Triplet Of Modes Research Articles

Nonlinear dynamics of multiple Alfvén modes driven by trapped energetic ions in tokamaks: A triplet paradigm

Non-linear dynamics of multiple infernal Alfvén eigenmodes—a subset of global Alfvén eigenmodes in tokamak plasmas with extended low-shear central core [Marchenko et al., Phys. Plasmas 16, 092502 (2009)]—is studied. The analysis is carried out for a mode triplet with toroidal mode-numbers n=1, 2, 3. It was assumed that the n = 1 mode was linearly unstable due to precession resonance with trapped fast ions, whereas the other modes were linearly damped. The modes were coupled due to a non-linearity in a bounce-averaged drift kinetic equation for the distribution function of fast ions. Nonlinear equations for the mode amplitudes and phases are derived and solved numerically. It is found that the temporal evolution of the amplitudes and the phase (responsible for the frequency chirping) of the modes exhibit Hopf bifurcations to stable limit cycles. This can explain a synchronous cyclic destabilization of multiple modes in Alfvén avalanches (sudden growth of amplitudes of the mode cluster with different n and approximately equal frequency spacing) in NSTX and bursting modes in MAST—events, which resulted in enhanced loss of fast ions.

Resonant Mode Coupling in δ Scuti Stars

Delta Scuti (δ Sct) variables are intermediate-mass stars that lie at the intersection of the main sequence and the instability strip on the Hertzsprung–Russell diagram. Various lines of evidence indicate that nonlinear mode interactions shape their oscillation spectra, including the particularly compelling detection of resonantly interacting mode triplets in the δ Sct star KIC 8054146. Motivated by these observations, we use the theory of three-mode coupling to study the strength and prevalence of nonlinear mode interactions in 14 δ Sct models that span the instability strip. For each model, we calculate the frequency detunings and nonlinear coupling strengths of ∼104 unique combinations of mode triplets. We find that all the models contain at least ∼100 well-coupled triplets whose detunings and coupling strengths are consistent with the triplets identified in KIC 8054146. Our results suggest that resonant mode interactions can be significant in δ Sct stars and may explain why many exhibit rapid changes in amplitude and oscillation period.

Evaluation of Plastic Deformation Considering the Phase-Mismatching Phenomenon of Nonlinear Lamb Wave Mixing

Nonlinear guided elastic waves have attracted extensive attention owing to their high sensitivity to microstructural changes. However, based on the widely used second harmonics, third harmonics and static components, it is still difficult to locate the micro-defects. Perhaps the nonlinear mixing of guided waves can solve these problems since their modes, frequencies and propagation direction can be flexibly selected. Note that the phenomena of phase mismatching usually occur due to the lack of precise acoustic properties for the measured samples, and they may affect the energy transmission from the fundamental waves to second-order harmonics as well as reduce the sensitivity to micro-damage. Therefore, these phenomena are systematically investigated to more accurately assessing the microstructural changes. It is theoretically, numerically, and experimentally found that the cumulative effect of difference- or sum-frequency components will be broken by the phase mismatching, accompanied by the appearance of the beat effect. Meanwhile, their spatial periodicity is inversely proportional to the wavenumber difference between fundamental waves and difference- or sum-frequency components. The sensitivity to micro-damage is compared between two typical mode triplets that approximately and exactly meet the resonance conditions, and the better one is utilized for assessing the accumulated plastic deformations in the thin plates.

Combined harmonic generation of feature guided waves mixing in a welded joint

In our previous research, the second harmonic generation (SHG) of feature guided waves (FGWs) in a welded joint with material nonlinearity was analyzed theoretically and observed numerically. However, some drawbacks including strict generation criteria, inability of distinguishing nonlinearity, and being unable to localize local defects of SHG-based nonlinear ultrasonic technique, limit the practical testing and application. Wave mixing is a promising alternative to overcome these shortcomings. In this paper, we investigated another acoustic nonlinear response of FGWs, so-called combined harmonic generation (CHG) caused by FGWs mixing in a welded joint. Theoretical analysis of second-order combined harmonic waves induced by two FGWs mixing in a welded joint was firstly implemented based on semi-analytical finite element (SAFE) and perfectly matched layer (PML) methods. Based on the theoretical analysis, four FGW mode triplets, which satisfy the internal resonant conditions, were selected for the observation of CHG caused by FGW mixing in a welded joint from a numerical perspective. CHG induced by FGW mixing with the cumulative growth effect was successfully observed within the mixing zone, which is consistent with the theoretical prediction. This study provides a straightforward analysis not previously available of the process of CHG of FGWs propagating and mixing in a welded joint.

