Models for instability in inviscid fluid flows, due to a resonance between two waves

In inviscid fluid flows instability arises generically due to a resonance between two wave modes. Here, it is shown that the structure of the weakly nonlinear regime depends crucially on whether the modal structure coincides, or remains distinct, at the resonance point where the wave phase speeds coincide. Then in the weakly nonlinear, long-wave limit the generic model consists either of a Boussinesq equation, or of two coupled Korteweg-de Vries equations, respectively. For short waves, the generic model is correspondingly either a nonlinear Klein-Gordon equation for the wave envelope, or a pair of coupled first-order envelope equations

Internal solitary waves in a variable medium

In both the ocean and the atmosphere, the interaction of a density stratified flow with topography can generate large-amplitude, horizontally propagating internal solitary waves. Often these waves are observed in regions where the waveguide properties vary in the direction of propagation. In this article we consider nonlinear evolution equations of the Kortewegde Vries type, with variable coefficients, and use these models to review the properties of slowly-varying periodic and solitary waves

Solitary wave solution for a non-integrable, variable coefficient nonlinear Schrodinger equation

A non-integrable, variable coefficient nonlinear Schrodinger equation which governs the nonlinear pulse propagation in an inhomogeneous medium is considered. The same equation is also applicable to optical pulse propagation in averaged, dispersion-managed optical fiber systems, or fiber systems with phase modulation and pulse compression. Multi-scale asymptotic techniques are employed to establish the leading order approximation of a solitary wave. A direct numerical simulation shows excellent agreement with the asymptotic solution. The interactions of two pulses are also studied

Nonlinear effects in wave scattering and generation

When a fluid flow interacts with a topographic feature, and the fluid can support wave propagation, then there is the potential for waves to be generated upstream and/or downstream. In many cases when the topographic feature has a small amplitude the situation can be successfully described using a linearised theory, and any nonlinear effects are determined as a small perturbation on the linear theory. However, when the flow is critical, that is, the system supports a long wave whose group velocity is zero in the reference frame of the topographic feature, then typically the linear theory fails and it is necessary to develop an intrinsically nonlinear theory. It is now known that in many cases such a transcritical, weakly nonlinear and weakly dispersive theory leads to a forced Korteweg- de Vries (fKdV) equation. In this article we shall sketch the contexts where the fKdV equation is applicable, and describe some of the most relevant solutions. There are two main classes of solutions. In the first, the initial condition for the fKdV equation is the zero state, so that the waves are generated directly by the flow interaction with the topography. In this case the solutions are characterised by the generation of upstream solitary waves and an oscillatory downstream wavetrain, with the detailed structure being determined by the detuning parameter and the polarity of the topographic forcing term. In the second class a solitary wave is incident on the topography, and depending on the system parameters may be repelled with a significant amplitude change, trapped with a change in amplitude, or allowed to pass by the topography with only a small change in amplitude

Generation of undular bores in the shelves of slowly-varying solitary waves

We study the long-time evolution of the trailing shelves that form behind solitary waves moving through an inhomogeneous media, within the framework of the variable-coeffecient Korteweg-de Vries equation. We show that the nonlinear evolution of the shelf leads typically to the generation of an undular bore and an expansion fan, which form apart but start to overlap and nonlinearly interact after a certain time interval. The interaction zone expands with time and asymptotically as time goes to infinity occupies the whole perturbed region. Its oscillatory structure strongly depends on the sign of the inhomogeneity gradient of the variable background medium. We describe the nonlinear evolution of the shelves in terms of exact solutions to the KdV-Whitham equations with natural boundary conditions for the Riemann invariants. These analytic solutions, in particular, describe the generation of small 'secondary' solitary waves in the trailing shelves, a process observed earlier in various numerical simulations

