Exploratory modelling of the formation of tidal bars under a propagating tidal wave using a linear stability analysis
White, R.D. (2021). Exploratory modelling of the formation of tidal bars under a propagating tidal wave using a linear stability analysis. MSc Thesis. University of Twente, Faculty of Engineering Technology: Enschede. 65 pp.
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Documenttype: Doctoraat/Thesis/Eindwerk
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| Abstract |
Tidal bars are rhythmic bed features that occur in tidal channels. These bars typically have heights of several metres and wavelengths of 1 to 15 km. The formation of rhythmic bed features is often studied using the linear stability concept, in which the formation of bedforms is explained as a free instability of a morphodynamic system. In linear stability models several assumptions are usually applied to simplify and schematise the modelled system. One of these is the so-called rigid lid assumption, through which the free surface effects are neglected. These are effects caused by changes in the position of the water surface due to tidal waves. This assumption is currently applied in most linear stability studies. In this study we develop a (numerical) linear stability model in which the rigid lid assumption is not applied, i.e., the NRL (non-rigid lid) model. This (numerical) NRL model aims to include (instead of neglect) the free surface effects caused by a propagating tidal wave. The NRL model can therefore be used to obtain physics-based insights into the influence of a propagating tidal wave on the formation of tidal bars. These insights are obtained by comparing the NRL model to a traditional (semi-analytical) linear stability model in which the rigid lid assumption is applied, i.e., the RL (rigid lid) model. This is done for two cases in the Western Scheldt: the standard friction case and the reduced friction case. First, a model is formulated for a propagating tidal wave in a tidal channel. This model consists of the depth-averaged hydrodynamic equations, a simple sediment transport formula and a sediment conservation equation for the bed evolution. Boundary conditions are defined that allow a tidal wave to propagate in the positive along-channel direction. Next, an expansion in the Froude number is used to obtain a solution to the basic state for the NRL model. The solutions for the basic flow show that the free water surface and depth-averaged velocity in the along-channel direction are described by sinusoidal waves that travel in the along-channel direction. Moreover, because the solution for the basic flow is spatially variant, the sensitivity of the bed evolution under the basic flow is analysed. This is done by comparing the bed evolution in the basic state to the approximated evolution of the bed in the perturbed state. If the former is small compared to the latter, it is valid to conclude that the bed in the basic state remains relatively flat. The results show that the solution to the basic state for the NRL model presented in this study is only valid for the reduced friction case. Thereafter, a numerical solution procedure for the perturbed state is developed. This is done because the traditional (semi-analytical) procedure breaks down when the solution to the basic state is spatially variant, as is the case for the NRL model. As part of this numerical procedure, a generalised eigenvalue problem for the evolution of this state is defined. The eigenvectors and eigenvalues that follow from this problem describe the structure of the bedforms and can be used to determine the other bedform characteristics (i.e., growth rate, cross-channel mode and wavelength). These characteristics are used to construct growth curves and to define the fastest growing mode (FGM). The FGM is considered the dominant bedform and therefore provides insight into the behaviour of the modelled system. The numerical solution procedure for the perturbed state is first applied to the RL model. This is done to validate the numerical procedure by comparing the numerical solutions to those obtained using the semi-analytical procedure. The results for both cases in the Western Scheldt show that the numerical and semi-analytical solutions are approximately equal when the number of discretisation points is sufficiently large. Finally, the numerical solution procedure is applied to the NRL model in which a propagating tidal wave is considered. The solutions for the NRL model are compared to those obtained using the RL model. The results for the reduced friction case show that the peaks of the growth curves (for higher cross channel modes) for the NRL model are higher and occur at larger wavenumbers compared to those for the RL model. We define two preconditions: the abovementioned difference is caused by a propagating tidal wave; and the influence of a propagating tidal wave is the same for both friction cases. Assuming these preconditions are true, we expect that a propagating tidal wave might have the following influence on the formation of tidal bars: (1) formation of shorter tidal bars (i.e., the wavelength of the FGM decreases); and (2) faster formation of tidal bars (i.e., the growth rate of the FGM increases). However, to ensure the validity of these statements, further research must be conducted in which the NRL model is improved and validated. |
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