Nonlinear Three-wave Coupling Research Articles

Use of differential plasma rotation to prevent disruptive tearing mode onset from 3-wave coupling

Abstract Plasma differential rotation is found to be capable of preventing disruptive neoclassical tearing modes (NTMs) seeded by nonlinear three-wave coupling. As tearing modes degrade confinement and can lead to disruptions, stabilization strategies are crucial to the successful operation of future devices. In ITER-relevant scenarios on DIII-D, rotationally coupled m / n = 1/1 and 3/2 modes have been observed to drive 2/1 islands through three-wave coupling. The frequency of the driven 2/1 mode is set by matching conditions and the frequencies of the driving modes. When the driven mode frequency matches the local plasma rotation frequency, e.g. at low differential rotation, the driven 2/1 island can grow into a disruptive NTM. Using neutral beam torque as an actuator to scan the differential rotation, these experiments demonstrate that a sufficiently large frequency mismatch prevents destabilization of disruptive 2/1 NTMs by three-wave coupling. This work indicates that differential rotation can be used as an actuator to prevent NTMs seeded by three-wave coupling.

Neoclassical Tearing Mode Seeding by Nonlinear Three-Wave Interactions in Tokamaks.

We report the experimental observation of seed magnetic island formation by nonlinear three-wave coupling of magnetic island triplets. In this experiment, disruptive 2,1 islands are seeded by the coupling of 4,3 and 3,2 tearing modes to a central 1,1 sawtooth precursor. Three-wave interactions between these modes are conclusively identified by bispectral analysis, indicating fixed phase relationships in agreement with theory. This new observation of this seeding mechanism has important implications for future reactors that must operate in stable plasma equilibria, free of disruptive 2,1 islands.

Pedestal dynamics prior to type-III ELM onset on HL-2A tokamak*

A dramatic increase of density gradient and a slight decrease of electron temperature gradient were observed in the pedestal just prior to each onset in a series of ELM bursts. An inward particle flux induced by a quasi-coherent mode was found to be responsible for such changes. The characteristics of the quasi-coherent mode, including frequency, wave vector, propagation direction and velocity were identified. The nonlinear three-wave coupling is a plausible mechanism for the generation of the mode. Decoupling of the mode induced density transport from energy transport was discovered. The gradient of density was observed always much higher than that of temperature for electrons at the starting point of the ELM burst and, therefore, the dominant role of density gradient over electron temperature gradient for ELM onset is demonstrated. Consistence and contrast of the results with confinement and transport in I-mode discharges are discussed.

Nonlocal resonances in weak turbulence of gravity-capillary waves.

We report a laboratory investigation of weak turbulence of water surface waves in the gravity-capillary crossover. By using time-space-resolved profilometry and a bicoherence analysis, we observe that the nonlinear processes involve three-wave resonant interactions. By studying the solutions of the resonance conditions, we show that the nonlinear interaction is dominantly one dimensional and involves collinear wave vectors. Furthermore, taking into account the spectral widening due to weak nonlinearity explains why nonlocal interactions are possible between a gravity wave and high-frequency capillary ones. We observe also that nonlinear three-wave coupling is possible among gravity waves, and we raise the question of the relevance of this mechanism for oceanic waves.

Phase locking of multi-helicity neoclassical tearing modes in tokamak plasmas

The attractive “hybrid” tokamak scenario combines comparatively high q95 operation with improved confinement compared with the conventional H98,y2 scaling law. Somewhat unusually, hybrid discharges often exhibit multiple neoclassical tearing modes (NTMs) possessing different mode numbers. The various NTMs are eventually observed to phase lock to one another, giving rise to a significant flattening, or even an inversion, of the core toroidal plasma rotation profile. This behavior is highly undesirable because the loss of core plasma rotation is known to have a deleterious effect on plasma stability. This paper presents a simple, single-fluid, cylindrical model of the phase locking of two NTMs with different poloidal and toroidal mode numbers in a tokamak plasma. Such locking takes place via a combination of nonlinear three-wave coupling and conventional toroidal coupling. In accordance with experimental observations, the model predicts that there is a bifurcation to a phase-locked state when the frequency mismatch between the modes is reduced to one half of its original value. In further accordance, the phase-locked state is characterized by the permanent alignment of one of the X-points of NTM island chains on the outboard mid-plane of the plasma, and a modified toroidal angular velocity profile, interior to the outermost coupled rational surface, which is such that the core rotation is flattened, or even inverted.

TEST FOR WAVEVECTOR ANISOTROPIES IN PLASMA TURBULENCE CASCADES

The frequency and wavevector matching conditions in nonlinear three-wave coupling are used to test whether the forward cascade of plasma turbulence may lead to wavevector anisotropies in a homogeneous, collisionless, magnetized plasma. Linear kinetic theory at β p = 0.01, 0.10, and 1.0 is used to determine the frequency-wavenumber dispersion of three normal modes: long-wavelength Alfven-cyclotron waves, long-wavelength magnetosonic waves, and intermediate-wavelength magnetosonic-whistler waves. Using linear dispersion in the nonlinear matching conditions, the test predicts with one exception that forward cascades are favored by fluctuations propagating nearly perpendicular to the background magnetic field B o . This is consistent with the typical development of wavevector anisotropies with k ⊥ k ∥ (subscripts refer to directions perpendicular and parallel to B o , respectively) in computer simulations of the forward cascade of various types of plasma turbulence. The exception is that, at β p = 1.0, the test predicts that the cascade of long-wavelength magnetosonic waves should be favored by modes at k ⊥ ~ k ∥.

