R. Grimshaw

Resonant flow of a stratified fluid over topography

The flow of a stratified fluid over topography is considered in the long-wavelength weakly nonlinear limit for the case when the flow is near resonance; that is, the basic flow speed is close to a linear long-wave phase speed for one of the long-wave modes. It is shown that the amplitude of this mode is governed by a forced Korteweg-de Vries equation. This equation is discussed both analytically and numerically for a variety of different cases, covering subcritical and supercritical flow, resonant or non-resonant, and for localized forcing that has either the same, or opposite, polarity to the solitary waves that would exist in the absence of forcing. In many cases a significant upstream disturbance is generated which consists of a train of solitary waves. The usefulness of internal hydraulic theory in interpreting the results is also demonstrated.

The solitary wave in water of variable depth

Equations are derived for two-dimensional long waves of small, but finite, amplitude in water of variable depth, analogous to those derived by Boussinesq for water of constant depth. When the depth is slowly varying compared to the length of the wave, an asymptotic solution of these equations is obtained which describes a slowly varying solitary wave; also differential equations for the slow variations of the parameters describing the solitary wave are derived, and solved in the case when the solitary wave evolves from a region of uniform depth. For small amplitudes it is found that the wave amplitude varies inversely as the depth.

Nonlinear instability at the interface between two viscous fluids

Co‐current flow of two viscous fluids in a channel is linearly unstable to long wavelength disturbances. The weakly nonlinear evolution of this instability is examined. It is shown that, because of surface tension and nonlinear effects, the interface can either return to its original undisturbed state or evolve to some finite amplitude steady state.

The solitary wave in water of variable depth. Part 2

This paper examines the deformation of a solitary wave due to a slow variation of the bottom topography. Differential equations which determine the slow variation of the parameters of a solitary wave are derived by a certain averaging process applied to the exact in viscid equations. The equations for the parameters are solved when the bottom topography varies only in one direction, and when the wave evolves from a region of uniform depth. The variation of amplitude with depth is determined and compared with some recent experimental results.

Nonlinear Ordinary Differential Equations

INTRODUCTION Preliminary Notions First-Order Systems Uniqueness and Existence Theorems Dependence on Parameters, and Continuation LINEAR EQUATIONS Uniqueness and Existence Theorem for a Linear System Homogeneous Linear Systems Inhomogeneous Linear Systems Second-Order Linear Equations Linear Equations with Constant Coefficients LINEAR EQUATIONS WITH PERIODIC COEFFICIENTS Floquet Theory Parametric Resonance Perturbation Methods for the Mathieu Equation The Mathieu Equation with Damping STABILITY Preliminary Definitions Stability for Linear Systems Principle of Linearized Stability Stability for Autonomous Systems Liapunov Functions PLANE AUTONOMOUS SYSTEMS Critical Points Linear Plane, Autonomous Systems Nonlinear Perturbations of Plane, Autonomous Systems PERIODIC SOLUTIONS OF PLANE AUTONOMOUS SYSTEMS Preliminary Results The Index of a Critical Point Van der Pol Equation Conservative Systems PERTURBATION METHODS FOR PERIODIC SOLUTIONS Poincare-Lindstedt Method Stability PERTURBATION METHODS FOR FORCED OSCILLATIONS Non-Resonant Case Resonant Case Resonant Oscillations for Duffings Equation Resonant Oscillations for Van der Pols Equation AVERAGING METHODS Averaging Methods for Autonomous Equations Averaging Methods for Forced Oscillations Adiabetic Invariance Multi-Scale Methods ELEMENTARY BIFURCATION THEORY Preliminary Notions One-Dimensional Bifurcations Hopf Bifurcation HAMILTONIAN SYSTEMS Hamiltonian and Lagrangian Dynamics Liouvilles Theorem Integral Invariants and Canonical Transformations Action-Angle Variables Action-Angle Variables: Perturbation Theory ANSWERS TO SELECTED PROBLEMS REFERENCES INDEX.

Long Nonlinear Surface and Internal Gravity Waves in a Rotating Ocean

Nonlinear dynamics of surface and internal waves in a stratified ocean under the influence of the Earths rotation is discussed. Attention is focussed upon guided waves long compared to the ocean depth. The effect of rotation on linear processes is reviewed in detail as well as the existing nonlinear models describing weakly and strongly nonlinear dynamics of long waves. The influence of rotation on small-scale waves and two-dimensional effects are also briefly considered. Some estimates of the influence of the Earths rotation on the parameters of real oceanic waves are presented and related to observational and numerical data.

Internal Solitary Waves

The basic theory of internal solitary waves is developed, with the main emphasis on environmental situations, such as the many occurrences of such waves in shallow coastal seas and in the atmospheric boundary layer. Commencing with the equations of motion for an inviscid, incompressible density-stratified fluid, we describe asymptotic reductions to model long-wave equations, such as the well-known Korteweg-de Vries equation. We then describe various solitary wave solutions, and propose a variable-coefficient extended Korteweg-de Vries equations as an appropriate evolution equation to describe internal solitary waves in environmental situations, when the effects of a variable background and dissipation need to be taken into account.

Slowly varying solitary waves. I. Korteweg-de Vries equation

The slowly varying solitary wave is constructed as an asymptotic solution of the variable coefficient Korteweg-de Vries equation. A multiple scale method is used to determine the amplitude and phase of the wave to the second order in the perturbation parameter. The structure ahead and behind the solitary wave is also determined, and the results are interpreted by using conservation laws. Outer expansions are introduced to remove non-uniformities in the expansion. Finally, when the coefficients satisfy a certain constraint, an exact solution is constructed.

A second‐order theory for solitary waves in shallow fluids

Solitary waves in density stratified fluids of shallow depth are described, to first order in wave amplitude, by the Korteweg–de Vries equation; the solution for a single solitary wave has the familiar ‘‘sech2’’ profile and a phase speed which varies linearly with the wave amplitude. This theory is here extended to second order in wave amplitude. The second‐order correction to the wave profile and the phase speed and the first‐order correction to the wavelength are all determined. Four special cases are discussed in detail. In certain special circumstances the first‐order theory may fail due to the vanishing of the nonlinear coefficient in the Korteweg–de Vries equation. When this occurs a different theory is required which leads to an equation with both quadratic and cubic nonlinearities.

Simulation of the Transformation of Internal Solitary Waves on Oceanic Shelves

Abstract Internal solitary waves transform as they propagate shoreward over the continental shelf into the coastal zone, from a combination of the horizontal variability of the oceanic hydrology (density and current stratification) and the variable depth. If this background environment varies sufficiently slowly in comparison with an individual solitary wave, then that wave possesses a soliton-like form with varying amplitude and phase. This stage is studied in detail in the framework of the variable-coefficient extended Korteweg–de Vries equation where the variation of the solitary wave parameters can be described analytically through an asymptotic description as a slowly varying solitary wave. Direct numerical simulation of the variable-coefficient extended Korteweg–de Vries equation is performed for several oceanic shelves (North West shelf of Australia, Malin shelf edge, and Arctic shelf) to demonstrate the applicability of the asymptotic theory. It is shown that the solitary wave may maintain its sol...