Noel F. Smyth

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.

Unsteady undular bores in fully nonlinear shallow-water theory

We consider unsteady undular bores for a pair of coupled equations of Boussinesq-type 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 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 wavefront 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.

Hydraulic Jump and Undular Bore Formation on a Shelf Break

Abstract The evolution of the semidiurnal internal tide as it propagates across the Australian North West Shelf is discussed analytically. As the tide is of long wavelength and small amplitude, this evolution is described by a perturbed extended Korteweg-de Vries equation. It is found that the flow is dominated by nonlinearity, and hence is hydraulic, except in the neighborhood of any shocks predicted by hydraulic theory. Hydraulic theory predicts the formation of two shocks in each period of the tide. The inclusion of dispersion in boundary layers around the shocks results in one of these shocks breaking up into an undular bore. Good agreement is found with observations by Holloway of the semidiurnal internal tide on the Australian North West Shelf.

Modulation Theory Solution for Resonant Flow Over Topography

The near-resonant flow of a stratified fluid over topography is considered in the weakly nonlinear, long-wave limit, this flow being governed by a forced Korteweg-de Vries equation. It is proved from the modulation equations for the Korteweg-de Vries equation, which apply away from the obstacle, that no steady state can form upstream of the obstacle. This has been noted from previous experimental and numerical studies. The solution upstream and downstream of the topography is constructed as a simple wave solution of the modulation equations. Based on similarities between the method by which this solution is found and the quarter plane problem for the Korteweg-de Vries equation, the solution to the quarter plane problem is found for the special case in which a positive constant is specified at x = 0.

The extended Korteweg-de Vries equation and the resonant flow of a fluid over topography

The extended Korteweg-de Vries equation which includes nonlinear and dispersive terms cubic in the wave amplitude is derived from the water-wave equations and the Lagrangian for the water-wave equations. For the special case in which only the higher-order nonlinear term is retained, the extended Korteweg-de Vries equation is transformed into the Korteweg-de Vries equation. Modulation equations for this equation are then derived from the modulation equations for the Korteweg-de Vries equation and the undular bore solution of the extended Korteweg-de Vries equation is found as a simple wave solution of these modulation equations. The modulation equations are also used to extend the solution for the resonant flow of a fluid over topography. This resonant flow occurs when, in the weakly nonlinear, long-wave limit, the basic flow speed is close to a linear long-wave phase speed for one of the long-wave modes. In addition to the effect of higher-order terms, the effect of boundary-layer viscosity is also considered. These solutions (with and without viscosity) are compared with recent experimental and numerical results.

ANALYSIS OF THE DYNAMICS AND TOUCHDOWN IN A MODEL OF ELECTROSTATIC MEMS

We study a reaction‐diffusion equation in a bounded domain in the plane, which is a mathematical model of an idealized electrostatically actuated microelectromechanical system (MEMS). A relevant feature in these systems is the “pull‐in” or “jump‐to contact” instability, which arises when applied voltages are increased beyond a critical value. In this situation, there is no longer a steady state configuration of the device where mechanical members of the device remain separate. It may present a limitation on the stable operation regime, as with a micropump, or it may be used to create contact, as with a microvalve. The applied voltage appears in the equation as a parameter. We prove that this parameter controls the dynamics in the sense that before a critical value the solution evolves to a steady state configuration, while for larger values of the parameter, the “pull‐in” instability or “touchdown” appears. We estimate the touchdown time. In one dimension, we prove that the touchdown is self‐similar and d...

Evolution of lump solutions for the KP equation

Abstract The two (space)-dimensional generalisation of the Korteweg-de Vries (KdV) equation is the Kadomtsev-Petviashvili (KP) equation. This equation possesses two solitary wave type solutions. One is independent of the direction orthogonal to the direction of propagation and is the soliton solution of the KdV equation extended to two space dimensions. The other is a true two-dimensional solitary wave solution which decays to zero in all space directions. It is this second solitary wave solution which is considered in the present work. It is known that the KP equation admits an inverse scattering solution. However this solution only applies for initial conditions which decay at infinity faster than the reciprocal distance from the origin. To study the evolution of a lump-like initial condition, a group velocity argument is used to determine the direction of propagation of the linear dispersive radiation generated as the lump evolves. Using this information combined with conservation equations and a suitable trial function, approximate ODEs governing the evolution of the isolated pulse are derived. These pulse solutions have a similar form to the pulse solitary wave solution of the KP equation, but with varying parameters. It is found that the pulse solitary wave solutions of the KP equation are asymptotically stable, and that depending on the initial conditions, the pulse either decays to a pulse of lower amplitude (shedding mass) or narrows down (shedding mass) to a pulse of higher amplitude. The solutions of the approximate ODEs for the pulse evolution are compared with full numerical solutions of the KP equation and good agreement is found.

Numerical shock propagation using geometrical shock dynamics

A simple numerical scheme for the calculation of the motion of shock waves in gases based on Whithams theory of geometrical shock dynamics is developed. This scheme is used to study the propagation of shock waves along walls and in channels and the self-focusing of initially curved shockfronts. The numerical results are compared with exact and numerical solutions of the geometrical-shock-dynamics equations and with recent experimental investigations.

Modelling the morning glory of the Gulf of Carpentaria

The morning glory is a meteorological phenomenon which occurs in northern Australia and takes the form of a series of roll clouds. The morning glory is generated by the interaction of nocturnal seabreezes over Cape York Peninsula and propagates in a south-westerly direction over the Gulf of Carpentaria. In the present work, it is shown that the morning glory can be modelled by the resonant flow of a two-layer fluid over topography, the topography being the mountains of Cape York Peninsula. In the limit of a deep upper layer, the equations of motion reduce to a forced Benjamin{Ono equation. In this context, resonant means that the underlying flow velocity of the seabreezes is near a linear long-wave velocity for one of the long-wave modes. The morning glory is then modelled by the undular bore (simple wave) solution of the modulation equations for the Benjamin{Ono equation. This modulation solution is compared with full numerical solutions of the forced Benjamin{Ono equation and good agreement is found when the wave amplitudes are not too large. The reason for the dierence between the numerical and modulation solutions for large wave amplitude is also discussed. Finally, the predictions of the modulation solution are compared with observational data on the morning glory and good agreement is found for the pressure jump due to the lead wave of the morning glory, but not for the speed and half-width of this lead wave. The reasons for this are discussed.

Spatial soliton evolution in nematic liquid crystals in the nonlinear local regime

Spatial solitons in nematic liquid crystals are considered in the regime of local response of the crystal, such solitons having been called nematicons in previous experimental studies. In the limit of low light intensity and local material response, it is shown that the full governing equations reduce to a single, higher-order nonlinear Schrodinger equation. Modulation equations are derived for the evolution of a nematiconlike pulse; these equations also include the dispersive radiation shed as the pulse evolves. The modulation equations show that a nematicon is (weakly) stable. Solutions of the modulation equations are compared with numerical solutions of the full governing equations, and good agreement is found when the light intensity is small.