Eigenmode Equation Research Articles

Effect of ion temperature anisotropy on ITG mode in reversed-field-pinch plasmas

The Ion Temperature Gradient (ITG) mode instability in the Reversed-Field-Pinch (RFP) plasmas with anisotropies in ion temperature and its gradient, is studied for the first time using the gyrokinetic integral eigenmode equation. Detailed numerical results indicate that ITG instability is reduced by ion temperature anisotropy, specifically when the temperature is higher in the direction perpendicular to the magnetic field or sufficiently high in the direction parallel to the magnetic field, with Landau damping playing a significant role. Moreover, the driving force of the ion temperature gradient in the parallel direction for the ITG mode is stronger than that in the perpendicular direction, particularly when the parallel ion temperature is higher. In addition, the study provides detailed insights into the synergistic effects of ion temperature anisotropy with magnetic shear, poloidal wave number, safety factor, and density gradient on ITG mode. It also elucidates the threshold of the ion temperature gradient necessary for mode excitation.

Kinetic Theory of Drift‐Mirror Mode

AbstractWe present a nonlocal gyrokinetic theory for the drift‐mirror mode in high‐ anisotropic plasmas. Here, represents the ratio of plasma pressure to magnetic pressure. The equilibrium distribution is established self‐consistently via guiding‐center Hamiltonian theory. To keep the nonuniformity and finite Larmor radius (FLR) effects on an equal footing, we derive the three‐field nonlocal eigenmode equations in Fourier space under the assumption of weak nonuniformity. It is found that the drift‐mirror mode is essentially a dissipative instability confined within the potential well due to the FLR effect. Numerical analyses demonstrate that the drift‐mirror mode originates from the coupling between shear Alfvén and mirror branch, where the ion‐sound wave component is negligible. Furthermore, the threshold of this nonlocal drift‐mirror mode is significantly lower than that of the conventional drift‐mirror mode.

ICE modes with high wave numbers

This work deals with high-k modes of the ion cyclotron emission (ICE)—the modes with frequencies close to ion cyclotron harmonics and wave numbers well exceeding those of fast magnetoacoustic modes (FMM). These modes exist due to finite Larmor radius of the ions. They are responsible for the ICE recently observed in the DIII-D tokamak [N. A. Crocker et al., Nucl. Fusion 62, 026023 (2022)]. The work is aimed at knowing the radial structure of high-k eigenmodes. Its analysis is relied on a comprehensive study carried out by employing a general dispersion equation for traveling waves, which allowed to group modes and determine conditions of their existence. Differential equations for various eigenmodes (standing waves) were obtained and analyzed. In particular, electrostatic modes and electromagnetic modes with various ratios of the longitudinal phase velocity to the electron thermal velocity were considered. A numerical code solving these equations was developed. It is found that high-k modes with sufficiently large poloidal mode numbers are located at the plasma periphery, and their amplitudes in the plasma core are very small. It is concluded that high-k modes are not an exceptional case, although presumably FMMs can explain most ICE experiments. They exist in plasmas with various magnitudes of β, representing different wave branches associated with finite Larmor radius of the ions. The mentioned experiment on DIII-D is discussed.

On equations for ion cyclotron modes in ‘warm’ bounded plasmas

An equation describing eigenmodes in the ion cyclotron frequency range in ‘warm’ bounded plasmas, i.e. eigenmodes which are absent in the two-fluid model but exist in kinetic theory due to finite Larmor radius of the ions, is derived for the first time. It is valid for electrostatic modes but the developed approach is generic. Calculations are carried out for two cases: first, for a homogeneous magnetic field; second, taking into account the effects of toroidicity in tokamaks. It is found that, in general, equations for eigenmodes in warm plasmas do not reduce to second-order differential equations (in contrast to those which are usually used to describe the radial structure of eigenmodes in fusion devices). The study of modes in warm plasmas is of interest, in particular, in connection with the recent observations of superthermal ion cyclotron emission in the NSTX-U spherical torus and DIII-D tokamak, which can be hardly explained by conventional theories employing fast magnetoacoustic modes.

