Tidal bars are repetitive estuarine bedforms with heights of several meters and wavelengths in the order 1–15 km. Understanding their formation and sensitivity to changes in channel characteristics is important as they hamper marine traffic and play a crucial role in the local ecosystem. Recent observations suggest that the local width of the channel is dominant in determining the tidal bar wavelength. However, theoretical studies could not reproduce the sensitivity of the tidal bar wavelength to channel width. This discrepancy between theory and observations suggests that a mechanism is missing. In this study, one of the theoretical models is extended and results in tidal bar wavelengths, lateral mode numbers, and growth rates that agree fairly well with those of natural tidal bars, including the wavelengths dependency on estuary width. An important extension of the model concerns the bed slope induced diffusive suspended load transport of sediments. With this, it is explained why previously, the modelled tidal bar wavelengths depend only weakly on estuary width and why in the extended model it does. This has, from a modelling point of view, general implications for morphological models using a total load sediment transport formulation with a so‐called bed slope parameter.

In a tidal channel with adjacent tidal flats, along-channel momentum is dissipated on the flats during rising tides. This leads to a sink of along-channel momentum. Using a perturbative method, it is shown that the momentum sink slightly reduces the M 2 amplitude of both the sea surface elevation and current velocity and favours flood dominant tides. These changes in tidal characteristics (phase and amplitude of sea surface elevations and currents) are noticeable if widths of tidal flats are at least of the same order as the channel width, and amplitudes and gradients of along-channel velocity are large. The M 2 amplitudes are reduced because stagnant water flows from the flats into the channel, thereby slowing down the current. The M 4 amplitudes and phases change because the momentum sink acts as an advective term during the fall of the tide, such a term generates flood dominant currents. For a prototype embayment that resembles the Marsdiep-Vlie double-inlet system of the Western Wadden Sea, it is found that for both the sea surface elevation and current velocity, including the momentum sink, lead to a decrease of approximately 2% in M 2 amplitudes and an increase of approximately 25% in M 4 amplitudes. As a result, the net import of coarse sediment is increased by approximately 35%, while the transport of fine sediment is hardly influenced by the momentum sink. For the Marsdiep-Vlie system, the M 2 sea surface amplitude obtained from the idealised model is similar to that computed with a realistic three-dimensional numerical model whilst the comparison with regard to M 4 improves if momentum sink is accounted for.

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The role of the Coriolis effect in the initial formation of bottom patterns in a tidal channel is studied by means of a linear stability analysis. The key finding is that the mechanism generating oblique tidal sand ridges on the continental shelf is also present in confined tidal channels. As a result, the Coriolis effect causes the fastest growing pattern to be a combination of tidal bars and oblique tidal sand ridges. Similar as on the continental shelf, the Coriolis-induced torques cause anticyclonic residual circulations around the ridges, which lead to the accumulation of sand above the ridges. Furthermore, an asymptotic analysis indicates that the maximum growth rate of the bottom perturbation is slightly increased by the Coriolis effect, while its preferred wavelength is hardly influenced.

<div>Tidal bars are bed forms in tidal channels that have a wave-like structure in both the along-channel and cross-channel direction. They are found in tidal channels all around the globe, for example, in the Western Scheldt in the Netherlands, the Exe Estuary in England, the Ord River Estuary in Australia and the Venice Lagoon in Italy. Typically, tidal bars are several meters high, have wavelengths of 1-15 km and migration speeds of meters per day. Understanding their dynamics is important as they are invaluable for many living organisms (e.g., migrating birds) but they hamper marine traffic.</div><div> </div><div> <div>It has been shown, by means of a linear stability analysis, that these bars emerge due to inherent feedbacks between the tidal currents and the erodible bed. When the bars mature, their dynamics becomes nonlinear. Schramkowski et al. (2004) applied a bifurcation analysis to analyse the bar dynamics, but their method was limited to small bottom friction. Here, we developed a numerical (time integration) model that simulates the nonlinear dynamics and the corresponding (stable) equilibrium patterns for realistic parameter values.</div> <div> </div> <div> <div>Using the output of the numerical model we derive a reduced order model with a method called SINDy (Brunton et al., 2016). Loiseau and Brunton (2018) showed that from output of complex numerical models simulating fully nonlinear fluid flows, SINDy can identify small systems of equations which govern the complex flows. Here we show that, for parameters regimes where the dynamics is weakly nonlinear, SINDy finds a Landau type equation that reproduces the tidal bar dynamics well. The Landau equation is a nonlinear ordinary differential equation in terms of the Fourier amplitude of the pattern that initially has the largest growth rate. The form of this equation corresponds with the one that is expected from the symmetry of the patterns. Also, the application of SINDy to the fully nonlinear dynamics of tidal bars will be discussed.</div> <div> </div> <div> </div> <div> <div>Brunton, S.L., Proctor, J.L., and Kutz, J.N. (2016). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113(15):3932-3937.</div> <div>Loiseau, J.-C. and Brunton, S.L. (2018). Constrained sparse Galerkin regression. Journal of Fluid Mechanics, 838:42-67.</div> <div>Schramkowski, G.P., Schuttelaars, H.M., and de Swart, H.E. (2004). Nonlinear channel-shoal dynamics in long tidal embayments. Ocean Dynamics, 54(3):399-407.</div> </div> </div> </div>