Modeling the M2 and M4 tidal wave propagation subjected to deepening, widening and friction change in an one dimensional basin system

Abstract

River estuaries are strongly modified by human interventions, such as dredging. For example the Ems river (Germany and Netherlands) has been deepened between in 1965 and 2005. As a result the sediment (sand and dirt) concentration has increased enormously \cite{Ems_data}. The muddiness of this sediment blocks sunlight and decreases the friction of the water with the bed level. The transport of sediment into the river is mainly caused by non-linearity in tidal waves. As a result of the elevation of the bed level and the friction change of the Ems river, the amplitude of the M2 and M4 tide has changed, which was concluded in previous research. In this study the M2 and M4 tide are studied individually, subjected to deepening, widening and friction change. The main research question in this study is '\textit{How does the propagation of the M2 and M4 tide change, subjected to deepening, change in width and change in friction?}' The result to this question can be applied to tidal rivers. To illustrate this, the results are compared to historical data of the Ems river. The sub question in this study is; '\textit{Can historical observations of the amplitude of the M2 and M4 tides in the Ems river over the years be explained with the change in propagation of the M2 and M4 tides, subjected to deepening and friction change?}'.

The questions are answered with help of a model. The model is constructed with the one dimensional water equations, where boundary conditions are used. Next the equations can be used in two cases. In the first case, which is used to answer the main question, the variables width, depth and friction of the river basin are taken to be length independent variables. In the second case, the width, depth and friction of the river basin are taken to be length dependent variables. This second case is used to model the propagation of the M2 and M4 tide for the Ems river. For the first case an analytical and numerical solution exist. From the error, the optimal grid size for the numerical model is obtained, which is taken as $N=100$. For the second case only a numerical solution exists.

With the model it is concluded that widening and an increasing friction cause a damping effect on the propagation of the M2 and M4 tide. Deepening, however, has a different effect on the M2 and M4 tide. It was seen that for certain value change in depth the M2 tide shows an amplification in amplitude, while the M4 tide shows an damping in amplitude for the same value change. The plots obtained in this part of the study can be used to see what is expected to happen to the amplitude of the M2 and M4 tide when a variable is varied in a tidal river basin. The latter is done for the Ems river.


An important observation made with the $x$-dependent model is that the amplification in amplitude from 1965 to 2005 can be explained by the decreased friction, due to increasing muddiness in the Ems river. Secondly, relatively, it was seen that the amplitude of the M4 tide shows a much greater amplification in amplitude at the beginning of the river than the amplitude of the M2 tide. This could be explained by the different change in amplitude of the M2 and M4 tide subjected to deepening, which was concluded from the $x$-independent model as well. However, the difference in M2 and M4 tide between 1965 and 2005 is not due to deepening on its own. Namely, the change in amplitude due to deepening highly depends on the value of the friction, which is different for the years 1965 and 2005. It is concluded that the exact amplitude of the M2 and M4 tide cannot be predicted with the $x$-independent model. However, it is concluded that the results from the $x$-independent model can be used to predict how the M2 and M4 tide will change due to deepening relatively to each other.


The results of this study are strongly influenced by the assumptions made to derive the model. Before the one dimensional, $x$-independent model was compared to the observations for the Ems river between 1965 and 2005 a few decisions had to be made regarding $x$-dependent to $x$-independent variables. This process needs to be further researched, before applying the x-independent model to other tidal rivers, which is left for further research.