Critical control in transcritical shallow-water flow over two obstacles

The nonlinear shallow-water equations are often used to model flow over topography. In this paper we use these equations both analytically and numerically to study flow over two widely separated localised obstacles, and compare the outcome with the corresponding flow over a single localised obstacle. Initially we assume uniform flow with constant water depth, which is then perturbed by the obstacles. The upstream flow can be characterised as subcritical, supercritical and transcritical, respectively. We review the well-known theory for flow over a single localised obstacle, where in the transcritical regime the flow is characterised by a local hydraulic flow over the obstacle, contained between an elevation shock propagating upstream and a depression shock propagating downstream. Classical shock closure conditions are used to determine these shocks. Then we show that the same approach can be used to describe the flow over two widely spaced localised obstacles. The flow development can be characterised by two stages. The first stage is the generation of upstream elevation shock and downstream depression shock from each obstacle alone, isolated from the other obstacle. The second stage is the interaction of two shocks between the two obstacles, followed by an adjustment to a hydraulic flow over both obstacles, with criticality being controlled by the higher of the two obstacles, and by the second obstacle when they have equal heights. This hydraulic flow is terminated by an elevation shock propagating upstream of the first obstacle and a depression shock propagating downstream of the second obstacle. A weakly nonlinear model for sufficiently small obstacles is developed to describe this second stage. The theoretical results are compared with fully nonlinear simulations obtained using a well-balanced finite-volume method. The analytical results agree quite well with the nonlinear simulations for sufficiently small obstacles

Nonlinear disintegration of the internal tide

The disintegration of a first-mode internal tide into shorter solitary-like waves is considered. Because observations frequently show both tides and waves with amplitudes beyond the restrictions of weakly nonlinear theory, the evolution is studied using a fully-nonlinear, weakly nonhydrostatic two-layer theory that included the effects of rotation. In the hydrostatic limit, the governing equations have periodic, nonlinear inertia-gravity solutions that are explored as models of the nonlinear internal tide. These are shown to be robust to weak nonhydrostatic effects. Numerical solutions show that the disintegration of an initially sinusoidal, linear internal tide is closely linked to the presence of these periodic waves. The initial tide steepens due to nonlinearity and sheds energy into short solitary waves. The disintegration is halted as the longwave part of the solution settles onto a state close to one of the nonlinear, hydrostatic solutions, with the short solitary waves superimposed. The degree of disintegration depends upon the initial amplitude of the tide and the properties of the underlying nonlinear solutions, which, depending on stratification and tidal frequency, exist only for a finite range of amplitudes (or energies). There is a lower threshold below which no short solitary waves are produced. However, for initial amplitudes above another threshold, given approximately by the energy of the limiting nonlinear inertia-gravity wave, most of the initial tidal energy goes into solitary waves. Recent observations of large amplitude solitary waves in the South China Sea are discussed in the context of these model results

Nonlinear geostrophic adjustment in the presence of a boundary

Nonlinear geostrophic adjustment in the presence of a boundar

Internal waves in a three-layer bubbly waveguide

In many oceanic situations, clouds, or layers of bubbles are present which may influence the larger-scale dynamics. A two-dimensional model of a diluted locally monodisperse mixture of an incompressible fluid with gas bubbles is used to model and study such situations. The problem is simplified by ignoring the horizontal variability in the structure of the bubble layer and by focusing instead on the vertical variability. We consider waves propagating horizontally in the oceanic waveguide, assuming that bubbles are confined to a thin upper layer of the otherwise homogeneous upper ocean, and using a three layer model to represent this situation. The presence of the depth-dependent distribution of bubbles introduces an effective stratification and considerably changes the value of the buoyancy frequency Nl in the absence of bubbles, replacing it with an effective value N, where N2 = N2 l +g g0 [(ln n0)z + 3 g0(lnR0)z] /(1− g0) (here g0 is the void fraction, n0 is the number density, and R0 is the radius of bubbles in the basic state). This leads to the possibility of existence of the internal waves in the otherwise homogeneous upper mixed layer. Also the presence of the bubbly layer causes significant changes to the dispersion relation for the usual internal waves

An integrable model for undular bores on shallow water

On the basis of the integrable Kaup-Boussinesq version of the shallow water equations, an analytical theory of undular bores is constructed. The problem of the decay of an initial discontinuity is considered