Unsteady undular bores in fully nonlinear shallow-water theory

Abstract

We consider unsteady undular bores for a pair of coupled equations of Boussinesqtype which contain the familiar fully nonlinear dissipationless shallow-water dynamics and the leading-order fully nonlinear dispersive terms. This system contains one horizontal space dimension and time and can be systematically derived from the full Euler equations for irrotational flows with a free surface using a standard long-wave asymptotic expansion. In this context the system was first derived by Su and Gardner. It coincides with the one-dimensional flat-bottom reduction of the Green-Naghdi 1 system and, additionally, has recently found a number of fluid dynamics applications other than the present context of shallow-water gravity waves. We then use the Whitham modulation theory for a one-phase periodic travelling wave to obtain an asymptotic analytical description of an undular bore in the Su-Gardner system for a full range of "depth" ratios across the bore. The positions of the leading and trailing edges of the undular bore and the amplitude of the leading solitary wave of the bore are found as functions of this "depth ratio". The formation of a partial undular bore with a rapidly-varying finite-amplitude trailing wave front is predicted for "depth ratios" across the bore exceeding 1.43. The analytical results from the modulation theory are shown to be in excellent agreement with full numerical solutions for the development of an undular bore in the Su-Gardner system. I INTRODUCTIONIn shallow water, the transition between two different basic states, each characterized by a constant depth and horizontal velocity, is usually referred to as a bore. For sufficiently large transitions, the front of the bore is often turbulent, but as noted in the classical work of Benjamin and Lighthill [1], transitions of moderate amplitude are accompanied by wave trains without any wave breaking, and are hence called undular bores. Well-known examples are the bores on the River Severn in England and the River Dordogne in France.Undular bores also arise in other fluid flow contexts; for instance they can occur as internal undular bores in the density-stratified waters of the coastal ocean (see, for instance, [2], [3]), and as striking wave-forms with associated cloud formation in the atmospheric boundary layer (see, for instance, [4], [5]). They can also arise in many other physical contexts, and in plasma physics for instance, are usually called collisionless shocks.The classical theory of shallow-water undular bores was initiated by Benjamin and Lighthill in [1]. It is based on the analysis of stationary solutions of the Kortewegde Vries (KdV) equation modified by a small viscous term [6]. Subsequent approaches to the same problem have been usually based on the Whitham modulation theory (see 2 [7], [8]), appropriately modified by dissipation; this allows one to study analytically the development of an undular bore to a steady state (see [9], [10], [11]). Most recently, this approach was used in [12] to study the development of an u...

Transformation of a shoaling undular bore

“…We also stress that the availability of the full modulation solution (an analog of the Gurevich-Pitaevskii solution (3.4)) for the initial "flat-bottom" undular bore is not a pre-requisite in our analysis, and a similar study can be undertaken for the systems where the initial evolution of the undular bore is described by a non-integrable dispersive equation (see El (2005) for the relevant generalisation of the Gurevich-Pitaevskii problem). In particular, one can consider the propagation of a fully nonlinear shallow water undular bore over a slope in the framework of the appropriate variable-coefficient Su-Gardner (Green-Naghdi, or Serre) equations (see El et al (2006) for the corresponding flat-bottom undular bore theory).…”

“…Smyth & Holloway (1988), El et al (2006), Esler & Pearce (2011)) or due to the resonant interaction of a fluid flow with variable topography (see e.g. Grimshaw & Smyth (1986), Baines (1995), El et al (2009)).…”

Changing forms and sudden smooth transitions of tsunami waves

“…In this sense it can be regarded as a unidirectional version of the two-way Su-Gardner equations (see El et al 2006 for a description of these equations). Nevertheless, we suggest that its solutions may be relevant, at least in part, to the study of down-up tsunami waves.…”

“…This model is admittedly phenomenological in that while the form of the nonlinear terms can be rigorously justified, the dispersive regularisation with a linear term is admittedly ad hoc and probably cannot be obtained with a systematic asymptotic expansion. This model is formally fully nonlinear, with weak linear dispersion, and in that sense can be regarded as a unidirectional version of the two-way Su-Gardner equations (see El et al 2006 for a description of these equations). This extended KdV equation contains families of 2-soliton solutions and also breather solutions, which demonstrate striking interactions between depression waves and elevation waves.…”

Transient waves generated by a moving bottom obstacle: a new near-field solution

“…Recently, El, Grimshaw & Smyth (2006) and El et al (2009) developed a far-field modulation solution for undular bores in the framework of the Boussinesq equations by Su & Gardner (1969). They were not able to describe the actual evolution of the surface elevation, but provided expressions for the height and speed of the leading and trailing edges of the upstream and downstream undular bores.…”