# Three-dimensional semi-idealized model for estuarine turbidity maxima in tidally dominated estuaries

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## Abstract

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.

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