Effect of radial electric field on ion-temperature gradient driven mode stability

To understand the effect of given radial electric field on ion-temperature gradient driven mode (ITG) stability in tokamak plasmas, we derive the eigenmode equation for ITG including the poloidal rotation and density modulation associated with radial electric field by using nonlinear gyrokinetic theory. The equation is solved for the eigenfrequency, growth rate and parallel mode structure of ITG both in short- and long-wavelength limit with energetic-particle-induced geodesic acoustic mode (EGAM) as a specific form. The eigenmode equation is not only solved analytically, but also solved numerically to validate the analytic solutions. It is found that, the radial electric field induced poloidal rotation can significantly stabilize ITG, while the density perturbation of the radial electric field may slightly distort the ITG parallel mode structure, but has little effect on ITG stability. The result is consistent with the common picture of turbulence suppression by poloidal shear flow. The general model is also applicable to the investigation of the indirect interaction of ITG and energetic particle driven Alfvén instabilities via zonal structures generation, by introducing poloidal rotation and density modulation associated with zonal structures spontaneously excited by Alfvén instabilities. The indirect channel is supplement to the direct interaction of microturbulences and energetic particle driven Alfvén instabilities.

Global simulations of energetic electron excitation of beta-induced Alfvén eigenmodes

The energetic electron (EE) excitation of beta-induced Alfvén eigenmodes is investigated by using the newly developed global eigenvalue code MAS, which is based on a hybrid model that consists of Landau fluid bulk plasma and drift kinetic EE. Specifically, the bulk plasma kinetic effects such as finite Larmor radius, diamagnetic drifts and Landau dampings, and the EE adiabatic fluid response of convection and non-adiabatic kinetic response of precessional drift resonance are incorporated in the simulations. The global eigenmode equation is solved for e-BAE mode structure and linear dispersion relation in tokamak non-perturbatively. The radial width of e-BAE mode structure becomes narrower as the toroidal mode number increases, which can be explained by the change of Alfvén continuous spectra that interact with kinetic Alfvén waves for corresponding eigenmode formation. The e-BAE growth rate exhibits a non-monotonic variation with toroidal mode number for precessional drift resonance destabilization, while the e-BAE real frequency is close to the continuum accumulation point that almost remains the same. The parametric dependence of e-BAE stability on EE density and that on temperature are analyzed by MAS non-perturbative simulations, which shows that the EE density can affect e-BAE real frequency and thus changes the resonance condition, resulting in e-BAE stabilization in the strong EE drive regime. Further, the EE non-perturbative effect on the symmetry breaking of e-BAE mode structure is reported. The poloidal symmetry breaking characterized by the ‘boomerang’ shape two-dimensional (2D) structure can be greatly enhanced by increasing EE temperature, together with the large radial variation of the poloidal phase angle of dominant principal poloidal harmonic. The radial symmetry breaking of e-BAE mode structure arises when EE density/temperature drive is not symmetric with respect to corresponding rational surface, which can lead to a net volume-averaged value of e-BAE parallel wave number which drives plasma intrinsic rotation. These results are helpful in understanding the e-BAE dynamics observed in recent experiments.

Impact of impurities on drift wave instabilities in reversed-field pinch plasmas.

The drift wave in the presence of impurity ions was investigated numerically in reversed-field pinch plasmas, using the gyrokinetic integral eigenmode equation. By comparing the results of regular and hollow plasma density profiles, it was found that the ion temperature gradient mode for the hollow density profile case is much harder to excite. For the impurity effects, when the impurity density gradient is opposite to the electrons, namely when L_{ez} (L_{ez}=L_{ne}/L_{nz} with 1/L_{n} being the density gradient scale length, and the subscript "e" and "z" indicates electrons and impurity ions, respectively) is negative, the impurities can enhance the instability. On the contrary, when L_{ez} is positive, the instability is stabilized. Regarding the trapped electron mode (TEM), the growth rate for plasmas with a hollow density profile remains smaller than that of the standard density gradient. There exists a threshold in L_{ez}. When L_{ez} is less than this value, the impurities destabilize the TEMs, while when L_{ez} is greater than this value, the impurities stabilize the TEMs. In addition, the influence of the collisionality on the TEMs was also studied.

Effect of electron temperature anisotropy on TEM in reversed-field-pinch plasmas

For the first time in the reversed-field-pinch configuration, trapped electron mode (TEM) with anisotropies of electron temperature and its gradient is studied by solving the gyrokinetic integral eigenmode equation. Detailed numerical analyses indicate that TEM is enhanced by the anisotropy with temperature in the direction perpendicular to the magnetic field that is higher than that in the direction parallel to the magnetic field when the latter is kept constant. However, the enhancement is limited, such that TEM is weakened and even stabilized when the anisotropy is higher than a critical value, due to strong Landau damping. In comparison with the isotropic case, the lower Landau damping with the higher parallel electron temperature makes TEM instability easier to excite, which expands the TEM unstable region in the diagram of density and temperature scale lengths. In addition, it is found that the electron temperature gradient in the perpendicular direction offers a stronger driving force on TEM instability than that in the parallel direction. The overall effects of the temperature gradients of electrons and ions, magnetic shear, safety factor and density gradient on TEM in the presence of the anisotropies are presented in detail.