Assessment of the 2S1 Mode Triplet Based on IGETS Data After 2010 Chile Earthquake

Assessment of the 2S1 Mode Triplet Based on IGETS Data After 2010 Chile Earthquake

Analytical and numerical investigations of one-way mixing of Lamb waves in a thin plate

This paper derives the resonance conditions for one-way resonant mixing of nonlinear Lamb waves. Two mode triplets are identified that satisfy such resonance conditions. It is also found that, when the primary waves are pulses of finite duration, the signal envelope of the corresponding resonant mixed wave is either a diamond or an elongated hexagon. The dimensions of these shapes are obtained explicitly in terms of the pulse lengths and group velocities of the Lamb modes in the mode triplet. These analytical results are verified by numerical simulations using the finite element method. Finally, a nondestructive evaluation method based on one-way mixing of Lamb waves is proposed for the inspection of a large area of a plate for damage distribution via a single access point. Numerical simulations of this nondestructive evaluation technique are conducted. It is found that using shorter pulses gives better spatial resolution, thus better suited for locating and sizing the damage zone, while using longer pulses gives a higher signal to noise ratio, thus better suited for quantifying the degree of damage in the damage zone. Results of this paper clarify some of the confusion in the existing literature regarding the resonance conditions for one-way mixing of nonlinear Lamb waves. They also provide a better understanding of the physical characteristics of the resonant mixed wave. Such understanding enables the design of optimal measurement systems for one-way mixing of Lamb waves for the purpose of conducting large area nondestructive evaluation of plate-like structures from a single access point. This proposed one-way mixing nondestructive evaluation technique can be extended to pipes.

Tidal Asteroseismology: Possible Evidence of Nonlinear Mode Coupling in an Equilibrium State in Kepler Eclipsing Binary KIC 3230227

Abstract Previously, a series of tidally-excited oscillations were discovered in the eccentric eclipsing binary KIC 3230227. The pulsation amplitudes and phases suggest the observed oscillations are prograde quadruple modes. In this paper, we refine the analysis and extract more oscillation frequencies. We also study the temporal variations of amplitudes and phases and show that almost all modes have stable phases and amplitudes. We then focus on the non-orbital-harmonic oscillations. We consider two formation mechanisms: (1) nonlinear response of the surface convective layer, and (2) nonlinear three/multi-mode coupling. Although the former can explain some of the observed features, we find the latter mechanism is more probable. Assuming that these are coupled modes, the constant amplitude/phase over four years can be explained by either an equilibrium state in the mode coupling or modes undergoing limit cycles with very long periods. The observed frequency detuning and the calculated damping rates of the daughter modes favor the equilibrium-state interpretation. This is verified by integrating the amplitude equations of three-mode coupling. We find that the steady-state relation derived in Weinberg et al., which relates the observed frequency detuning, phase detuning, and mode damping rates, is approximately satisfied for one mode triplet. We also try to identify the azimuthal number of the modes based on the observed mode amplitude ratios and the selection rules in nonlinear three-mode coupling. We discuss further implications of these observations on nonlinear tidal asteroseismology.

Second order harmonic guided wave mutual interactions in plate: Vector analysis, numerical simulation, and experimental results

