How tides and river flows determine estuarine bathymetries

For strongly tidal, funnel-shaped estuaries, we examine how tides and river flows determine size and shape. We also consider how long it takes for bathymetric adjustment, both to determine whether present-day bathymetry reflects prevailing forcing and how rapidly changes might occur under future forcing scenarios.Starting with the assumption of a 'synchronous' estuary (i.e., where the sea surface slope resulting from the axial gradient in phase of tidal elevation significantly exceeds the gradient in tidal amplitude ), an expression is derived for the slope of the sea bed. Thence, by integration we derive expressions for the axial depth profile and estuarine length, L, as a function of and D, the prescribed depth at the mouth. Calculated values of L are broadly consistent with observations. The synchronous estuary approach enables a number of dynamical parameters to be directly calculated and conveniently illustrated as functions of and D, namely: current amplitude Û, ratio of friction to inertia terms, estuarine length, stratification, saline intrusion length, flushing time, mean suspended sediment concentration and sediment in-fill times.Four separate derivations for the length of saline intrusion, L_I, all indicate a dependency on (U_o is the residual river flow velocity and f is the bed friction coefficient). Likely bathymetries for `mixed' estuaries can be delineated by mapping, against and D, the conditions L<sub>I</sub>/L<1,E<sub>X</sub>/L<1 (E<sub>X</sub> is the tidal excursion) alongside the Simpson-Hunter criteria D/U<sup>3</sup><50 m<sup>−2</sup> s<sup>3</sup>. This zone encompasses 24 out of 25 `randomly' selected UK estuaries.However, the length of saline intrusion in a funnel-shaped estuary is also sensitive to axial location. Observations suggest that this location corresponds to a minimum in landward intrusion of salt. By combining the derived expressions for L and L_I with this latter criterion, an expression is derived relating D_i, the depth at the centre of the intrusion, to the corresponding value of U_o. This expression indicates U_o is always close to 1 cm s⁻¹, as commonly observed. Converting from U_o to river flow, Q, provides a morphological expression linking estuarine depth to Q (with a small dependence on side slope gradients).These dynamical solutions are coupled with further generalised theory related to depth and time-mean, suspended sediment concentrations (as functions of and D). Then, by assuming the transport of fine marine sediments approximates that of a dissolved tracer, the rate of estuarine supply can be determined by combining these derived mean concentrations with estimates of flushing time, F_T, based on L_I. By further assuming that all such sediments are deposited, minimum times for these deposition rates to in-fill estuaries are determined. These times range from a decade for the shortest, shallowest estuaries to upwards of millennia in longer, deeper estuaries with smaller tidal ranges.