Surface waves over currents and uneven bottom

Abstract

The propagation of surface water waves interacting with a current and an uneven bottom is studied. Such a situation is typical for ocean waves where the winds generate currents in the top layer of the ocean. The role of the bottom topography is taken into account since it also influences the local wave and current patterns. Specific scaling of the variables is selected which leads to approximations of Boussinesq and KdV types. The arising KdV equation with variable coefficients, dependent on the bottom topography, is studied numerically when the initial condition is in the form of the one soliton solution for the initial depth. Emergence of new solitons is observed as a result of the wave interaction with the uneven bottom.

Introduction

Surface waves are examples of geophysical waves found ubiquitously in the oceans. They usually result from distant winds and may travel thousands of miles before they decay or break. Typically surface wave heights are 1–2 m but extreme wave heights approaching nearly 20 m (excluding rogue waves) have been observed.

The complete dynamics of the fluid, in many situations, can be recovered from the surface wave motion. Therefore the wave propagation is an important research topic, with possible coastal and oceanic engineering applications in areas like oil platform stability and designs of structures like breakwaters and artificial reefs.

Various factors play a role in the formation and the dynamics of the surface waves. Some of these are rotational effects (for example currents) and the bottom topography. The propagation of waves over a current field is, in general, a complicated phenomenon and an active research topic. Currents exist due to many factors but primarily due to wind and tides. One of the most significant currents is the Equatorial Undercurrent (EUC) which can be observed at approximately 20 degrees south of the Equator. The EUC flows eastwards, at a speed of around one meter per second, one hundred meters below the ocean surface and is approximately 300 kilometers wide. Some more details about the complexity of the dynamics of EUC which pertain to the analysis of solutions can be found in the papers by Constantin and Johnson (2016a, 2017b). A general overview on mathematical tools employed for the study of problems in physical oceanography is given in Johnson (2018). We would also like to point out the recent emergence of solutions modelling large-scale ocean currents with distinctive persistence patterns like gyres (Constantin and Johnson, 2017a) and the Antarctic Circumpolar Current (Constantin and Johnson, 2016b). An analysis of a solution describing nonlinear surface waves propagating zonally on a zonal current in the presence of Coriolis effects and exhibiting two modes of wave motion was presented by Constantin and Monismith (2017).

For the aim of our investigations we consider two-dimensional flows in the f-plane approximation. Such an approach is reasonable since the Equator acts as a wave guide (Fedorov and Brown, 2009) and the depth-dependent currents are confined to a shallow near-surface layer. For a rigorous existence proof of rotational water flows in the f-plane approximation we refer the reader to the paper by Constantin (2013) see also the paper by Martin (2018).

In the review chapters by Peregrine (1976) and Jonsson (1990), the physical circumstances in which interactions among water waves and currents occur are discussed and also mathematical models of these interactions are outlined. Thomas (1981), Thomas (1990) studied the horizontal velocity distribution and surface elevation in a wave-current environment. A method has been developed for the measurement of the strength of the wave-current interaction by Thomas and Klopman (1997). A variety of other aspects has been an active research topic (Teles da Silva and Peregrine, 1988, Constantin and Strauss, 2004, Constantin and Escher, 2004, Constantin and Escher, 2011, Constantin et al., 2006, Henry, 2013, Constantin and Johnson, 2015, Constantin and Johnson, 2017a, Constantin and Johnson, 2017b, Escher et al., 2016). Recent developments in the mathematical aspects of wave-current interactions are presented in the monograph by Constantin (2011).

In physical reality, the seabed can be a complex undulating structure with trenches, underwater mountains and sediment, not to mention the multitude of marine flora and fauna. In some parts of the ocean, due to volcanic activity, the seabed is not even a stationary structure. Numerous studies are dedicated to the wave motion over an uneven bottom (Johnson, 1973a, Johnson, 1973b, Johnson, 1997, Nachbin, 2003, Craig et al., 2005a, Dejak and Sigal, 2006, de Bouard et al., 2008 Compelli et al., 2017; Nachbin and Ribeiro, 2018). In the majority of the existing publications however, the fluid is irrotational, i.e. the underlying current is not taken into account.

In our studies we are going to apply the Hamiltonian approach to derive models for surface wave propagation in the presence of a current and of an uneven bottom. The significance of the work of Hamilton in developing Hamiltonian mechanics in the early 19th century became apparent in the context of fluid systems following the publication of Vladimir Zakharov (1968) with additional constructions by Milder (1977) and Miles (1977). In his work Zakharov demonstrated the Hamiltonian structure of an infinitely deep vorticity-free single fluid system. Further significant studies which followed on include the works of Benjamin and Olver (1982), Craig and Groves (1994), Craig et al., 2005b, Craig et al., 2006 among others. The Hamiltonian approach in the analysis of irrotational single media systems, was then extended to include systems with nonzero vorticity. For instance a rotational single fluid system, of finite-depth, was shown by Constantin et al. (2008) to exhibit nearly canonical Hamiltonian structure and Wahlén (2007) showed that a fully canonical structure could indeed be achieved via a variable transformation. Moreover, Constantin et al. (2016) showed that wave-current interactions in stratified rotational flows also possess a Hamiltonian formulation.

Our aim is to derive and investigate a model, which takes into account both the effect of the current and the non-flat bottom. Such a situation is common for ocean waves and waves over the EUC are just one particular realistic example. To fulfil our goal we show first that the governing equations for gravity water flows in the f-plane approximation (near the Equator) that exhibit constant (non-vanishing) vorticity and propagate over a non-flat bed can be written in Hamiltonian form. Subsequently, we derive from the Hamiltonian formulation a small amplitude and a long-wave approximation, typically related to the Boussinesq and KdV propagation regimes. We use several mathematical results explained previously in Constantin et al. (2016) and Compelli et al. (2018). We would like to mention that the Hamiltonian formulation for two-layered stratified water flows over a flat bed and displaying piecewise constant vorticity in the equatorial f-plane approximation was accomplished by Ionescu-Kruse and Martin (2017).