Exponential traveling-wave on a viscous fluid flowing down an inclined plane

The evolution of a single long wave of finite amplitude at the interface of two immiscible fluids of different viscosities and densities, between two horizontal plates is solved, using a boundary layer flow approximation for the equation of motion in each fluid layer. It is found that when the nonlinear inertial effects are taken into account in a moderate manner, at least in the frame of the boundary layer approximation, the initial unperturbed flow with smooth interface is stable to a single wave perturbation at the interface, even in the presence of adverse density and viscosity stratifications. However, when the nonlinear effects are increased in a specific way, and the magnitudes of the parameters involved are kept within the order of magnitude established for the present theory, an unstable flow configuration can be obtained.

Two models of coastal currents are described that allow fully nonlinear wavelike solutions for the limit of long waves. The first model is an adaptation of a model used by Yi and Warn for finite-amplitude βplane Rossby waves in a channel. It utilizes a particular choice of continental shelf topography to obtain a nonlinear evolution equation for long waves of finite amplitude. The second model describes the waves that form at the vorticity interface between two regions of constant potential vorticity. Again a nonlinear evolution equation is obtained for long waves of finite amplitude. For both model equations, numerical results are presented and compared with the corresponding results for the BDA equation, which is the weakly nonlinear limit for both models.

Propagation of long waves of finite amplitude at the interface of two viscous fluids

To facilitate the theoretical prediction of the evolution of a short gravity wave on a long wave of finite amplitude, we consider a model where the long wave is represented by Gerstner's exact but rotational solution in Lagrangian coordinates. Analytical formulae for the modulation of an infinitesimal irrotational short wave are shown to be qualitatively accurate in comparison with the numerical results by Longuet-Higgins (1987) and with the analytical results by Henyey et al. (1988) for irrotational long waves. Discrepancies are generally of second order in the long-wave steepness, consistent with the vorticity in Gerstner's solution. Weakly nonlinear short waves are shown to be parametrically excited by the long wave over a long time. In particular, multiple bands of modulational instability appear in the parameter space. Numerical calculations of the nonlinear evolution equation show the onset of chaos for sufficiently large parameter $\alpha = \epsilon (k\overline{A})^2/2\Omega/\sigma $, where $\epsilon k\overline{A} $ is the short-wave steepness and (εΩ, σ) the frequency of the (long, short) wave. Furthermore, if the short-wave amplitude A is approximated by a two-mode truncated Fourier series, the evolution equation reduces to a non-autonomous Hamiltonian system. The numerical solutions confirm that the onset of chaos is an inherent feature of the parametrically excited nonlinear system.

The propagation of long waves of finite amplitude at the interface of two viscous fluids in the presence of interfacial tension is examined. The effect of capillarity on the shape of the waves at the interface of two superposed fluids is investigated for a wide range of density differences, viscosity ratios and imposed pressure gradients. It is found that in planar geometry surface tension stabilizes the interfacial disturbances. Attention is given to the case in which the upper fluid is more dense and comprises a thin film above the lower fluid. With the heavier fluid on the top the flow pattern is always unstable when surface tension effects are neglected. In this case the interfacial waves do not grow forever and reach a finite amplitude only when the interfacial tension is greater than a critical value.

Abstract. Nonlinear transformation and runup of long waves of finite amplitude in a basin of variable depth is analyzed in the framework of 1-D nonlinear shallow-water theory. The basin depth is slowly varied far offshore and joins a plane beach near the shore. A small-amplitude linear sinusoidal incident wave is assumed. The wave dynamics far offshore can be described with the use of asymptotic methods based on two parameters: bottom slope and wave amplitude. An analytical solution allows the calculation of increasing wave height, steepness and spectral amplitudes during wave propagation from the initial wave characteristics and bottom profile. Three special types of bottom profile (beach of constant slope, and convex and concave beach profiles) are considered in detail within this approach. The wave runup on a plane beach is described in the framework of the Carrier-Greenspan approach with initial data, which come from wave deformation in a basin of slowly varying depth. The dependence of the maximum runup height and the condition of a wave breaking are analyzed in relation to wave parameters in deep water.

Analysis is made of the effects of a long wave of finite amplitude on the oscillation of a particle in a rotating, viscous fluid. It is shown that the zeroth-order solution of the Lagrangian equations of motion consists of the pressure wave and inertial frequencies, whereas the first-order solution gives three frequencies, namely the sum and difference of the pressure-wave and inertial frequencies, and a frequency of twice the pressure-wave frequency. The effects of the frictional force are to damp out the inertial oscillation but leave the other physically significant oscillations intact. Furthermore, it eliminates the solution which contributes to the amplification of the amplitude of particle trajectories in an inviscid fluid. A comparison of the frequencies predicted by the theory with those obtained from the flights of constant-pressure balloons is made.

The effects of a long wave of finite amplitude on the oscillation of a particle in a rotating fluid are examined by solving the Lagrangian equations of motion. It is shown that the motion of the particle is nonoscillatory or oscillatory, depending respectively on whether the forcing frequency kUr is equal to the multiples of the natural frequency or not. In the former case, the zeroth-order solution consists of the natural and forcing frequencies, whereas the first-order solution gives three new frequencies. In the latter case, the zeroth-order solution consists of the natural frequency with linearly growing amplitude, whereas the first-order solution gives a high frequency oscillation of twice the natural frequency with linearly growing amplitude. A comparison of the frequencies predicted by the zeroth-order and first-order solutions and those calculated from the trajectories of 300-mb constant-pressure balloons is made.

Derived herein is a set of partial differential equations governing the propagation of an arbitrary, long-wave disturbance of small, but finite amplitude. The equations reduce to that of Boussinesq (1872) when the assumption is made that the disturbance is propagating in one direction only. The equations are hyperbolic with characteristic curves of constant slope. The initial-value problem can be solved very readily by numerical integration along characteristics. A few examples are included.

Propagation of long waves of finite amplitude in a liquid with polydispersed gas bubbles

Diagnostics of the medium structure by long wave of finite amplitude

To study rapid and laminar flow of a mud layer down a gentle slope, a shallow layer of Bingham fluid is considered. Attention is focused on long waves of finite amplitude. Convective inertia is kept in the approximate equations which are simplified by Kármán’s momentum integral approach with an assumed velocity profile. A linearized instability analysis is first carried out for periodic disturbances. The nonlinear development of unstable disturbances is computed numerically to study the formation of periodic shocks, or roll waves. Nonlinear motion caused by external influx to an otherwise steadily flowing mud layer is also investigated.

The propagation of long waves of finite amplitude in water with depth to wavelength ratios less than about one-tenth and greater than about one-fiftieth can be described by cnoidal wave theory. To date little use has been made of the theory because of the difficulties involved in practical application. This paper presents the theory necessary for predicting the transforming characteristics of long waves based on cnoidal theory. Basically the method involves calculating the power transmission for a wave train m shallow water from cnoidal theory and equating this to the deep water power transmission assuming no reflections or loss of energy as the waves move into shoaling water. The equations for wave power have been programmed for the range of cnoidal waves, and the results are plotted in non-dimensional form.

Long waves of finite amplitude in polydispersed gas suspensions

This study is concerned with long internal gravity waves in a stratified fluid contained between rigid horizontal boundaries. For a general stratification, long waves of finite amplitude will tend to distort, and no permanent wave shape will result. In certain important cases, however, steady waveforms are found to be possible. The properties of such waves are investigated, and their relationship to the solutions provided by the weakly nonlinear theory is studied.