Daughter Waves Research Articles

Utilizing parametric instabilities to diagnose edge density fluctuations in TCV

Abstract Two-plasmon decay instabilities (TPDIs) are a class of parametric instabilities where a gyrotron beam, polarized in X-mode, decays nonlinearly into two upper hybrid waves at approximately half frequency. Here, we present the design of a simple radiometer that was installed at Tokamak á configuration variable to diagnose TPDIs as well as a novel technique that was used to calibrate the radiometer using thermal electron cyclotron emissions. We show examples of measured radiation from TPDIs that occur when the gyrotron beam intersects blob filaments in the scrape-off layer and demonstrate that the frequency separation of the TPDI daughter waves can be used to assess the densities of the blob filaments. This technique could be utilized to diagnose density fluctuations due to both blobs, ELMs, and rotating magnetic islands.

Spectral line broadening of the Raman scattered waves in laser plasmas

Parametric instabilities are of a great concern especially in the context of laser fusion. Electromagnetic waves scattered by these instabilities can reach high amplitudes, thereby diverting a significant portion of the energy from the incident laser beam, which is essential for compressing and heating the fuel capsule. In addition, the emerging electrostatic wave can trap and accelerate electrons, which, upon impact, can preheat the target and thus prevent its effective compression. Here, we focus on the stimulated Raman scattering and the associated processes. Spectral broadening of the backward Raman scattering daughter wave is theoretically described process, which is caused by so-called trapped particle instability. The existence of trapped and freely moving electrons in plasma influenced by high amplitude electron plasma wave leads also to the anomalous dispersion of this wave. It is reflected in the shifts in the electrostatic spectrum which, on the other hand, must be visible also in the electromagnetic spectrum via the feed-back loop of the instability. In the present paper, we discuss scattered wave modes spectral broadening using results of numerical simulations.

Cascades of Parametric Instabilities in the Tokamak Plasma Edge during Electron Cyclotron Resonance Heating.

We report observations of nonlinear two-plasmon decay instabilities (TPDIs) of a high-power microwave beam, a process similar to half-harmonic generation in optics, during electron cyclotron resonance heating in a tokamak. TPDIs are found to occur regularly in the plasma edge due to wave trapping in density fluctuations for various confinement modes, and the frequencies of both observed daughter waves agree with modeling. Emissions from a cascade of subsequent decays, which indicate a generation of ion Bernstein waves, are correlated with fast-ion generation. This emphasizes the limitations of standard linear microwave propagation models and possibly paves the way for novel microwave applications in plasmas.

Nonlinear beat wave decay of Kelvin/diocotron waves on a two-dimensional vortex

We describe theory and experiments investigating nonlinear beat wave decay of diocotron modes on a nonneutral plasma column (or Kelvin waves on a vortex). Specifically, a Kelvin/diocotron pump wave varying as Ap exp [i(lpθ−ωpt)] decays into two waves: a Kelvin/diocotron daughter wave with exponentially growing amplitude Ad(t), mode number ld<lp, and frequency ωd; and an exponentially growing “beat wave” with mode number lb and frequency ωb. Nonlinear wave–wave coupling requires lb=lp−ld and ωb=ωp−ωd. The new theory simplifies and extends a previous weak-turbulence theory for the exponential growth rate of this instability, by instead using an eigenmode expansion to describe the beat wave as a wavepacket of continuum (Case/van Kampen) modes. The new theory predicts the growth rate, the nonlinear frequency shift (both proportional to Ap2), and the functional form of the beat wave, with amplitude proportional to ApAd*(t). Experiments observe beat wave decay on electron plasma columns for a range of mode numbers up to lp=5 and ld = 4, with results in quantitative agreement with the theory, including the ld = 1 case for which measured growth rates are negligible, as expected theoretically.

Broadening of the power deposition profile in on-axis X2 ECRH experiments in the L-2M stellarator and the low-threshold parametric decay instability of the pump wave

The effect of considerable broadening of the power deposition profile in on-axis X2 electron cyclotron resonance heating experiments observed in the L-2M stellarator is interpreted in the paper as one of the consequences of the low-threshold parametric decay instability of the pump wave, resulting in excitation of ion and electron Bernstein daughter waves. The instability is considered under the specific L-2M experimental conditions and shown to be absolute. The threshold of this instability is exceeded in the experiment. This finally allows explaining, at least qualitatively, the effect of nonlocal microwave power deposition detected in L-2M.

