Harmonic generation by nonlinear self-interaction of a single internal wave mode

2017 ◽  
Vol 828 ◽  
pp. 630-647 ◽  
Author(s):  
Scott Wunsch
Keyword(s):  
Steady State ◽  
Small Amplitude ◽  
Internal Tide ◽  
Single Mode ◽  
Steady State Solution ◽  
Wave Mode ◽  
Internal Tides ◽  
Weakly Nonlinear ◽  
Harmonic Response ◽  
Weakly Nonlinear Theory

Weakly nonlinear theory is used to explore the dynamics of a single-mode internal tide in variable stratification with rotation. Nonlinear self-interaction in variable stratification generates a perturbation which is forced with double the original frequency and wavenumber. The dynamics of the perturbation is analogous to a forced harmonic oscillator, with the steady-state solution corresponding to a bound harmonic matching the forcing frequency and wavenumber. When the forcing frequency is near a natural frequency of the system, even a small-amplitude (nearly linear) internal tide may induce a significant harmonic response. Idealized stratification profiles are utilized to explore the relevance of this effect for oceanic $M_{2}$ baroclinic internal tides, and the results indicate that a rapidly growing harmonic may occur in some environments near the Equator, but is unlikely at higher latitudes. The results are relevant to recent observations of $M_{4}$ (harmonic) internal tides in the South China Sea and elsewhere. More generally, nonlinear self-interaction may contribute to the transfer of energy to smaller scales and the dissipation of baroclinic internal tides, especially in equatorial waters.

Global energetics of fast magnetic reconnection

1988 ◽  
Vol 40 (3) ◽  
pp. 505-515 ◽  
Author(s):  
M. Jardine ◽  
E. R. Priest
Keyword(s):  
Steady State ◽  
Total Energy ◽  
Second Order ◽  
Weakly Nonlinear ◽  
First Order ◽  
Weakly Nonlinear Theory ◽  
Special Cases ◽  
Pile Up ◽  
Different Types ◽  
Converging Flow

We examine the global energetics of a recent weakly nonlinear theory of fast steady-state reconnection in an incompressible plasma (Jardine & Priest 1988). This is itself an extension to second order of the Priest & Forbes (1986) family of models, of which Petschek-like and Sonnerup-like solutions are special cases. While to first order we find that the energy conversion is insensitive to the type of solution (such as slow compression or flux pile-up), to second order not only does the total energy converted vary but so also does the ratio of the thermal to kinetic energies produced. For a slow compression with a strongly converging flow, the amount of energy converted is greatest and is dominated by the thermal contribution, while for a flux pile-up with a strongly diverging flow, the amount of energy converted is smallest and is dominated by the kinetic contribution. We also find that the total energy flowing out of the downstream region can be increased either by increasing the external magnetic Mach number Me or the external plasma beta βe Increasing Me also enhances the variations between different types of solutions.

Weakly nonlinear theory of fast steady-state magnetic reconnection

1988 ◽  
Vol 40 (1) ◽  
pp. 143-161 ◽  
Author(s):  
M. Jardine ◽  
E. R. Priest
Keyword(s):  
Magnetic Field ◽  
Steady State ◽  
Boundary Conditions ◽  
Free Parameter ◽  
Diffusion Region ◽  
Slow Mode ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Special Cases ◽  
Field Lines

A family of models for fast steady-state reconnection has recently been presented by Priest and Forbes, of which the Petschek-like and Sonnerup-like solutions are special cases. This essentially linear treatment involves expanding about a uniform flow and field in powers of the external Alfvén Mach number Me, and hence is valid for small values of that parameter. To lowest order, the discrete slow-mode compressions attached to the diffusion region are straight, while downstream of them the plasma flows at simply the external Alfvén speed vAe and the field lines are straight. Here we present an extension of these solutions to the next order, which not only reveals that the wave itself is curved (as are the downstream magnetic field lines), but also that the downstream solution is sensitive to changes in the upstream boundary conditions. In the downstream solution there is a free parameter, which may be specified as a downstream boundary condition. Thus the boundary conditions at both the inflow and the outflow boundaries are crucial in determining the nature of the reconnection.

scholarly journals Nonlinear Disintegration of the Internal Tide

2008 ◽  
Vol 38 (3) ◽  
pp. 686-701 ◽  
Author(s):  
Karl R. Helfrich ◽  
Roger H. J. Grimshaw
Keyword(s):  
Solitary Waves ◽  
Nonlinear Waves ◽  
Numerical Solutions ◽  
Internal Tide ◽  
Tidal Energy ◽  
Initial Amplitude ◽  
Governing Equations ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Hydrostatic Limit

