Suspended sediment transport under seiches in circular and elliptical basins

Abstract

Enclosed bodies of water such as lakes or harbours often experience large-scale oscillatory motions (seiching). As a simple model of such flow, we investigate exact solutions to the shallow-water equations which represent oscillatory flow in an elliptical basin with parabolic cross section. Specifically, we consider two fundamental modes of oscillation, in one of which the flow is parallel to the axis of the ellipse, while in the other it is radial. We obtain periodic analytical solutions for sediment transport, including erosion, deposition and advection, under either mode of oscillation, and present a method for obtaining such solutions for a more general class of flow fields and sediment transport models.

Our solutions provide estimates of the morphodynamical importance of seiching motions and also reveal a characteristic pattern of net erosion and deposition associated with each mode. In particular, we find that a net flux of suspended sediment can be transported from the deeper to the shallower regions of the basin. These transport patterns, which are driven essentially by settling lag, are highly robust to the formulation of the sediment transport relation and appear not to be substantially affected by the omission of frictional terms in the hydrodynamics: they should thus provide considerable insight into sediment transport in less-idealised systems.

Introduction

The processes by which sediment is mobilised and redistributed in confined bodies of water such as harbours, lagoons and lakes are naturally of considerable interest to the coastal and civil engineer, as well as to the sedimentologist or limnologist. In particular, an understanding of these processes is necessary in order to predict and control the morphological evolution of such systems, with potentially important ecological and navigational consequences. To date, these topics have been addressed almost exclusively through empirical and numerical studies, and the object of this paper is to complement such studies with a theoretical investigation of some of the mechanisms involved.

We approach the problem by developing exact solutions for the suspended sediment field in the water body under specific types of hydrodynamic forcing. These solutions are useful in two respects. Firstly, they offer insight into the physical processes which control sediment redistribution, in a clearer form than may readily be obtained from field data or from extensive numerical simulations. Secondly, they provide easily reproduced test cases which may be used to validate numerical models for the prediction of sediment transport. We now discuss each of these points in turn.

In a confined body of water, bedload transport, in the form of either granular flows of noncohesive material (see, e.g. Seminara, 2001) or fluid mud layers (e.g. Roberts, 1993, Ali et al., 1997), is directed preferentially downslope into the deeper regions in the middle of the basin. Other mechanisms such as sediment focussing by secondary circulation are also known to contribute to this net inward movement of sediment (Bloesch, 1995). However, it is less obvious how or whether there is some return flux of suspended sediment into the shallower regions of the basin—a question which is clearly of considerable importance for the morphodynamics of the system.

One obvious candidate mechanism for such transport is wind-generated waves and the associated currents, which may be expected to be particularly important where the water is relatively shallow. However, when considering transport in deeper regions, it is necessary to examine the possible effects of other hydrodynamic processes. In particular, both internal and surface seiching motions have been proposed as possible mechanisms for sediment reworking and transport in studies of contemporary and historical sedimentation (see, for example, Bloesch, 1995, Shteinman et al., 1997, Chapon et al., 1999), and there is therefore a useful role to be played by a theoretical investigation of such transport. The current study is complementary to recent investigations of the role of suspended sediment transport on tidal flats Pritchard et al., 2002, Pritchard and Hogg, 2003a and in tidal inlets Schuttelaars and de Swart, 1996, Schuttelaars and de Swart, 1999, Schuttelaars, 1998.

In the current study, we confine ourselves to considering surface seiches in a somewhat idealised geometry. The advantage of this approach is that we are able to employ the exact solutions for the nonlinear normal modes of oscillation of fluid in a basin of parabolic cross section which were derived by Thacker (1981) and thus to obtain exact analytical solutions for suspended sediment transport. This allows a clearer and somewhat more thorough investigation of the transport mechanisms than a purely numerical study.

A particular difficulty encountered in numerical models of nearshore hydrodynamics and sediment transport is the representation of the variables in shallow water close to the moving shoreline. In order to obtain reliable results in this region, it is essential that numerical methods be validated both against field data and against exact solutions where these are available. There is, however, a scarcity of available solutions for such validation even for the hydrodynamic fields and even fewer for the suspended sediment field: this therefore represents an important secondary motivation for the work described here. A particular feature of the Lagrangian approach, which we employ to develop our solutions, is that the moving shoreline boundary may be handled without the need to impose extra physical or numerical conditions on the suspended sediment concentration here.

In this paper, we compare the sediment transport under two basic modes of oscillation of the fluid body. In one, the basin is elliptical in plan view, and the fluid motion is parallel to one axis of the ellipse; in the other, the basin is circular in plan view, and the fluid motion is radial. A distinct pattern of net sediment transport is associated with each mode: axial flow transports material outwards from the centre and deposits it in the shallower parts of the basin, while radial flow erodes material from an annular region and deposits it principally landwards of this region. The robustness of these results suggests that they offer genuine insight into how such oscillations may contribute to long-term patterns of sediment movement and thus to morphological change.

In Section 2, we introduce the shallow-water equations which describe the fluid motion and sediment transport. In particular, we describe a model for the erosion and deposition of sediment which is principally intended to represent coarse sand, but which may easily be adapted to finer and even to cohesive sediment. In Section 3, we consider axial oscillations in an elliptical basin, and in Section 4, we consider radial oscillations in a circular basin. In both sections, exact periodic solutions to the sediment transport equation are constructed using the general method described in Appendix A. Finally, in Section 5, we summarise our results and discuss their implications for more general bathymetries. Appendix B deals briefly with the extent to which the neglect of friction in obtaining these results may be justified.