Control of laser-plasma instabilities by non-collinear polychromatic light

Normal broadband lasers with collinear polychromatic components have immense potential for mitigating laser plasma instabilities (LPIs). However, the projection complexity of collinear polychromatic light (CPL) is a significant challenge owing to the demand for a large bandwidth and beamlet number. Here, we propose a theoretical LPI model and optical design for non-collinear polychromatic light (NCPL), which has a small angle ∼4∘ and large frequency difference ∼1% between the double-color beamlets. LPI models of the NCPL demonstrate a decoupling threshold for the shared daughter waves under a multibeam configuration. Compared with the CPL, the wavevector couplings of LPIs are further reduced by the introduced angle. Therefore, both the growth rate and saturation level of LPIs are greatly mitigated by using the NCPL. The two- and three-dimensional simulation results indicate that the NCPL reduces the absolute and convective decoupling thresholds of the CPL and is sufficient to effectively mitigate the reflectivity, hot-electron generation, and intensity of cross-beam energy transfer. An optical design for the efficient generation of ultraviolet NCPL has been presented based on the unsaturated optical parametric amplification and non-collinear sum-frequency generation.

Four-wave resonant interaction of surface gravity waves in finite water depth

We investigated the four-wave resonant and quasi-resonant interactions in a special degenerated case, wherein bichromatic mother waves are generated to give birth to a daughter wave. Through theoretical and numerical analyses, we first discover the threshold value of water depth when the four-wave resonance diminishes. Our work can shed light on the limiting shallow water depth where the energy exchange due to four wave resonance creases.

Absolute parametric decay instabilities of X2 microwave beams in reduced models and fully kinetic codes

Nonlinear wave interactions known as parametric decay instabilities (PDIs) have been known to occur in fusion plasmas for many years. In the past, they have generally been considered of little importance in the context of high power microwave beams aimed for X2 heating due to high thresholds. Experimental evidence suggests that non-monotonic density structures, such as islands and ELMs, can lead to low thresholds absolute PDIs. The consequences include degraded electron cyclotron resonance heating performance, absorption into unintended modes and the production of strong scattering, which may damage microwave sensitive equipment. We present a reduced analytical 1D model for absolute PDI of an X-mode microwave beam into a cascade of electrostatic eigenmodes of a non-monotonic density structure. The model is solved numerically and compared with fully kinetic particle-in-cell (PIC) simulations which are computationally much more expensive. The primary decay, which initiates cascade, shows good agreement with the PIC simulations in terms of daughter frequencies, growth rates and saturation, in particular in a weakly nonlinear pump intensity regime. The secondary daughter waves of the reduced model, while visible, do not appear to be the dominant processes in the PIC simulations.

Resonant and near-resonant internal wave triads for non-uniform stratifications. Part 2. Vertically bounded domain with mild-slope bathymetry

Weakly nonlinear internal wave–wave interaction is a key mechanism that cascades energy from large to small scales, leading to ocean turbulence and mixing. Oceans typically have a non-uniform density stratification profile; moreover, submarine topography leads to a spatially varying bathymetry ( $h$ ). Under these conditions and assuming mild-slope bathymetry, we employ multiple-scale analysis to derive the wave amplitude equations for weakly nonlinear wave–wave interactions. The waves are assumed to have a slowly (rapidly) varying amplitude (phase) in space and time. For uniform stratifications, the horizontal wavenumber ( $k$ ) condition for waves (1, 2, 3), given by ${k}_{(1,a)}+{k}_{(2,b)}\!+\!{k}_{(3,c)}\!=\!0$ , is unaffected as $h$ is varied, where $(a,b,c)$ denote the mode number. Moreover, the nonlinear coupling coefficients (NLC) are proportional to $1/h^2$ , implying that triadic waves grow faster while travelling up a seamount. For non-uniform stratifications, triads that do not satisfy the condition $a=b=c$ may not satisfy the horizontal wavenumber condition as $h$ is varied, and unlike uniform stratification, the NLC may not decrease (increase) monotonically with increasing (decreasing) $h$ . NLC, and hence wave growth rates for weakly nonlinear wave–wave interactions, can also vary rapidly with $h$ . The most unstable daughter wave combination of a triad with a mode-1 parent wave can also change for relatively small changes in $h$ . We also investigate higher-order self-interactions in the presence of a monochromatic, small-amplitude topography; here, the topography behaves as a zero-frequency wave. We derive the amplitude evolution equations and show that higher-order self-interactions might be a viable mechanism of energy cascade.

