a paper relating to “shallow, well-mixed estuaries.”) The associated scatter of the observed values is shown to be, in part, related to the influence of their prevailing sediment regime on their respective bed friction coefficients. This scatter is used to suggest a new expression relating the bed friction coefficient to the surficial mud content. Sensitivity of the solutions to changes in bed friction coefficient, mean sea level, and tidal conditions are also examined. These analyses indicate what changes in the shape and size of an estuary might follow from variations in the prevailing tidal and sedimentary regimes. These new results are assessed in the context of extensive earlier studies into tidal asymmetry in estuaries.

Corresponding author address: Dr. D. Prandle, Proudman Oceanographic Laboratory, Bidston Observatory, Merseyside CH43 7RA, United Kingdom. Email: dp@pol.ac.uk

Abstract

Localized analytical solutions are derived for the propagation of a single (predominant) tidal constituent in estuaries with strongly convergent triangular cross sections. The advective term is neglected, and the friction term is linearized. The solutions apply at any locality where the cross-sectional shape remains sensibly congruent and the spatial gradient of tidal elevation amplitude is small (i.e., “synchronous”). Then, for specified tidal elevation amplitude ζ̂ and water depth D, these solutions indicate values of tidal current amplitude û, ratio of friction to inertial terms, slope of the estuarine bed, rate of energy dissipation, and phase difference θ between ζ̂ and û. Illustrations of these results over a range of ζ̂ and D provide reference frameworks to assess observed conditions at any position in any estuary satisfying the criteria cited. They may also indicate the range of parameters likely to be consistent with bathymetric equilibrium in dynamic sedimentary regimes. The ratio of friction to inertial terms is shown to be represented by tanθ. Moreover, this ratio is further shown to be approximated by 10ζ̂/D, the latter demarking the classification of shallow and deep estuaries. For shallow estuaries, by specifying a constant value of ζ̂, an expression for estuarine length is determined approximating 2350D5/4/ζ̂1/2 (m). This expression is consistent with observed values from some 50 estuaries spanning, without selection, both the entire U.K. coastline and that of the U.S. eastern seaboard. (The U.S. data are extracted from a paper relating to “shallow, well-mixed estuaries.”) The associated scatter of the observed values is shown to be, in part, related to the influence of their prevailing sediment regime on their respective bed friction coefficients. This scatter is used to suggest a new expression relating the bed friction coefficient to the surficial mud content. Sensitivity of the solutions to changes in bed friction coefficient, mean sea level, and tidal conditions are also examined. These analyses indicate what changes in the shape and size of an estuary might follow from variations in the prevailing tidal and sedimentary regimes. These new results are assessed in the context of extensive earlier studies into tidal asymmetry in estuaries.

Corresponding author address: Dr. D. Prandle, Proudman Oceanographic Laboratory, Bidston Observatory, Merseyside CH43 7RA, United Kingdom. Email: dp@pol.ac.uk

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