Eigenmode Equation Research Articles

Unstable Rayleigh–Taylor modes in the ionosphere in the presence of dusts

We develop and apply non-local theoretical and numerical model to study the stability of Rayleigh–Taylor (RT) modes in the Earth's atmosphere in the presence of dusty plasma. We use fluid equations and the dynamics of charged dust grains to obtain an eigenmode equation to study the stability of the RT mode. Typical ionospheric parameters are used to investigate the stability of the mode. The growth rate of the RT is found to be considerably higher in a dusty plasma than in a pure plasma. This might be the clue in explaining the origin of low-frequency modes in the ionosphere. We also study the effects of the dust charge, the dust mass, the density scale length and the dust density on the growth rate of the instability.

Torsional Alfvén waves in small scale current threads of the solar corona

Context. The magnetic field structuring in the solar corona occurs on large scales (loops and funnels), but also on small scales. For instance, coronal loops are made up of thin strands with different densities and magnetic fields across the loop. Aims. We consider a thin current thread and model it as a magnetic flux tube with twisted magnetic field inside the tube and straight field outside. We prove the existence of trapped Alfven modes in twisted magnetic flux tubes (current threads) and we calculate the wave profile in the radial direction for two different magnetic twist models. Methods. We used the Hall MHD equations that we linearized in order to derive and solve the eigenmode equation for the torsional Alfven waves. Results. We show that the trapped Alfven eigenmodes do exist and are localized in thin current threads where the magnetic field is twisted. The wave spectrum is discrete in phase velocity, and the number of modes is finite and depends on the amount of the magnetic field twist. The phase speeds of the modes are between the minimum of the Alfven speed in the interior and the exterior Alfen speed. Conclusions. Torsional Alfven waves can be guided by thin twisted magnetic flux-tubes (current threads) in the solar corona. We suggest that the current threads guiding torsional Alfven waves, are subject to enhanced plasma heating due to wave dissipation.

Modified finite-difference frequency-domain method for two-dimensional metallic photonic crystal analysis

In contrast to dielectric photonic crystals, the propagation characteristics of metallic photonic crystals are of great importance in millimeter wave and submillimeter wave applications. It is convenient and reliable to get the band diagrams and field distributions of photonic crystals after solutions of the eigenmode equations, which is derived from the Yee-mesh-based finite-difference frequency-domain method. However, this method cannot be used for the analysis of metallic photonic crystals because of the essential distinctions between metal and dielectric. Based on this method, we derive eigenmode equations for two-dimensional metallic photonic crystals by introducing the metal surface boundary conditions. And then, after some numerical calculations, the transverse electric mode and the transverse magnetic mode global band gaps of different lattice structures are obtained, including both square lattice and triangular lattice. Finally, we discuss the advantages of metallic periodic structures in mode selection and device integration by the comparison between metallic photonic band gap and dielectric photonic band gap.

Flute-Mode Stability of Quadrupole-Anchored Tandem Mirror Plasmas

We study the flute-mode stability of quadrupole-anchored tandem mirror plasmas. The present analysis is based on Newcomb's Lagrangian density with an assumption of small Larmor radius of ions for paraxial approximation. A radial eigenmode equation for flute perturbations is derived without an eikonal approximation in flux coordinates, where effects of E × B plasma rotation due to an ambipolar electric field are considered. The obtained eigenmode equation is applicable to a mode with an arbitrary azimuthal mode number and can describe interchange, E × B rotational, and Kelvin-Helmholtz modes driven by the shear effect of E × B plasma rotation. A quadratic dispersion equation in perturbation frequency ω is derived based on a simplified cylindrical plasma model and is used to discuss the flute stability of quadrupole-anchored GAMMA 10 tandem mirror plasmas, where the anchor beta required for plasma stability is calculated for the m = 1 and 2 modes for different values of central-cell ambipolar potential.

Beta-induced Alfvén-acoustic eigenmodes in stellarator plasmas with low shear

The coupling of low-frequency Alfvénic modes with acoustic oscillations due to curvature of the background magnetic field is considered for stellarator plasmas with low shear. Magnetohydrodynamic (MHD) analysis demonstrates that the interaction between these branches can generate gaps in the continua with a width proportional to beta as well as the magnitude of the Fourier harmonics of the magnetic field strength which cause the coupling. The gaps can provide a habitat for beta-induced Alfvén-acoustic eigenmodes (BAAEs). Using the causality principle, a technique is developed to resolve the singular behavior of the MHD BAAE eigenmode equation at the points of resonance with the acoustic continuum. Alternatively, the singularities arising in the reduced MHD description can be resolved by accounting for the finite parallel electrical field. Both approaches yield consistent continuum damping rate, which proves to be small. Numerical calculations for analytically fitted experimental profiles of electron-dominated plasma in Helically Symmetric eXperiment (HSX) facility yield two weakly damped BAAE modes with different frequencies: one is close to the maximum of the lower-frequency Alfvén-acoustic continuum, and the other is located well within the BAAE gap. The numerically found BAAEs have frequencies in the same range as the experimentally observed electromagnetic modes in HSX, even when the finite diamagnetic frequency effects are considered.