Nonlinear decay of high-power microwaves into trapped modes in inhomogeneous plasma

We present novel experimental evidence of parametric decay instability of microwave beams in the plasma edge of the Wendelstein 7-X stellarator. We propose that the instability is sustained by trapping of only one daughter wave in the non-monotonic density profile measured with high spatial resolution within a stationary magnetic island. The power levels and spectral shapes of the detected microwave signal are reproduced by numerical modelling and a theoretical power threshold is predicted around 300 kW, comparable with observations. We predict a fraction of power drained by daughter waves around 4% in the experiments, potentially increasing above 50% for more hollow edge density profiles. Such absorption levels could significantly reduce the efficiency of the microwave heating and current-drive system in tokamaks and stellarators.

Collective stimulated Brillouin scattering with shared ion acoustic wave under the action of two overlapping laser beams

The overlapping of multiple beams is common in inertial confinement fusion (ICF), making the collective stimulated Brillouin scattering (SBS) with shared ion acoustic wave (IAW) potentially important because of the effectively larger laser intensities to drive the instability. In this work, based on a linear kinetic model, an exact analytic solution for the convective amplification of collective SBS with shared IAW stimulated by two overlapped beams is presented. From this solution, effects of the wavelength difference, crossing angle, polarization states, and finite beam overlapping volume of the two laser beams on the collective SBS modes with shared IAW are studied. It is found that a wavelength difference of several nanometers between the laser beams has negligible effects, except for a very small crossing angle about one degree. However, the crossing angle, beam polarization states, and finite beam overlapping volume can have significant influences. Furthermore, the out-of-plane modes, in which the wavevectors of daughter waves lie in different planes from the two overlapped beams, are found to be important for certain polarization states and crossing angles of the laser beams. This work is helpful to comprehend and estimate the collective SBS with shared IAW in ICF experiments.

Collective stimulated Brillouin scattering modes of two crossing laser beams with shared scattered wave

In inertial confinement fusion (ICF), overlapping of laser beams is common. Owing to the effective high laser intensity of the overlapped beams, the collective mode of stimulated Brillouin scattering (SBS) with a shared scattered light wave is potentially important. In this work, an exact analytic solution for the convective gain coefficient of the collective SBS modes with shared scattered wave is presented for two overlapped beams based on a linear kinetic model. The effects of the crossing angle, polarization states, and finite beam overlapping volume of the two laser beams on the shared light modes are analyzed for cases with zero and nonzero wavelength difference between the two beams. It is found that all these factors have a significant influence on the shared light modes of SBS. Furthermore, the out-of-plane modes, in which the wavevectors of daughter waves lie in different planes from the two overlapped beams, are found to be important for certain polarization states and especially for obtuse crossing angles. In particular, adjusting the polarization directions of the two beams to be orthogonal to each other or tuning the wavelength difference to a sufficiently large value (of the order of nanometers) are found to be effective methods to suppress the shared light modes of SBS. This work will be helpful for comprehending and suppressing collective SBS with shared scattered waves in ICF experiments.

The characteristics of daughter waves emerging from colliding solitary waves in astrophysical plasma media

ABSTRACT The aim is to state the properties of ion acoustic solitary waves in course of collision and extract characteristics of the daughter wave in a magnetized electron–ion plasma. The magnetized plasma medium that is a constituent of white dwarfs and astrophysical plasmas that possesses relativistically degenerate electrons and thermal ions in the presence of a constant background magnetic field. The model is based on the extended Poincaré–Lighthill–Kuo (ePLK) method where a set of Korteweg–de Vries equations is obtained to show the phase shifts of colliding waves together with the amplitude and width of the born daughter solitary waves. The numerical results and presented figures regarding the amplitude and width of solitons provide a description of the influence of plasma parameters on soliton interactions, namely ion to electron temperature ratio (σi), ion cyclotron frequency (ωci), and angle between magnetic field and collision line (θ) together with their interplay in shaping the character of solitary waves. It is concluded that only rarefactive electrostatic non-linear waves are able to propagate in such plasma media. The daughter wave amplitude possesses a scaling behaviour regarding the impact angle. Interplay of the parameters on the phase shifts is presented. Ratio of amplitude and width of the daughter wave is directly proportional to the background field, the impact angle controls its maximum. It is observed that the magnetic field elevates ratio of the solitary wave amplitude to width leading it to a shorter life and hence interaction range with neighbouring sites.