Abstract The disintegration of a first-mode internal tide into shorter solitary-like waves is considered. Since 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 includes 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 long waves are shown to be robust to weak nonhydrostatic effects. Numerical solutions show that the disintegration of an initial sinusoidal linear internal tide is closely linked to the presence of these nonlinear 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 is a function of initial amplitude of the tide and the properties of the underlying nonlinear hydrostatic 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 hydrostatic inertia–gravity wave, most of the initial tidal energy goes into solitary waves. Recent observations in the South China Sea are briefly discussed.

scholarly journals Nonlinear hydroelastic waves on a linear shear current at finite depth

2019 ◽  
Vol 876 ◽  
pp. 55-86 ◽  
Author(s):  
T. Gao ◽  
Z. Wang ◽  
P. A. Milewski
Keyword(s):  
Solitary Waves ◽  
Small Amplitude ◽  
Modulational Instability ◽  
Finite Depth ◽  
Time Dependent ◽  
Mapping Technique ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Constant Vorticity

This work is concerned with waves propagating on water of finite depth with a constant-vorticity current under a deformable flexible sheet. The pressure exerted by the sheet is modelled by using the Cosserat thin shell theory. By means of multi-scale analysis, small amplitude nonlinear modulation equations in several regimes are considered, including the nonlinear Schrödinger equation (NLS) which is used to predict the existence of small-amplitude wavepacket solitary waves in the full Euler equations and to study the modulational instability of quasi-monochromatic wavetrains. Guided by these weakly nonlinear results, fully nonlinear steady and time-dependent computations are performed by employing a conformal mapping technique. Bifurcation mechanisms and typical profiles of solitary waves for different underlying shear currents are presented in detail. It is shown that even when small-amplitude solitary waves are not predicted by the weakly nonlinear theory, we can numerically find large-amplitude solitary waves in the fully nonlinear equations. Time-dependent simulations are carried out to confirm the modulational stability results and illustrate possible outcomes of the nonlinear evolution in unstable cases.

Richtmyer–Meshkov instability on a quasi-single-mode interface

2019 ◽  
Vol 872 ◽  
pp. 729-751 ◽  
Author(s):  
Yu Liang ◽  
Zhigang Zhai ◽  
Juchun Ding ◽  
Xisheng Luo
Keyword(s):  
Nonlinear Model ◽  
Small Amplitude ◽  
Linear Growth ◽  
Single Mode ◽  
High Order ◽  
Initial Amplitude ◽  
Previous Theory ◽  
Weakly Nonlinear ◽  
Nonlinear Stage ◽  
Amplitude Growth

Experiments on Richtmyer–Meshkov instability of quasi-single-mode interfaces are performed. Four quasi-single-mode air/$\text{SF}_{6}$ interfaces with different deviations from the single-mode one are generated by the soap film technique to evaluate the effects of high-order modes on amplitude growth in the linear and weakly nonlinear stages. For each case, two different initial amplitudes are considered to highlight the high-amplitude effect. For the single-mode and saw-tooth interfaces with high initial amplitude, a cavity is observed at the spike head, providing experimental evidence for the previous numerical results for the first time. For the quasi-single-mode interfaces, the fundamental mode is the dominant one such that it determines the amplitude linear growth, and subsequently the impulsive theory gives a reasonable prediction of the experiments by introducing a reduction factor. The discrepancy in linear growth rates between the experiment and the prediction is amplified as the quasi-single-mode interface deviates more severely from the single-mode one. In the weakly nonlinear stage, the nonlinear model valid for a single-mode interface with small amplitude loses efficacy, which indicates that the effects of high-order modes on amplitude growth must be considered. For the saw-tooth interface with small amplitude, the amplitudes of the first three harmonics are extracted from the experiment and compared with the previous theory. The comparison proves that each initial mode develops independently in the linear and weakly nonlinear stages. A nonlinear model proposed by Zhang & Guo (J. Fluid Mech., vol. 786, 2016, pp. 47–61) is then modified by considering the effects of high-order modes. The modified model is proved to be valid in the weakly nonlinear stage even for the cases with high initial amplitude. More high-order modes are needed to match the experiment for the interfaces with a more severe deviation from the single-mode one.

Harmonic Response of a Piping System Considering Pipe - Support Friction via HFT Method

2011 ◽  
Author(s):  
Jose´ Argu¨elles ◽  
Euro Casanova ◽  
Miguel Asuaje
Keyword(s):  
Steady State ◽  
Initial Conditions ◽  
Computational Cost ◽  
Steady State Solution ◽  
Integration Scheme ◽  
Piping System ◽  
System Construction ◽  
Harmonic Response ◽  
Linear Behavior ◽  
Piping Systems

The harmonic response of piping systems is in general estimated considering linear behavior, i.e. neglecting pipe-support dry friction, shocks between adjacent pipes or between pipe and supports, unidirectional supports, gaps, etc. Under this hypothesis, harmonic analysis is straight forward for a forcing frequency and not specially demanding in terms of computational cost. On the contrary, if any nonlinearity is taken into account, integration in time is required until the system achieves a steady-state solution, which demands dealing with initial conditions, intensive computation and long calculation times. In the particular case of support friction, vibration amplitudes calculated using the linear assumption may be well overestimated for some systems or conditions, thus misguiding designers and analysts, which traduces in excessive cost in system construction. In this work, the Hybrid Frequency-Time method (HFT) is used to calculate the steady state amplitudes of a piping system subjected to harmonic excitations and considering pipe-support friction. Comparisons between a full integration scheme and the proposed methodology are presented and discussed for a typical system. Results show that the HFT method is a valid practical option to estimate the harmonic response of a piping system when considering support friction.