TEM in toroidal plasmas with anisotropic electron temperature

Trapped electron modes (TEMs) in tokamak plasmas with anisotropies of electron temperature and its gradient are studied by solving the gyrokinetic integral eigenmode equation. Detailed numerical analyses indicate that, in comparison with that in plasmas of isotropic electron temperature, TEMs are enhanced (weakened) by the anisotropy with temperature in the direction perpendicular to magnetic field higher (lower) than that in the direction parallel to the magnetic field when the latter is kept constant. However, the enhancement is limited such that TEMs are weakened rapidly and even stabilized when the anisotropy is higher than a critical value owing to an effective reduction of bounce movement of the trapped electrons. In addition, it is found that the gradients of perpendicular and parallel temperatures of electrons have driving and suppressing effects on the TEMs, respectively. The overall effects of the temperature gradients of electrons and ions, magnetic shear, safety factor, and density gradient on TEMs in the presence of the anisotropies are presented in detail.

Characteristics and mechanism of coupling effects in parallel-cladded acoustic waveguides

The characteristics and mechanism of coupling effects between parallel cladded acoustic waveguides (PCAWs) are essential when considering their applications in acoustic wave control and signal processing. We investigated its characteristics and revealed the nature of the coupling effect using a theoretical model of two-dimensional PCAWs and simulation experiments. We derived the eigenmode equation describing the behavior of a single waveguide based on the wave acoustic theory and derived analytic expressions for the coupling effects in the PCAWs using the coupled mode theory. Using the finite-element method, we analyzed the waveguide coupling exhibited by this structure given different configurational and acoustic parameter settings. Both theoretical and simulated results indicate that the input wave directed into one of four ports of this structure propagates and tunnels alternately between the two waveguides. Our theoretical model established yields analytic relations between the coupling lengths as well as the dependence on parameters of the evanescent wave and the structure. Analyses indicate wave coupling in the two PCAWs is essentially mediated by the evanescent wave. The unique evolution of the acoustic wave in PCAWs can be employed to develop pure acoustic devices such as frequency-selective filters, directional couplers, and acoustic switches.

The intermittent excitation of geodesic acoustic mode by resonant Instanton of electron drift wave envelope in L-mode discharge near tokamak edge

There are two distinct phases in the evolution of drift wave envelope in the presence of zonal flow. A long-lived standing wave phase, which we call the Caviton, and a short-lived traveling wave phase (in radial direction) we call the Instanton. Several abrupt phenomena observed in tokamaks, such as intermittent excitation of geodesic acoustic mode (GAM) shown in this paper, could be attributed to the sudden and fast radial motion of Instanton. The composite drift wave — zonal flow system evolves at the two well-separate scales: the micro-scale and the meso-scale. The eigenmode equation of the model defines the zero-order (micro-scale) variation; it is solved by making use of the two-dimensional (2D) weakly asymmetric ballooning theory (WABT), a theory suitable for modes localized to rational surface like drift waves, and then refined by shifted inverse power method, an iterative finite difference method. The next order is the equation of electron drift wave (EDW) envelope (containing group velocity of EDW) which is modulated by the zonal flow generated by Reynolds stress of EDW. This equation is coupled to the zonal flow equation, and numerically solved in spatiotemporal representation; the results are displayed in self-explanatory graphs. One observes a strong correlation between the Caviton-Instanton transition and the zero-crossing of radial group velocity of EDW. The calculation brings out the defining characteristics of the Instanton: it begins as a linear traveling wave right after the transition. Then, it evolves to a nonlinear stage with increasing frequency all the way to 20 kHz. The modulation to Reynolds stress in zonal flow equation brought in by the nonlinear Instanton will cause resonant excitation to GAM. The intermittency is shown due to the random phase mixing between multiple central rational surfaces in the reaction region.

Two-dimensional gyrokinetic integral eigenmode equation for drift instabilities in tokamak

Based on the Vlasov equation, a set of two-dimensional gyrokinetic integral eigenmode equations for drift instabilities in tokamak plasmas is derived. All kinetic effects of ions, including transience parallel to the magnetic field, drifts induced by the curvature and gradient of the magnetic field, and drifts induced by the equilibrium electric potential, and the effects of finite Larmor radius are taken into account. Not only the radial structure but also the poloidal structure of the modes as well as coupling of neighboring poloidal modes induced by toroidicity of the magnetic field can be obtained from these equations. This is an alternative to ballooning mode representation treatment, which can give mode structures along magnetic field lines only. The equations provide a theoretical base for relevant numerical simulation investigations.

