Mathematical modeling of tidal hydrodynamics in shallow lagoons: A review of open issues and applications to the Venice lagoon

Abstract

Although considerable progress has been made in the application of two- and three-dimensional shallow water models to simulate flow in estuaries and coastal lagoons, a number of outstanding problems still remain in this branch of computational fluid dynamics.

These problems mainly deal with the proper representation of physical processes and arise when dealing with very shallow flow, temporary submergence and time-dependent flow domains, and complex morphology, and are often related to inaccuracies in modeling the geometry of the domain. Among others, wetting and drying of salt marshes and tidal flats, inclusion of the small scale drainage networks which often strongly affect hydrodynamics, turbulence closure schemes, and accurate friction evaluation are discussed in this paper.

To better illustrate these problems and their possible solutions some examples are presented which use the Venice lagoon as a benchmark shallow tidal basin characterized by a complex interplay of channels and marshes.

Introduction

Current developments of two-dimensional shallow water models and the availability of refined numerical techniques and of powerful and accessible computers offer increased opportunities to simulate tidal flow in estuaries and coastal basins. Available numerical techniques to solve the shallow water equations are now consolidated. However, a number of issues related to physical processes representation are still open or, rather, unsatisfactorily closed and require further theoretical and experimental study.

In dealing with flow in shallow tidal lagoons one has often to cope with very small water depths and with flooding and drying of marshes and tidal flats. In this case the accuracy in the evaluation of both mass balance and flow resistance is essential if phase shifts and amplitudes of the tide propagating inside the lagoon are to be accurately predicted. In the recent past the problem has received considerable attention (Braschi et al., 1994; Balzano, 1998; Ji et al., 2001; Oey, 2005). However, most of the proposed algorithms have been developed to tackle numerical instabilities arising in moving boundary hydrodynamics problems with minor or no attention to the physical processes actually involved.

In the presence of rapidly varying topography (e.g., transition from tidal channels to tidal flats or marshes), which do not conform with the gradually varying topography assumed in deriving the two–dimensional shallow flow equations, some serious modeling problems arise which involve convective accelerations and horizontal shear stresses. Importantly, the latter are also affected by the chosen turbulence model. Considerable progress has been made in the development of turbulence closure schemes; however, their application in two-dimensional shallow-water models at the geophysical scale have never been properly assessed and a number of turbulence-related problems, discussed in this paper, remain unanswered.

It is also worthwhile to stress that different problems, and as a consequence, different modeling approaches and simplifications can be adopted when simulating flow over morphologically different areas (i.e., marshes, shallows, and tidal channels of different sizes). Clearly, the choice also depends on the required accuracy. For example, modeling problems affecting the inner part of a tidal basin, related to rapidly varying topography and complex morphology, generally differs from problems related to the hydrodynamic modeling of lagoon inlets and the main channels departing from them. As a consequence, accurate solutions for the former numerical problems do not easily match with optimal solutions for the latter problems. This matching is indeed a further modeling problem. To better specify the above statements, it is worth recalling that in strongly dissipative and morphologically complex systems, such as the Venice lagoon, convective accelerations and horizontal Reynolds stresses play a minor role compared to gravity and bed shear stresses. Moreover, an accurate description of these terms requires a very accurate resolution of the velocity field, which is a hard task when the domain is large and the numerical discretization is comparably rough. Therefore, the inclusion of these terms in the model may lead to inaccuracies which can be far greater than those one has when these terms are neglected. Indeed, it can be shown that effects produced by convective acceleration can be treated, from a large scale point of view, as an additional energy dissipation produced by small scale, i.e., not resolved by the model, momentum mixing and can be accounted for by adjusting the friction coefficient (Defina, 2000b; D’Alpaos and Defina, 1993; D’Alpaos et al., 1995; Umgiesser et al., 2004). However, when simulating tidal flow through the inlets and along the main channels the above simplification is quite crude. In most cases it is not acceptable and the inclusion of convective acceleration and Reynolds stress in the model is strictly required.

Adequate solutions to the above problems cannot be just given by adopting accurate and well-structured numerical schemes, or else extremely refined computational grids. An important effort should instead be addressed toward modeling the relevant physical phenomena, which are neglected or drastically filtered by the numerical solution. This can be accomplished through the construction of suitable subgrid models, i.e. by setting up a phenomenological representation of the overall processes which ensures a statistically equivalent description of the actual physics.

In the present contribution we review and discuss the main modeling problems which arise when dealing with very shallow flows and morphologically complex bottom topography. Possible solutions based on subgrid modeling approach are proposed. To better illustrate the problems and their possible solutions some examples are presented which use the Venice lagoon as typical irregular and shallow tidal basin.

The lagoon of Venice is a wide shallow basin crossed by a network of deep channels departing from three inlets, namely Lido, Malamocco and Chioggia (Fig. 1). The lagoon is also characterized by the presence of small islands and wide salt marshes and tidal flats. Salt marshes exhibit a dendritic structure of channels of varying sizes. These channels perform a drainage function, often continuing to flow long after the tide has receded and the marshes are exposed.

These morphologic units play different roles in affecting tidal propagation within the lagoon, but all have comparable importance. Quite different behavior characterizes the flow over the marshes, along the channels, and over the tidal flats. Therefore, different assumptions are required to account for the local morphology and hydrodynamics in order to maximize the accuracy and minimize the computational effort.

The paper is organized as follows. Section 2 discusses the wetting and drying problem in shallow flows over irregular topography. Section 3 provides the description of a 1D–2D model which couples 2D elements for the flow over shallow areas and large channels, and 1D elements for the flow in the small tidal channels and creeks. Section 4 focuses on problems related to the numerical treatment of convective acceleration. In Section 5 the problem of assessing the eddy viscosity coefficient, to accurately estimate the horizontal Reynolds stresses is discussed. Finally, some concluding remarks are made in Section 6.