Triads and Rogue Events for Internal Waves in Stratified Fluids with a Constant Buoyancy Frequency

Internal waves in a stratified fluid with a constant buoyancy frequency were studied, with special attention given to rogue modes, extreme waves, dynamical evolution, and Fermi–Pasta–Ulam–Tsingou type recurrence phenomena. Rogue waves for triads in a general physical setting have recently been derived analytically, but the implications in a fluid mechanics context have not yet been fully assessed. Numerical simulations were conducted for cases of coupled triads where the common member is a daughter wave mode. In sharp contrast with previous studies, rogue modes instead of plane waves were used as the initial condition. Furthermore, spatial dependence was incorporated. Rogue or extreme waves in one set of triads provided a possible mechanism for significant energy transfer among modes of the internal wave spectrum, in addition to the other known theories, e.g., weak turbulence. Remarkably, Fermi–Pasta–Ulam–Tsingou recurrence types of growth and decay cycles arose, similar to those observed for surface gravity wave groups governed by the cubic nonlinear Schrödinger equation. These mechanisms will enhance our understanding of transport processes in oceans.

Trapped upper hybrid waves as eigenmodes of non-monotonic background density profiles

Non-monotonic plasma density structures such as blobs and magnetic islands give rise to trapped upper hybrid (UH) waves. Trapped UH waves which satisfy Bohr–Sommerfeld quantization can be thought of as eigenmodes of a cavity. Using fully kinetic particle-in-cell simulations, we verify the existence of these UH eigenmodes and demonstrate their significance as only eigenfrequencies become unstable to three-wave interactions. The eigenmodes can be excited through parametric decay instabilities (PDIs) of an X-mode pump wave at approximately twice the UH frequency, as could be the case for a gyrotron beam traversing a blob in a magnetically confined fusion plasma. We derive a closed expression for the wavenumber of UH waves, which is accurate both close to the UH layer and to the electron cyclotron resonance. This allows for fast analysis of eigenmodes in a non-monotonic structure. An expression for the amplification of PDI daughter waves in an inhomogeneous plasma is extended to a decay region where the first several derivatives vanish. From the amplification in a convective PDI, we estimate the growth rate of the absolute PDI involving the trapped waves. We show that the excitation of eigenmodes through PDIs in our simulations are indeed absolute rather than convective due to the trapping of the daughter waves. Additionally, we show that only eigenmodes get excited through the PDIs, and that we are able to predict the growth rates of the daughter waves and how they scale with the pump wave intensity. This is evidence supporting a fundamental assumption of analytical theory describing low threshold strong scattering observed in magnetically confined fusion experiments during second harmonic electron cyclotron resonance heating (ECRH). Such low threshold instabilities can degrade ECRH performance but also offer novel uses for ion heating or as diagnostics.

Multi-dimensional kinetic simulations of laser radiation absorption and electron acceleration in inhomogeneous underdense plasma

The sub-relativistic laser beam interaction with an underdense plasma is investigated via two-dimensional (2D) numerical simulations with respect to the laser polarization direction. Different parametric instabilities dominate the interaction depending on the propagation direction of the daughter waves in the simulation plane with respect to the laser field polarization and laser propagation direction. In the plane containing the laser electric field (p-polarization) the interaction is dominated by the two-plasmon decay instability and the beating of large amplitude electron plasma waves produces periodic ion density perturbations suppressing stimulated Raman scattering in the quarter critical density zone. A stronger absorption and heating of hot electrons is observed in the case where laser polarization is perpendicular to the simulation plane (s-polarization). Furthermore, by comparing a plane laser wave with a narrow beamlet, the effect of the initial transverse laser profile is proven to play an important role in exciting the filamentation instability, which competes with stimulated Brillouin scattering and affects the laser absorption and hot electron generation. A dedicated three-dimensional simulation indicates that a 2D simulation with p-polarization produces a more reliable results while the case of s-polarization overestimates the laser absorption and hot electron generation.

