Observations of barrier coasts around the world suggest that some systems do not conform to the O'Brien-Jarret law. Here we explain this by investigating how resonance and bottom friction affect the response of tidal inlets to variations in basin geometry. Therefore, we develop a morphodynamic barrier coast model that is based on the stability concept of Escoffier for the morphological evolution of the inlets, coupled with an idealized hydrodynamic model that describes the water motion in the outer sea, inlets, and arbitrarily shaped back-barrier basin. We find that the total tidal prism through all inlets is predominantly determined by the (cross-shore) width of the basin and identify three regimes for this. First, a linear regime for narrow basins (i.e., basin width (Formula presented.) tidal wavelength) where a larger basin leads to a linear increase in total tidal prism. Second, a resonant regime for basins with a width around the resonant condition in which the total tidal prism reaches a peak. This resonance condition is a quarter tidal wavelength for basins without friction, which shifts to narrower basins as friction becomes stronger, down to 0.15 tidal wavelength. Third, a dissipative regime for wide basins (i.e., the cross-shore basin dimension or basin width (Formula presented.) resonant condition) with sufficiently strong bottom friction in which the total tidal prism does not change for wider basins, because the tidal wave completely dissipates in the basin.

We develop a three-dimensional idealized model that is specifically aimed at gaining insight in the physical mechanisms resulting in the formation of estuarine turbidity maxima in tidally dominated estuaries. First, the three-dimensional equations for water motion and suspended sediment concentration together with the so-called morphodynamic equilibrium condition, are scaled. Next, surface elevation, velocity and sediment concentration are expanded in a small parameter ϵ=A¯_M2 /H, where A¯_M2 is the mean amplitude of the M₂ tide and H is the mean water depth at the seaward side. This results in a system of equations at each order in this small parameter. This ordering allows solving for the vertical structure of the velocity and suspended sediment concentration, independently of the horizontal dimension. After obtaining these vertical structures, the horizontal dependencies of the physical variables follow from solving a two-dimensional elliptic partial differential equation for the surface elevation. The availability of fine sediments in the estuary follows from a two-dimensional elliptic partial differential equation which results from requiring the system to be in morphodynamic equilibrium, and prescribing the total amount of easily erodible sediments available in the estuary. These elliptic equations for the surface elevation and sediment availability are solved numerically using the finite element method with cubic polynomials as basis functions. As a first application, the model is applied to the Ems estuary using a simplified geometry and bathymetric profiles characteristic for the years 1980 and 2005. The availability of fine sediments and location of maximum concentration are investigated for different lateral depth profiles. In the first experiment, a uniform lateral depth is considered. In this case, both the sediment availability and suspended sediment concentration are, as expected, uniform in the lateral direction. In 1980, the sediment is mainly trapped near the entrance, while in 2005, the sediment is mostly trapped in the freshwater zone. In the next experiment, the lateral bathymetry is varied parabolically while keeping the mean depth unchanged. In this case, the fine sediment is mainly found at the shallow sides, but the maximum sediment concentration is found in the deeper channel where the bed shear stress is much larger than on the shoals. As a final experiment, a more realistic (but smoothed) geometry and bathymetry for the Ems estuary are considered, showing the possibilities of applying the newly developed model to complex geometries and bathymetries.