How tides and river flows determine estuarine bathymetries

Abstract 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 ζ ), 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 ζ 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 ζ and D, namely: current amplitude U, 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 D 2 /f U U o ( 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 ζ and D, the conditions L I / L E X / L E X is the tidal excursion) alongside the Simpson–Hunter criteria D / U 3 −2  s 3 . 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 ζ 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.