The extraordinary sensitivity of nonlinear ultrasonic waves to the early stages of material degradation makes them excellent candidates for nondestructive material characterization. However, distinguishing weak material nonlinearity from instrumentation nonlinearity remains problematic for second harmonic generation approaches. A solution to this problem is to mix waves having different frequencies and to let their mutual interaction generate sum and difference harmonics at frequencies far from those of the instrumentation. Mixing of bulk waves and surface waves has been researched for some time, but mixing of guided waves has not yet been investigated in depth. A unique aspect of guided waves is their dispersive nature, which means we need to assure that a wave can propagate at the sum or difference frequency. A wave vector analysis is conducted that enables selection of primary waves traveling in any direction that generate phase matched secondary waves. We have tabulated many sets of primary waves and phase matched sum and difference harmonics. An example wave mode triplet of two counter-propagating collinear shear horizontal waves that interact to generate a symmetric Lamb wave at the sum frequency is simulated using finite element analysis and then laboratory experiments are conducted. The finite element simulation eliminates issues associated with instrumentation nonlinearities and signal-to-noise ratio. A straightforward subtraction method is used in the experiments to identify the material nonlinearity induced mutual interaction and show that the generated Lamb wave propagates on its own and is large enough to measure. Since the Lamb wave has different polarity than the shear horizontal waves the material nonlinearity is clearly identifiable. Thus, the mutual interactions of shear horizontal waves in plates could enable volumetric characterization of material in remote regions from transducers mounted on just one side of the plate.

A 220/247.5/275-GHz, 1.0-MW, Triple Frequency Regime Gyrotron

In this paper, a complete design methodology of a triple frequency gyrotron is presented, which can also be further applied to multifrequency operations. Frequencies for operation are selected as 220, 247.5, and 275 GHz for the localized and intense heating of magnetically confined plasmas (i.e., electron cyclotron heating and current drive) for future fusion reactors. A cautious selection procedure of the mode triplet is portrayed in accordance with the all possible physical and technical constraints. Typical cold cavity (beam absent) and single-mode (beam present) calculations are performed and presented with extended interaction structure (including an optimized nonlinear taper section). A triode type configuration is adopted for magnetron injection gun to produce the electron beam with desired characteristics as required for RF behavior. Time-dependent multimode calculations are presented with nonuniform magnetic field and beam parameters optimized by gun simulations. These rigorous calculations affirm proper working of the design with $\approx 1$ -MW continuous wave power for chosen mode triplet and efficiency $\approx 35$ %.

Resolving fine spectral features in lattice vibrational modes using femtosecond coherent spectroscopy

We show resolution of fine spectral features within several Raman active vibrational modes in potassium titanyl phosphate (KTP) crystal. Measurements are performed using a femtosecond time-domain coherent anti-Stokes Raman scattering spectroscopy technique that is capable of delivering equivalent spectral resolution of 0.1 cm−1. The Raman spectra retrieved from our measurements show several spectral components corresponding to vibrations of different symmetry with distinctly different damping rates. In particular, linewidths for unassigned optical phonon mode triplet centered at around 820 cm−1 are found to be 7.5 ± 0.2 cm−1, 9.1 ± 0.3 cm−1, and 11.2 ± 0.3 cm−1. Results of our experiments will ultimately help to design an all-solid-state source for sub-optical-wavelength waveform generation that is based on stimulated Raman scattering.

Spinning down newborn neutron stars: Nonlinear development of ther-mode instability

We model the nonlinear saturation of the r-mode instability via three-mode couplings and the effects of the instability on the spin evolution of young neutron stars. We include one mode triplet consisting of the r-mode and two near-resonant inertial modes that couple to it. We find that the spectrum of evolutions is more diverse than previously thought. We start our evolutions with a star of temperature {approx}10{sup 10} K and a spin frequency close to the Kepler break-up frequency. We assume that hyperon bulk viscosity dominates at high temperatures (T{approx}10{sup 9}-10{sup 10} K) and boundary layer viscosity dominates at lower temperatures ({approx}a fewx10{sup 8} K). To explore possible nonlinear behavior, we vary properties of the star such as the hyperon superfluid transition temperature, the strength of the boundary layer viscosity, and the fraction of the star that cools via direct URCA reactions. The evolution of the star is dynamic and initially dominated by fast neutrino cooling. Nonlinear effects become important when the r-mode amplitude grows above its first parametric instability threshold. The balance between neutrino cooling and viscous heating plays an important role in the evolution. Depending on the initial r-mode amplitude, and on the strength of the viscosity and more » of the cooling this balance can occur at different temperatures. If thermal equilibrium occurs on the r-mode stability curve, where gravitational driving equals viscous damping, the evolution may be adequately described by a one-mode model. Otherwise, nonlinear effects are important and lead to various more complicated scenarios. Once thermal balance occurs, the star spins down oscillating between thermal equilibrium states until the instability is no longer active. The average evolution of the mode amplitudes can be approximated by quasistationary states that are determined algebraically. For lower viscosity we observe runaway behavior in which the r-mode amplitude passes several parametric instability thresholds. In this case more modes need to be included to model the evolution accurately. In the most optimistic case, we find that gravitational radiation from the r-mode instability in a very young, fast spinning neutron star within about 1 Mpc of Earth may be detectable by advanced LIGO for years, and perhaps decades, after formation. Details regarding the amplitude and duration of the emission depend on the internal dissipation of the modes of the star, which would be probed by such detections. « less