Measurements of impulsive reconnection driven by nonlinear Hall dynamics

The magnetic fields associated with reconnection in the edge of the reversed field pinch configuration have been measured in the Madison Symmetric Torus. The measured magnetic field structure is compared with theoretical predictions computed in both toroidal and cylindrical geometries. The summation of multiple modes has been accomplished to reveal a complex but still coherent edge structure. Key terms of relevant Ohm’s law are accessible from magnetic field measurement and reveal the ordering [(1/ne)J×B⪢E>ηJ], which implies that two fluid effects are important in the physics governing this reconnection. Further, it is seen that the nonlinear three-wave coupling of the Hall term acts as a driving mechanism for this linearly stable mode.

Turbulence and zonal flows in edge plasmas of the HL-2A tokamak

Measurements with a toroidally and poloidally displaced three-dimensional set of Langmuir probe arrays have revealed details of turbulence, geodesic acoustic modes (GAMs), zonal flows and their interactions in the edge region of HL-2A tokamak plasmas. The coexistence of intensive low frequency zonal flows (LFZFs) of f < 4 kHz and the GAMs of 7 kHz < fGAM < 20 kHz has been unambiguously demonstrated. The poloidal and toroidal symmetries of the flows, including the GAMs, are verified experimentally. In particular, the coherency of the flows over a large toroidal scale of 2100 mm at a magnetic flux surface is emphasized. The LFZF packets are shown to propagate outward and inward as equally likely events, whereas the predominantly outward propagation of the GAM packets is analyzed. The nonlinear three-wave coupling of the flows with ambient turbulence is shown with a bicoherency analysis and an envelope modulation of the latter by the former. The intensity of the LFZFs is observed to increase and decrease with increases in ECRH power (∼300–700 kW) and safety factor q ∼ (3.5–6.2), respectively, whereas the intensity of the GAMs increases with increases in both ECRH power and q.

Direct measurement of poloidal long-wavelength E x B flows in the HT-7 tokamak.

The poloidal long-wavelength E x B time-varying flows were directly measured using a forked Langmuir probe in the HT-7 tokamak. Low-frequency (<10 kHz) E x B flows were observed at the plasma edge, which possess many of the characteristics of zonal flows, including a poloidal long-wavelength (k(theta)rho(i) approximately 0) and narrow radial extent (k(r)rho(i) approximately 0.1). The cross bicoherence of turbulent Reynolds stress indicates the existence of nonlinear three-wave coupling processes and the generation of low-frequency E x B flows. The estimated flow-shearing rate is of the same order of magnitude as the turbulence decorrelation rate and may thus regulate the fluctuation level and thereby the turbulence-driven transport.

Nonlinear three- wave interaction among barotropic rossby waves in a large- scale forced barotropic flow

In this paper, the coupling equations describing nonlinear three-wave interaction among Rossby waves including the forcing of an external vorticity source are obtained. Under certain conditions, the coupling equations with a constant amplitude forcing, the stability analysis indicates that when the amplitude of the external forcing increases to a certain extent, a pitchfork bifurcation occurs. Also, it is shown from numerical results that the bifurcation can lead to chaotic behavior of “ strange≓ attractor. For the obtained three-variable equation, when the amplitude of modulated external forcing gradually increases, a period-doubling bifurcation is found to lead to chaotic behavior. Thus, in a nonlinear three-wave coupling model in the large-scale forced barotropic atmospheric flow, chaotic behavior can be observed. This chaotic behavior can explain in part 30-60-day low-frequency oscillations observed in mid-high latitudes.

Order and chaos in nonlinear wave interactions in astrophysical and space plasmas

We discuss the observation and theory of nonlinear wave interactions in astrophysical and space plasmas. Observational evidence of nonlinear mode-mode coupling in solar type III radio bursts, auroral whistler-mode radiation and active experiments in space is reviewed. The transition from order to chaos in nonlinear plasma wave interactions is analysed. For Hamiltonian four-wave interactions, the nonlinear dynamical process may undergo a chaotic transition via the quasi-periodicity route. For dissipative three-wave interactions, the nonlinear system may evolve from order to chaos via two different routes: period doubling and intermittency. It is shown that chaos in dissipative nonlinear three-wave coupling can be controlled by applying a small sinusoidal wave to the system.

