Makoto Umeki

Nonlinear Dynamics and Chaos in Parametrically Excited Surface Waves

Surface waves in a closed container subject to vertical oscillation are studied. Nonlinear dynamical equations of two nearly degenerate subharmonic modes responding to the external forcing are derived, using the averaged Lagrangian method for slowly varying amplitudes. Stability and bifurcation diagrams are shown for the system with linear damping. Period-doubling bifurcation and chaotic solutions with one positive Lyapunov characteristic exponent are obtained numerically. It is shown that some of the period-doubling bifurcations are related to the symmetry of the dynamical system.

Scattering of dislocated wave fronts by vertical vorticity and the Aharonov-Bohm effect. I. Shallow water.

When a surface wave interacts with a vertical vortex in shallow water the latter induces a dislocation in the incident wave fronts that is analogous to what happens in the Aharonov-Bohm effect for the scattering of electrons by a confined magnetic field. In addition to this global similarity between these two physical systems there is scattering. This paper reports a detailed calculation of this scattering, which is quantitatively different from the electronic case in that a surface wave penetrates the inside of a vortex while electrons do not penetrate a solenoid. This difference, together with an additional difference in the equations that govern both physical systems, lead to a quite different scattering in the case of surface waves, whose main characteristic is a strong asymmetry in the scattering cross section. The assumptions and approximations under which these effects happen are carefully considered, and their applicability to the case of the scattering of acoustic waves by vorticity is noted.

Faraday resonance in rectangular geometry

The motion of subharmonic resonant modes of surface waves in a rectangular container subjected to vertical periodic oscillation is studied based on the weakly nonlinear model equations derived by both the average Lagrangian and the two-timescale method. Explicit estimates of the nonlinearity of some specific modes are given. The bifurcations of stationary states including a Hopf bifurcation are examined. Numerical calculations of the dissipative dynamical equations show periodic and chaotic attractors. Theoretical parameter-space diagrams and numerical results are compared in detail with Simonelli & Gollubs (1989) surface-wave modecompetition experiments. It is shown that the average Hamiltonian system for the present 2:1:1 external-internal resonance with suitable coefficients has homoclinic chaos, which was mathematically proven by Holmes (1986) for the specific case of 2:1:2 external-internal resonance.

Nonlinear dynamics of two-mode interactions in parametric excitation of surface waves

Parametric excitation of surface waves in a container under vertical forcing is investigated in detail, by an averaged Lagrangian method due to John Miles, and a system of evolution equations of third-order nonlinearity is presented for the case that the forcing frequency is chosen to be near twice the frequencies of two nearly degenerate free modes. The system of first-order differential equations in four variables which are derived from an averaged Hamiltonian is considered in a unified fashion, and the analytical results are compared with three experimental observations. It is found with the help of numerical integration that this dynamical system yields not only excitation of a single-mode state, but also interaction between two modes in which each mode oscillates either periodically or chaotically. These results are in good agreement with the observations, except for one case in which nonlinearity is considered to be too strong. As a fourth case, homoclinic chaos in the Hamiltonian system of two-degrees of freedom without damping is studied numerically. It is suggested that the chaotic mode competition observed in the experiments is different from the homoclinic chaos.

Parametric Dissipative Nonlinear Schrödinger Equation

Nonlinear Schrodinger equation with complex-conjugate-type (parametric) forcing and linear damping is studied both analytically and numerically. Bifurcations of stationary and time-dependent solutions are investigated. A periodic motion arising via a Hopf bifurcation of the cnoidal solution explains well the soliton-oscillation phenomena of water wave in a vertically forced long container discovered by Wu et al . The analogy is pointed out in an explicit way between the mode competition of surface waves in a container and the soliton-oscillation. The existence of spatiotemporal chaotic states is illustrated by numerical simulations.

Spirals and dislocations in wave-vortex systems

The two-dimensional interaction between sound waves and a vortex is studied. When the mach number defined by the ratio of the typical velocity due to the vortex to the speed of sound is small and the ratio of the size of the vortex to the wavelength is large, a differential equation for the sound waves is derived. Some classes of spiral solutions of the equation are obtained by relating their phase function to the background flow due to the vortex. Using the analogies between the Aharonov-Bohm effect in quantum mechanics, shallow water waves, and sound waves, the scattering problem of an incident dislocated wave is discussed.

Clustering Analysis of Periodic Point Vortices with the L Function

The motion of point vortices with periodic boundary conditions was studied by using Weierstrass zeta functions. The scattering and recoupling of a vortex pair by a third vortex becomes remarkable when the vortex density is large. The clustering of vortices with various initial conditions is quantitated by the L function used in the point process theory in spatial ecology. It is shown that clustering persists if the initial distribution is clustered like an infinite row or a checkered pattern.

Stream patterns of an isothermal atmosphere over an isolated mountain

Abstract The three-dimensional steady flow of an air stream over an isolated mountain in an isothermal atmosphere is studied based on a linear perturbation analysis. Displacements of fluid particles in a uniform flow are computed in detail. From the dispersion relation of internal gravity waves for the stratified air stream, one can construct lee wave patterns. Changes of relative humidity in the stream are represented simply by a term proportional to the vertical displacement. It is inferred that the “cap-cloud” and the “wing-like” cloud that are often formed over the top and in the lee of an isolated mountain, respectively, are related to the upward displacement of thin moist layers by the presence of the mountain.

Pattern Selection in Faraday Surface Waves

Pattern selection phenomena in parametrically excited surface waves are studied by the weakly-nonlinear system of three modes derived by the average Lagrangian method. The third-order coefficients of nonlinear interaction between two line patterns intersecting at an arbitrary angle are obtained in the gravity-capillary waves of arbitrary depth. Classification, stability analysis and bifurcation study of the fixed points of the dynamical equations with linear damping are performed in the cases of three symmetric line modes and general modes. Investigating the stability of various patterns to the disturbance of line mode with an arbitrary intersecting angle and the internal stability, it is shown that squares are the most preferred pattern in capillary waves, while lines are selected in gravity waves.

Bifurcations and Chaos in a Six-dimensional Turbulence Model of Gledzer

Gledzers cascade shell model of turbulence with six real variables is studied numerically using mathematica 5.1. The Poincare plot of the first mode v 1 is used to determine periodic, doubly periodic and chaotic solutions and the routes to chaos via both frequency locking and period doubling. The circle map on the torus is well approximated by the summation of several sinusoidal functions. The dependence of the rotation number on the viscosity parameter is in accordance with that of the sine circle map. The complicated bifurcation structure and the revival of a stable periodic solution at a smaller viscosity parameter in the present model indicates that the turbulent state may be very sensitive to the Reynolds number.