Single-fluid stability of stationary plasma equilibria with velocity shear and magnetic shear

By using incompressible single-fluid equations with a generalized Ohm’s law neglecting the electron inertia, a linear eigenmode equation for a magnetic field perturbation is derived for stationary equilibria in a slab geometry with velocity and magnetic shears. The general eigenmode equation contains a fourth-order derivative of the perturbation in the highest order and contains Alfvén and whistler mode components for a homogeneous plasma. The ratio of the characteristic ion inertia length to the characteristic inhomogeneity scale length is chosen as a small parameter for expansion. Neglecting whistler mode in the lowest order, the eigenmode equation becomes a second-order differential equation similar to the ideal magnetohydrodynamic eigenmode equation except for the fact that the unperturbed perpendicular velocity contains both electric and ion diamagnetic drifts. A sufficient condition for stability against the Kelvin–Helmholtz instability driven by shear in the ion diamagnetic drift velocity is derived and then applied to tokamaks.

Helicon waves in the magnetotail

[1] The helicon modes are studied in Earth's magnetotail using the Harris sheet as a model of its equilibrium. The plasma sheet is divided into two regions: In region 1 both the electrons and the ions are treated as magnetized, whereas in region 2 the electrons are treated as magnetized but the ions are unmagnetized. The eigenmode equation for the low-frequency, parallel propagating right-hand polarized modes is solved for the two cases, namely, (1) the uniform regions and (2) the inhomogeneous case. The eigenvalues and the eigenfunctions are obtained for the symmetric and antisymmmetric helicon modes. The case of uniform regions predicts lower frequencies for both the symmetric and antisymmetric modes than that of the inhomogeneous case. For typical plasma sheet parameters, the wave frequencies are found to be in the range of ∼(1–50) mHz for the symmetric modes, and ∼(5–100) mHz for the antisymmetric helicon modes. The wavelengths (along x direction) associated with the helicon modes are in the range of ∼(0.3–9) RE. The antisymmetric mode does not allow flux crossing across the neutral sheet axis. The symmetric helicon mode does allow flux crossing leading to the change in the magnetic topology, and thus, it must involve reconnection.

Compressional Alfvén eigenmode structure in spherical tokamaks

The two-dimensional structure of compressional Alfvén eigenmodes (CAEs) below the ion cyclotron frequency is studied numerically by solving the cold plasma Hall-MHD equations for a realistic spherical tokamak equilibrium. The simplest of the computed eigenmodes have a standing wave-like structure and the higher frequency solutions show a more travelling wave-like behaviour. The effects of the equilibrium current and the Hall terms in the eigenmode equation are investigated, and these terms are found to shift the frequencies of eigenmodes with different sign of the toroidal mode number away from each other. A classification scheme is proposed which relates each spherical tokamak eigenmode to the three mode numbers of the corresponding large aspect ratio circular cross section eigenmode obtained by gradually decreasing the elongation and increasing the aspect ratio.

Modelling of graded index waveguide fabricated by ion exchange on Er3+ doped glass

A general numerical tool, based on thermal diffusion equation and full-vectorial eigen-mode equation, has been presented for the systematic analysis of graded index channel waveguide fabricated by ion exchange on Er3+ doped glass. Finite difference method with full-vectorial formulation (FV-FDM) is applied to solving the full-vectorial modes of graded index channel waveguide for the first time. The coupled difference equations based on magnetic fields in FV-FDM are derived from the Taylor series expansion and accurate formulation of boundary conditions. Hybrid nature of vectorial guided modes for both pump (980 nm) and signal light (1550 nm) are demonstrated by the simulation. Results show that the fabrication parameters of ion exchange, such as channel opening width and time ratio of second step to first step in ion exchange, have large influence on the properties of waveguide. By optimizing the fabrication parameters, maintenance of monomode for signal light and improvement of the gain dynamics can be achieved in Er3+ doped waveguide amplifier (EDWA) fabricated by ion exchange technique. This theoretical model is significant for the design and fabrication of EDWA with ion exchange technique. Furthermore, a single polarization EDWA, which operates at wavelength from 1528 nm to 1541 nm for HE polarization, is numerically designed.