Mitigation of multibeam stimulated Raman scattering with polychromatic light

A theoretical analysis is presented for the stimulated Raman scattering (SRS) driven by multiple beams with certain frequency differences or bandwidths. The concomitant electrostatic modes are excited by the scattering lights when the pump beams share a common electron plasma wave to drive SRS. This coupling can be suppressed when the frequency difference of the two beams is three times larger than the SRS growth rate of a single beam, where the beams develop their scattering lights independently. To mitigate SRS further under the multibeam irradiation, polychromatic pumps are introduced to reduce the SRS saturation level, where the weakly coupled common modes are heavily damped in hot plasmas. The sidescattering driven by polychromatic light in inhomogeneous plasmas is significantly reduced due to the wavevector mismatch of daughter waves. Particle-in-cell simulations confirm our theoretical analysis and demonstrate that the strength of multibeam SRS and the hot electron production in inhomogeneous plasmas can be controlled at a low level by polychromatic light with a few percentage bandwidth (), even in the non-linear regime.

Multiple Alfvén eigenmodes induced by energetic electrons and nonlinear mode couplings in EAST radio-frequency heated H-mode plasmas

Multiple electromagnetic coherent modes with frequencies f ∼ 20–300 kHz and toroidal mode numbers n = 1 and n = 2 have been observed and investigated in radio-frequency heated H-mode plasmas of the EAST tokamak. The experimental results show that the two main branches of these coherent modes are driven by energetic electrons (EEs), which are produced in the processes of radio-frequency current drive and heating. Bicoherence analysis indicates that there are strong nonlinear mode interactions between the two branches (mother waves), i.e. one is in the low-frequency range of f ∼ 20–50 kHz and the other one is in the high-frequency range of f ∼ 120–250 kHz, and their nonlinear couplings can generate many harmonics (daughter waves). Both coherent modes propagate poloidally along the electron diamagnetic drift direction. The gyrokinetic eigenvalue simulations support the view that both the low-frequency and the high-frequency coherent modes observed in EAST are Alfvén eigenmode (AE) type, and the kinetic effects of background plasmas and EEs are responsible for the formation and excitation of AEs, respectively. The low-frequency coherent mode is identified as the kinetic beta-induced Alfvén eigenmode located in the edge, and the high-frequency coherent mode is radially global, which is characterized by a toroidal Alfvén eigenmode (TAE) in the core and also has the components of a kinetic TAE and ellipticity-induced Alfvén eigenmode in the outer region due to the large downshift of the Alfvén continuum gap from the core to the edge in H-mode discharges.

A frequency filter of backscattered light of stimulated Raman scattering due to the Raman rescattering in the gas-filled hohlraums

The coupling evolutions of stimulated Raman scattering (SRS) and Raman rescattering (re-SRS) are investigated under the parameter conditions of relevance to the gas-filled hohlraum experiments at the National Ignition Facility by using the nonenveloped fluid code FLAME. It is found that re-SRS works as a frequency filter of the backscattered light of SRS in the gas region. The low frequency modes of scattered light originated from a higher density region would stimulate re-SRS and be heavily depleted by re-SRS near the region of their quarter critical density. The energy in daughter waves of re-SRS is deposited in the gas plasmas. The large amplitude of the daughter Langmuir wave of re-SRS would stimulate cascade Langmuir decay instabilities and induce obvious low frequency density modulations, which can further result in the inflation of high frequency modes of scattered light of SRS at densities lower than the growth region of re-SRS.

Density Region of Low-Threshold Parametric Decay Instabilities in Electron Cyclotron Resonance Heating Experiments in Tokamak

The density conditions under which low-threshold parametric decay instabilities occur are analyzed by a physical model. Calculations show that low-threshold parametric decay instabilities occur in the density region. Magnetic field effects on the density region are analyzed. The maximum width of the density region is found. The plasma temperature has little effect on the density region. In order to avoid the low-threshold parametric decay instabilities, the daughter wave’s frequency must be far from the UH frequency inside magnetic island. It’s the key to control the magnetic field inside the magnetic island.