Nonlinear water waves

Adrian Constantin

Abstract

This introduction to the issue provides a review of some recent developments in the study of water waves. The content and contributions of the papers that make up this Theme Issue are also discussed.

1. Introduction

The propagation of water waves has fascinated laymen and scientists for centuries. Water waves come in a seemingly endless array of forms, shaped by ever-changing influences (e.g. the topography of the sea floor, the speed and direction of the wind, and the presence of an underlying current) and yet they are to some extent mathematically predictable.

The most significant research activity on water waves can be traced back to the nineteenth century, all the way to the investigations of solitons that started in the early 1960s and to studies of waves of large amplitude in the past two decades. In recent years, the scientific literature has been enriched by thousands of contributions to the study of water waves. It would be an impossible task to report on this immense activity in a Theme Issue. However, during the past decade the research on tsunamis stands out by the importance to better understand the mechanisms of these water waves, which can have such disastrous effects. Two other themes stand out for the profound and exciting developments that have taken place within these specific areas. One concerns wave–current interactions and the other deals with the flow pattern beneath a regular surface water wave (particle trajectories, and the properties of the velocity field and of the pressure). These last two themes are capable of extension, being at the interface of theory and applications, and they both furnish striking examples of mathematically elegant results with relevance to physics.

This Theme Issue collects various papers from some leading experts with the aim to offer a snapshot of the current activity in these three areas of research within the broad field of water waves. The authors consider water waves from different points of view. Some are concerned with approximations of small-amplitude motions, while others tackle the nonlinear governing equations for water waves. A wide range of mathematical methods and modelling approaches (analytical, numerical, as well as experimental) is covered, giving a broad overview of these topics of current interest. All these papers were first presented in seminars during the three-month programme ‘Nonlinear Water Waves’ that took place between April and June 2011 at the Erwin Schrödinger International Institute for Mathematical Physics in Vienna, Austria.

2. Content of the present Theme Issue

Tsunamis are intriguing and frightening manifestations and the most recent disasters (December 2004 and March 2011) have taught us that we are unable to predict with accuracy the appearance of a tsunami. We refer to the recent publications [1,2,3] for detailed information about the December 2004 tsunami, and to Stimpson [4] for a discussion of reports on the March 2011 tsunami. For the mathematical modelling of tsunami waves, it is important to highlight the essential mechanisms by which these waves, once initiated, move, evolve and eventually become such destructive forces of nature. Owing to the different regimes, it is convenient to split the process of predicting tsunamis into two parts. One concerns the propagation of these waves in the open sea and the other in coastal regions. The first part starts from the assumption that the variations of the bed are on a scale comparable to those of the water's free surface, so that linear theory becomes relevant; cf. the considerations made in Constantin & Germain [5]. The second part requires an analysis of the wave dynamics in each particular region, and the specific features of the bottom topography are essential. A case study is the content of Tobias & Stiassnie [6]. The third paper devoted to tsunamis is the survey paper by Arcas & Segur [7]. Written in non-technical language to make it understandable to a broad audience, the authors summarize our current knowledge of the dynamics of tsunamis and illustrate how this knowledge is now being used to forecast tsunamis and to protect people better from the dangers of future tsunamis.

In recent years, major theoretical advances have been made in understanding the flow pattern in an irrotational travelling water wave (periodic, as well as solitary). Qualitative features of the particle paths were elucidated in the papers [811] and results on the symmetry of the surface waves were obtained by Constantin et al. [12], while the behaviour of the pressure was investigated by Constantin and co-workers [11,13]. In this Theme Issue, the papers by Hsu et al. [14] and Umeyama [15] are remarkable because of the numerical and experimental confirmation of these theoretical results. On the other hand, Clamond [16] formulates on the basis of numerical simulations some conjectures that refine the present understanding of the velocity field beneath a travelling water wave in irrotational flow. Chen et al. [17] and Okamoto & Shōji [18] also contribute to the study of particle paths, the first being concerned with numerical and experimental studies, and the second with analytical and numerical approaches.

In wave–current interactions, it is necessary to study rotational water waves since the hallmark of a non-uniform current is a flow of non-zero vorticity. The existence theory of rotational periodic travelling waves of small amplitude goes back to Dubreil-Jacotin [19] but only recently have existence results for waves of large amplitude been obtained [2022]. For solitary waves, the state of the art for flows with vorticity is restricted to waves of small amplitude; cf. the discussion in Hur [23] and Strauss [24]. For numerical simulations, we refer to earlier research [2527]. A crucial role in understanding nonlinear phenomena is played by regularity results. For the free boundary problem of water waves with vorticity, this information is an inherent part of the solution. Also, from the point of view of numerical computations, regularity results are of particular importance for the development of approximation schemes and the application of higher-order discretization techniques. A recent breakthrough regularity result for classical solutions to the governing equations for rotational water waves in flows without stagnation points was obtained by Constantin & Escher [28]. This opened up courses of action that have been pursued vigorously and led to various important results in a wider hydrodynamical context, as one can see from the contributions by Escher [29] and Henry [30] in this issue. The regularity of weak solutions in flows without stagnation points is the object of the contribution by Varvaruca & Zarnescu [31]. The issue of flows with stagnation points remains to be explored, especially due to the availability of existence results for waves of small amplitude (cf. [32,33]). One of the most fascinating flow patterns that arise in this context is Kelvin's streamline pattern of the cat's eye [34]. The paper by Johnson [35] addresses the possibility of generating critical layers beneath the surface of water in rotational flow, assuming that initially the flow does not contain one. Allowing for changes in the pressure at the surface, a transfer of energy to/from the flow at the surface becomes possible and two scenarios are investigated by asymptotic methods.

Footnotes

References

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