Effects of Er shear on edge turbulence of the China Tokamak-6B

On the China Tokamak (CT-6B) [Nucl. Fusion 36, 1669 (1996)], application of negative limiter bias resulted in enhanced Er shear in the naturally occurring Er shear layer near the fixed limiter radius with negligible change of plasma density, electron temperature, the parallel plasma flow, and the impurity ion radiation power. In the layer, decreased turbulent fluctuation, reduced poloidal correlation and increased nonlinear coupling of the turbulence were observed to be very possibly correlated with enhanced Er shear. The results suggest that there exists interaction of Er shear with turbulence, and an Er shear-induced shift in the phase angle between density and poloidal electric field fluctuations and nonlinear three-wave coupling may play an important role in suppressing edge turbulence.

Lorenz-type chaos in instability saturation by nonlinear three-wave mode coupling

It is shown that resonant three-wave coupling equations in plasma physics can be recast in a form which is isomorphic to the Lorenz turbulence equations. The possibility of Lorenz-type chaos is predicted

Spectral transform and solitons for the three-wave coupling model with nontrivial boundary conditions

A nonlinear three-wave coupling model with nontrivial boundary conditions is analyzed in the framework of the inverse spectral transform. Spectral data are determined and their evolution is derived. It is shown that there exists mutual cancellation of divergences of the scattering matrix elements in branching points on the plane of a spectral parameter. A regular dark-soliton-type solution is obtained.

Controlling chaos in nonlinear three-wave coupling.

It is shown that the chaotic orbits in the saturated states of nonlinear three-wave coupling can be controlled by applying a new resonance, via a small sinusoidal anti-Stokes wave, to the system. For a given set of linear frequency mismatch and growth-damping parameters, any desired periodic orbit can be achieved by a proper adjustment of the amplitude of the control wave. This method is illustrated by numerical simulations. \textcopyright{} 1996 The American Physical Society.

Nonlinear wave coupling and subharmonic resonance in planar jet shear layer transition

An experimental investigation of the nonlinear spectral development associated with the transition of an initially laminar planar jet shear layer is presented. The local shear layer spectral dynamics are modeled as a nonlinear system containing both linear and quadratically nonlinear system elements. A novel digital signal processing technique is applied which allows the linear and quadratically nonlinear wave coupling coefficients that characterize the local spectral dynamics to be estimated directly from time-series velocity fluctuation data. The method utilized is noniterative and explicitly includes fourth-order spectral moments. From the linear coupling coefficient, the local spatial growth rate and dispersion relation may be obtained. Of particular interest in the work reported is the associated nonlinear wave coupling coefficient which provides a measure of the efficiency of local nonlinear three-wave coupling. These show that the jet shear layer exhibits a strong preference for difference mode interactions. With the estimation of the related nonlinear power transfer function, the total, linear, and quadratic nonlinear spectral energy transfer may be locally estimated. When these measures are used in conjunction with the local quadratic bicoherency and local linear–quadratic coupling bicoherency, the local nonlinear spectral dynamics of the flow may be completely and unambiguously quantified. Particular emphasis is placed upon discerning the mechanism which gives rise to subharmonic resonance with the fundamental. Results show that the amplification of the subharmonic occurs via a parametric resonance with the fundamental rather than by direct quadratic difference interaction. That is, the subharmonic is boosted by a weakly nonlinear mechanism involving a linear growth rate modification due to a resonant phase lock with the fundamental instability.

Propagation and interaction of asymptotic waves in ideal magnetofluiddynamics

Nonlinear three-wave coupling coefficients are derived by a systematic application of nonlinear high frequency geometric optics to the equations of ideal magnetofluiddynamics. These coefficients concern all the possible interactions between Alfven waves and the two kinds of magnetoacoustic waves: the fast magnetosonic and the slow ones. The propagation of off-resonance waves is also treated and critical times for two specific initial data are found.

Bifurcation and ’’strange’’ behavior in instability saturation by nonlinear three-wave mode coupling

A simple model of nonlinear saturation of an unstable mode is studied. The model consists of the resonant three-wave coupling equations with the highest frequency wave linearly unstable and the two lower frequency waves damped. As the damping of the stable waves is increased, the solutions go through an interesting succession of qualitative changes, including bifurcations to increasingly complex periodic solutions and the appearance of apparently chaotic (i.e., aperiodic) solutions. These qualitative features are shown to be explainable on the basis of a one-dimensional mapping which is numerically derivable from the original system of differential equations.

An approach to linear and non-linear wave coupling using dispersion and energy relations †

Abstract By inclusion of an. external driving force, wave motion of any kind can be characterized by a dispersion function. This function is closely related to the energetic properties of wave motion, and then also to the averaged Lagrangian density. Linear and nonlinear wave interaction can be analysed by inclusion of internal driving forces. Normalization procedures for the amplitudes can be avoided and time and space perturbations studied simultaneously. This analysis is further connected to slowly varying amplitudes and quasi-monochromatic waves. This paper presents the above-mentioned method and applies it to linear two-wave coupling, and non-linear three-wave coupling between positive and negative energy waves, and finally to amplitude modulation. The general equations obtained by this procedure are useful for general discussions. The simplicity of the method may prove useful in different applications.