Spin evolution of accreting neutron stars: Nonlinear development of ther-mode instability

The nonlinear saturation of the $r$-mode instability and its effects on the spin evolution of low mass x-ray binaries (LMXBs) are modeled using the triplet of modes at the lowest parametric instability threshold. We solve numerically the coupled equations for the three modes in conjunction with the spin and temperature evolution equations. We observe that very quickly the mode amplitudes settle into quasistationary states that change slowly as the temperature and spin of the star evolve. Once these states are reached, the mode amplitudes can be found algebraically and the system of equations is reduced from eight to two equations: spin and temperature evolution. The evolution of the neutron star angular velocity and temperature follow easily calculated trajectories along these sequences of quasistationary states. The outcome depends on whether or not the star will reach thermal equilibrium, where the viscous heating by the three modes is equal to the neutrino cooling ($H=C$ curve). If, when the $r$-mode becomes unstable, the star spins at a frequency below the maximum of the $H=C$ curve, then it will reach a state of thermal equilibrium. It can then either (1) undergo a cyclic evolution with a small cycle size with a frequency change of at most 10%, (2) evolve toward a full equilibrium state in which the accretion torque balances the gravitational radiation emission, or (3) enter a thermogravitational runaway on a very long time scale of $\ensuremath{\approx}{10}^{6}\text{ }\text{ }\mathrm{years}$. If the star does not reach a state of thermal equilibrium, then a faster thermal runaway (time scale of $\ensuremath{\approx}100\text{ }\text{ }\mathrm{years}$) occurs and the $r$-mode amplitude increases above the second parametric instability threshold. Following this evolution requires more inertial modes to be included. The sources of damping considered are shear viscosity, hyperon bulk viscosity, and viscosity within the core-crust boundary layer. We vary proprieties of the star such as the hyperon superfluid transition temperature ${T}_{c}$, the fraction of the star that is above the threshold for direct URCA reactions, and slippage factor, and map the different scenarios we obtain to ranges of these parameters. We focus on ${T}_{c}\ensuremath{\gtrsim}5\ifmmode\times\else\texttimes\fi{}{10}^{9}\text{ }\text{ }\mathrm{K}$ where nonlinear effects are important. Wagoner [R. Wagoner, Astrophys. J. 578, L63 (2002).] has shown that a very low $r$-mode amplitude arises at smaller ${T}_{c}$. For all our bounded evolutions the $r$-mode amplitude remains small $\ensuremath{\sim}{10}^{\ensuremath{-}5}$. The spin frequency of accreting neutron stars is limited by boundary layer viscosity to ${\ensuremath{\nu}}_{\mathrm{max}}\ensuremath{\approx}800\text{ }\text{ }\mathrm{Hz}[{S}_{\mathrm{ns}}/({M}_{1.4}{R}_{6}){]}^{4/11}{T}_{8}^{\ensuremath{-}2/11}$. Fast rotators are allowed for $[{S}_{\mathrm{ns}}/({M}_{1.4}{R}_{6}){]}^{4/11}{T}_{8}^{\ensuremath{-}2/11}\ensuremath{\sim}1$ and we find that in this case the $r$-mode instability would be active for about 1 in 1000 LMXBs and that only the gravitational waves from LMXBs in the local group of galaxies could be detected by advanced LIGO interferometers.