Impurity effects on trapped electron modes in tokamak plasmas with inverted electron density profile

Impurity effects on trapped electron modes (TEMs) in tokamak plasmas with inverted electron density profile (IEDP) are numerically investigated with a gyrokinetic integral eigenmode equation. It is found that different from the negative gradient of normal electron density profile, the positive gradient of the IEDP has a stabilizing effect on TEM in the presence of impurity ions. The electron temperature gradient threshold for TEM excitation increases not only with the increasing absolute value of IEDP but also with increasing impurity content. Furthermore, the effects of different impurity species and different impurity peaking profiles on TEMs with the IEDP are analyzed in detail. It is shown that there is a transition point of impurity density profile, on both sides of which the impurity has opposite effects on TEM. The dependence of such a transition point on electron temperature and density gradients is obtained numerically. Besides, the synergistic effects of ion temperature gradient and impurity density gradient are studied, in which a similar transition point of the ion temperature gradient is also identified in the case of outwardly peaked impurity density profile. In addition, impurity effects on the characteristics of mode structure and on the radial transport coefficients in positive and negative magnetic shear regions are discussed as well based on quasi-linear mixing length estimation.

η i-mode in toroidal plasmas with anisotropic ion temperature and its gradient

The gyrokinetic integral eigenmode equation is applied to study ion-temperature-gradient (ITG) mode in toroidal plasmas with magnetic shear, where the ion temperature and its gradient are anisotropic. The numerical studies demonstrate that instability of ITG mode is reduced by an ion temperature anisotropy of higher perpendicular temperature T i⊥ or high enough parallel temperature T i∥, in which Landau damping plays an important role. The temperature gradient in the perpendicular (parallel) direction is stronger to drive ITG mode for large T i⊥ (T i∥). These effects are directly related to the temperature gradient threshold for excitation of ITG mode. In addition, the synergy effect of magnetic shear and anisotropy of ion temperature and its gradient is studied in detail, where the combination of magnetic shear and large parallel temperature has the most obvious inhibitory effect on ITG mode.

Group velocity in spatiotemporal representation of collisionless trapped electron mode in tokamak

The multiple scale derivative expansion method is used to manipulate the electron drift kinetic equation, following the theoretical framework of drift wave–zonal flow system developed by Zhang et al. [Zhang Y Z, Liu Z Y, Mahajan S M, Xie T, Liu J <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1063/1.4995302"> 2017 <i>Phys. Plasmas</i> <b>24</b> 122304 </ext-link>]. At the zeroth order it is the linear eigenmode equation describing the trapped electron mode on a mirco-scale. At the first order it is the envelop equation for trapped electron mode modulated by the zonal flow on a meso-scale. The eigenmode equation has been solved by Xie et al. [Xie T, Zhang Y Z, Mahajan S M, Wu F, He Hongda, Liu Z Y <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="https://doi.org/10.1063/1.5048538"> 2019 <i>Phys. Plasmas</i> <b>26</b> 022503 </ext-link>] to obtain the eigenvalue and two-dimensional mode structure of trapped electron mode. These are essential components in calculating group velocities contained in the envelop equation. The radial group velocity arises from the geodesic curvature of magnetic field in tokamak. The poloidal group velocity stems from the normal curvature and diamagnetic drift velocity, which yields the mapping between the poloidal angle and time. Since the radial group velocity is also a function of poloidal angle, it is mapped to a periodic function of time with a period of milliseconds. The numerical results indicate the rapid zero-crossing, which is significant in the drift wave – zonal flow system and provides a sound foundation for studying zonal flow driven by trapped electron mode.

Stability of a tokamak plasma with diffuse toroidal rotation

The present work considers the stability of a high- $\beta$ , large aspect ratio, circular plasma with diffuse profiles for the safety factor and the angular toroidal frequency (López & Guazzotto, Phys. Plasmas, vol. 24, 032501). An application of the Frieman–Rotenberg formalism results in a system of scalar eigenmode equations whose coupling is retained at the plasma–vacuum transition but is disregarded across the plasma column, which is a standard practice. The solution technique consists of a multidimensional shooting method for the poloidal harmonics; robust initial guesses are constructed by solving the dispersion relation in the static scenario with vanishing magnetic shear. Flow shear appears as a high- $\beta$ toroidal contribution, and we illustrate its destabilizing influence on $n=1$ external kink modes in the presence of ideal and resistive walls. Internal resonances are avoided by means of the selection of appropriate equilibrium parameters. The stabilizing influence of a finite positive average magnetic shear is also exemplified.