A Harmonic Analysis for the Steady Analysis of Non-Linear Systems

2012 ◽  
Vol 433-440 ◽  
pp. 5536-5541
Author(s):  
Shan Chai ◽  
Can Chang Liu ◽  
Hong Yan Li
Keyword(s):  
Steady State ◽  
Linear Systems ◽  
Steady State Solution ◽  
Response Analysis ◽  
Time Interval ◽  
Harmonic Response ◽  
Non Linear ◽  
Non Linear Systems ◽  
Harmonic Response Analysis ◽  
Linear Differential

A numerical analysis is used to investigate the response of non-linear systems under aperiodic excitations based on the harmonic response analysis method. An idea of fine discretization is proposed to turn the aperiodic excitations into the superposition of a series of periodic excitations in a tiny time interval. The method of perturbation is employed to transform the non-linear governing equation into a series of linear differential equations. Harmonic response analysis can be applied in the solution of aperiodic steady response. The algebraic algorithm of direct steady-state analysis can improve computational efficiency. The defect that the steady-state solution can be gotten out until the free vibration attenuates is avoided. The examples show that the numerical results match well with the analytic data.

scholarly journals Generation of weakly nonlinear nonhydrostatic internal tides over large topography: a multi-modal approach

2008 ◽  
Vol 15 (2) ◽  
pp. 233-244 ◽  
Author(s):  
R. Maugé ◽  
T. Gerkema
Keyword(s):  
Evolution Equations ◽  
Numerical Models ◽  
Internal Tide ◽  
Nonlinear Effects ◽  
Internal Tides ◽  
Internal Solitary Waves ◽  
Weakly Nonlinear ◽  
Seasonal Thermocline ◽  
Generation Problem ◽  
Local Generation

Abstract. A set of evolution equations is derived for the modal coefficients in a weakly nonlinear nonhydrostatic internal-tide generation problem. The equations allow for the presence of large-amplitude topography, e.g. a continental slope, which is formally assumed to have a length scale much larger than that of the internal tide. However, comparison with results from more sophisticated numerical models show that this restriction can in practice be relaxed. It is shown that a topographically induced coupling between modes occurs that is distinct from nonlinear coupling. Nonlinear effects include the generation of higher harmonics by reflection from boundaries, i.e. steeper tidal beams at frequencies that are multiples of the basic tidal frequency. With a seasonal thermocline included, the model is capable of reproducing the phenomenon of local generation of internal solitary waves by a tidal beam impinging on the seasonal thermocline.

scholarly journals Can weakly nonlinear theory explain Faraday wave patterns near onset?

2015 ◽  
Vol 777 ◽  
pp. 604-632 ◽  
Author(s):  
A. C. Skeldon ◽  
A. M. Rucklidge
Keyword(s):  
Nonlinear Theory ◽  
Stokes Equations ◽  
Single Mode ◽  
Theoretical Work ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Temporal Modes ◽  
Pattern Forming

The Faraday problem is an important pattern-forming system that provides some middle ground between systems where the initial instability involves just a single mode, and in which complexity then results from mode interactions or secondary bifurcations, and cases where a system is highly turbulent and many spatial and temporal modes are excited. It has been a rich source of novel patterns and of theoretical work aimed at understanding how and why such patterns occur. Yet it is particularly challenging to tie theory to experiment: the experiments are difficult to perform; the parameter regime of interest (large box, moderate viscosity) along with the technical difficulties of solving the free-boundary Navier–Stokes equations make numerical solution of the problem hard; and the fact that the instabilities result in an entire circle of unstable wavevectors presents considerable theoretical difficulties. In principle, weakly nonlinear theory should be able to predict which patterns are stable near pattern onset. In this paper we present the first quantitative comparison between weakly nonlinear theory of the full Navier–Stokes equations and (previously published) experimental results for the Faraday problem with multiple-frequency forcing. We confirm that three-wave interactions sit at the heart of why complex patterns are stabilised, but also highlight some discrepancies between theory and experiment. These suggest the need for further experimental and theoretical work to fully investigate the issues of pattern bistability and the role of bicritical/tricritical points in determining bifurcation structure.

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