Nonlinear Couplings betweenr‐Modes of Rotating Neutron Stars

The r-modes of neutron stars can be driven to instability by gravitational radiation. While linear perturbation theory predicts the existence of this instability, linear theory cannot provide any information about the nonlinear development of the instability. The subject of this paper is the weakly nonlinear regime of fluid dynamics. In the weakly nonlinear regime, the nonlinear fluid equations are approximated by an infinite set of oscillators that are coupled together so that terms quadratic in the mode amplitudes are kept in the equations of motion. In this paper, the coupling coefficients between the r-modes are computed. The stellar model assumed is a polytropic model in which a source of buoyancy is included so that the Schwarzschild discriminant is nonzero. The properties of these coupling coefficients and the types of resonances possible are discussed in this paper. It is shown that no exact resonance involving the unstable l = m = 2 r-mode occurs and that only a small number of modes have a dimensionless coupling constant larger than unity. However, an infinite number of resonant mode triplets exist that couple indirectly to the unstable r-mode. All couplings in this paper involve the l > |m| r-modes that only exist if the star is slowly rotating. This work is complementary to that of Schenk and coworkers in 2002, who consider rapidly rotating stars that are neutral to convection.

Momentum transport from nonlinear mode coupling of magnetic fluctuations

A cause of observed anomalous plasma momentum transport in a reversed-field pinch is determined experimentally. Magnetohydrodynamic theory predicts that nonlinear interactions involving triplets of tearing modes produce internal torques that redistribute momentum. Evidence for the nonlinear torque is acquired by detecting the correlation of momentum redistribution with the mode triplets, with the elimination of one of the modes in the triplet, and with the external driving of one of the modes.

On the propagation and interaction of ultralow-frequency waves in a waveguide with gyrotropic plasma

The propagation of ultralow-frequency (ULF) electromagnetic waves in a waveguide filled with magnetoactive plasma is consieered on the basis of the monoaxial crystal approximation which is more general than the MHD approximation. The formulas for the phase and group velocities of the ordinary (magnetoacoustic) and extraordinary waves (similar to Alfven waves in the MHD approximation) are derived. It is shown that analysis outside the scope of the MHD approximation results in a finite (nonzero) velocity of energy propagation of the extraordinary wave (Alfven wave at the limit) along the waveguide. The possible waveguide mode triplets satisfying the conditions of synchronism are found. Three-wave resonant parametric and nonlinear interaction between waves with such characteristics leads to effective spectral transformation of the ULF radiation.

Land� factors of collective mode multiplets in3He-B and coupling strengths to sound waves

In a previous paper we predicted a linear splitting of collective modes in3He-B with magnetic field, in particular, a fivefold splitting of the (8/5)1/2 Δ mode. A fivefold splitting of about the same size has been recently observed by Avenel, Varoquaux, and Ebisawa for the new sound attenuation peak. In this paper all 18 collective modes are classified in terms of total angular momentumJ=0, 1, 2, andJz by means of a unitary transformation to the nine eigenstates¦J, Jz〉. The Lande factors for the field splitting of the real and imaginary mode triplets and quintuplets [including the (12/5)1/2Δ mode quintuplet] are calculated for all temperatures. We also determine the sound dispersion relation. The angular dependence of the coupling constants of theJ=2 quintuplets is given by vY 2 2 Jz (θ)v, in agreement with the results of Avenel, Varoquaux, and Ebisawa for the new mode. We find a coupling constant of the (8/5)1/2 Δ mode to sound which is an order of magnitude smaller than that given by Koch and Wolfle. By comparing this with the experimental strengths we conclude that the particle-hole asymmetry parameter is about four times larger than that derived from the free-particle density of states.

On the kinetic equation for resonant three-wave coupling

The kinetic equation for resonant three-wave coupling is solved exactly in the one-dimensional case assuming that only a single triplet of modes in interacting. No assumptions are made as to the froms of the interaction kernel or of the oscillation frequencies as functions of wave-number. Both positive- and negative- energy waves are allowed to be present simultaenously. In explosively unstable situations the time, t∞, for divergent behaviour to occur is calculated in terms of the interaction keral and intial conditions. In situations where the system is non-linearly stable, the density of quasi-particles is found to tend time-asymptotically to a steady state which depends in detail on the initial preparation of the system.For the case of two or three dimensions, the asymptotic state is obtained when the system is stable, the k-space is finite, and only one triplet of modes interact. If these modes all have the same sign of energy, the Rayleigh—Jeans distribution is valid; but if one of the modes has the oppsoite sign of energy, an additional conservation law leads to a modification of the Rayleigh-Jeans distribution.