The Crystal Field Plasmon Splitting

We present an electromagnetic analogue of crystal (or ligand) field theory that describes geometric eigenmodes and the resonant plasmon wavelength in plasmonic nanocrystals in terms of a simple shape descriptor. Our model, crystal field plasmon splitting, is based on secular equations for geometric eigenmodes, allowing for the separation of pure shape from materials effects. As an example, the model is implemented to the experimental gold nanoparticles that undergo cube-to-sphere transition, showing that geometric eigenvalues change correspondingly in analogy to the atomic energy levels in Tanabe–Sugano correlation diagrams.

Effects of trapped electrons on the ion temperature gradient mode in tokamak plasmas with hollow density profiles

The ion temperature gradient (ITG) mode in the presence of impurity ions and trapped electrons (TEs) is numerically investigated in tokamak plasmas with hollow density profiles, using the gyrokinetic integral eigenmode equation. It is found that in the inverted density plasma, the increase of the ITG enhances the growth rate and frequency of the ITG, and the density gradient plays an important role in the ITG modes. For weak density gradient situations, the trapped electron effects increase the instability of the ITG, while the impurity has an obviously stabilizing effect. For the strong density gradient cases, both the impurities and trapped electrons enhance the ITG instabilities. In addition, it is shown that the growth rate of the ITG decreases with positive magnetic shear s while the real frequency increases with positive magnetic shear. The growth rate of the ITG increases with negative magnetic shear s while the real frequency decreases with negative magnetic shear. The length of the calculated mode structure in the positive and negative magnetic shear intervals is also presented.

Short wavelength ion temperature gradient mode in tokamak plasmas with hollow density profiles

By the gyrokinetic integral eigenmode equation, we numerically investigate the short-wavelength ion-temperature-gradient (SWITG) mode in tokamak plasmas with hollow density profiles. This paper finds that the critical ion temperature gradient (ITG) R/LTiC exists for the hollow density profile. Over such a gradient, the SWITG mode is unstable. The R/LTiC for a hollow density profile is slightly lower than that for a peaked density profile, though far away from the threshold, the mode in the latter case is more unstable. Furthermore, the scaling of the ITG threshold increasing with ion-to-electron temperature ratio Ti/Te has been calculated, suggesting that the SWITG mode is more difficult to excite in hot ion plasmas as compared with that in hot electron plasmas. For the slightly hollow density profiles, it is shown that in the case of flat (peaked) electron temperature profiles, the SWITG modes are destabilized (stabilized) by the trapped electrons. For the steep hollow density profiles, however, the trapped electrons reduce the growth rate substantially, and thus, the SWITG stability threshold is raised significantly. In addition, the dependence of the threshold on local plasma parameters is analyzed in detail. It is found that the SWITG modes become more unstable for the high safety factor, small Larmor radius, weak magnetic shear, and high electron-to-ion temperature ratio.

Impurity effects on ion temperature gradient driven multiple modes in transport barriers

The impurity effects on ion temperature gradient (ITG) driven instability in transport barriers (TBs) are numerically investigated with the gyrokinetic integral eigenmode equations in tokamak plasmas. In particular, the effects of temperature and density gradients of the main ions ( and ) are analyzed independently to understand the physical mechanisms better, instead of keeping their ratio as carried out in previous works, when the parameters of impurity ions vary. It is found that the effect of impurity ions with outwardly peaked density profiles on ITG modes depends on the competition between the destabilizing effect of the impurity density gradient and the stabilizing effect induced by the dilution of main ions from impurity ions when is fixed, which is in significant contrast with the results for a fixed . The destabilizing effects include enhancement of ITG modes and coupling to the impurity mode (IM) in weak ITGs (big ) and strong impurity density gradient regimes. In addition, the stability boundaries for ITG modes, including high-order modes, are discussed in detail, and compared with previous works (Fröjdh et al 1992 Nucl. Fusion 32 419). Furthermore, the impurity ions with either inwardly or slightly outwardly peaked density profiles have weaker and stronger stabilizing effects on small and big poloidal wave vector modes, respectively. However, the impurity ions with steeper outwardly peaked density profiles have stronger stabilizing effects on big modes. Moreover, the inwardly peaked impurity ion density profiles are beneficial for main ion confinement and impurity decumulation, due to the main (impurity) ions flowing inwardly (outwardly). Finally, analyses of eigenmode structure and the quasi-linear particle flux are performed in detail. The results show that impurity ions have non-negligible effects, especially on higher-order ITG modes.