Ideal instabilities in a high-β rotating cylindrical plasma in the presence of an azimuthal magnetic field and a gravitational field

Magnetohydrodynamic (MHD) theory of ideal instabilities in a high-β rotating cylindrical plasma with an azimuthal magnetic field and a radial gravitational field is developed (β is the ratio of the plasma and magnetic field pressures). The basis of this theory is a system of two first-order differential equations for the Frieman–Rotenberg variable (the sum of the perturbed plasma and magnetic field pressures) and the radial plasma displacement, which leads to the second-order differential equation for the displacement. The sausage instability criterion is derived which generalizes the earlier results for a plasma without gravitation. It is shown that this instability can occur for both the decreasing and increasing plasma pressures. The non-axisymmetric modes are also considered. This analysis is related to the MHD instability theory in a nonrotating plasma dealing with snake instabilities. A number of rotational and gravitational effects on both the m = 1 and m > 1 modes are revealed, where m is the azimuthal mode number. The eigenmode equation describing the Suydam modes in the presence of rotational and gravitational effects is derived. These modes can be responsible, in particular, for the Velikhov and rotational-convective instabilities.

Ion temperature gradient driven instability in high beta plasmas of a sheared slab

A set of simplified integral eigenmode equations with neglecting of the magnetic gradient drift effects are derived and employed to investigate the ion temperature gradient modes in plasmas of sheared slab geometry. Perturbations of both perpendicular and parallel components of magnetic vector potential are considered, as well as the perturbation of the electrostatic potential. The characteristics of the ion temperature gradient instability are studied with the equations and the corresponding computer code, and compared with those with the drift effects included. The results show that the magnetic gradient drift has a block effect on ion temperature gradient mode stabilization, and the perpendicular component of the perturbed magnetic vector potential has strong stabilizing effects on ion temperature gradient modes. Nevertheless, the results from the simplified equations are qualitatively in agreement with those from the complete equations and the differences between them are not significant quantitatively. The simplified equations have the advantages of saving computer time and being well suited to analytic methods.

Effect of Elongation on Critical Gradient for Toroidal Electron Temperature Gradient Modes

The electron temperature gradient mode is investigated in elongated toroidal plasmas with a gyrokinetic integral eigenmode equation code. Dependence of the critical electron temperature gradient on the elongation is calculated. It is found that when the elongation increases, the growth rate spectrum is greatly shifted towards shorter poloidal wavelength, and then the poloidal wavenumber at which the mode is destabilizing critically in elongated plasmas will be larger than that in circular plasmas.

A semi-analytic method for solving rib-type waveguide problems

A new semi-analytic method for solving optical rib-type waveguide problems is presented. In the method, the cross-section of a rib-type waveguide is divided into several regions. In each region, the refractive index profile and field distribution are expanded into Fourier cosine series, and then are substituted in the wave equation. A second-order differential matrix equation is then derived for each region, with a closed-form solution obtainable. With the boundary conditions used, an eigenmode equation for the rib waveguide can be derived and solved numerically to give the modal indices. Here, the presented method is used to deal with two rib waveguides in different geometric dimensions and/or compositions, respectively. Computational results show that the presented method is quite efficient, in terms of CPU time, in finding the modal indices accurately. The relative error in computing the modal index with the method is about 10− 5–10− 6.

Effects of Finite Aspect Ratio and Noncircular Plasma Flux Surface on Electron Temperature Gradient Driven Modes

A gyrokinetic model with integral eigenmode equations is developed based on the local equilibrium of shaped tokamak plasmas. Effects of main geometric parameters (finite aspect ratio, elongation, triangularity, and Shafranov shift gradient) on the electrostatic electron temperature gradient (ETG) driven modes are investigated numerically. It is found that the finite aspect ratio has a general stabilizing effect, while the elongation can be either stabilizing or destabilizing, depending on the poloidal wavelength of the mode and other parameters. It is shown that a low aspect ratio enhances the stabilizing effect of elongation, and weakens its destabilizing effect as well.

Lateral leakage in symmetric SOI rib-type slot waveguides

We theoretically investigate the lateral leakage in fully symmetric horizontal rib-type slot waveguides caused by coupling between the TM-like slot mode and a TE slab mode. The leakage mechanism is described and the dependence on the geometry parameters is studied using the variational mode-matching method. In addition, the effective-index method to solve the transcendent eigenmode equation of rib-type slot waveguides is applied and compared with the aforementioned rigorous numerical method. We show that this semi-analytical approach yields results with sufficiently high accuracy. For design purposes the influence of structural deviations caused by the fabrication process is studied. We show that the usable range of the geometry parameters can be widely extended by exploiting leakage suppressing mechanisms.

Spectral Effect of Zonal Flows and Enhanced Growth Rate

The effect of the spectrum of the radial wave number of zonal flows on zonal flow generation is theoretically investigated using the Hasegawa-Mima turbulence model by representing the spectrum by means of two monochromatic waves. Based on this method, we explored a ten-wave coupling model of modulational instability. We found that the zonal flow generation is qualitatively different in cases with and without such a spectral effect, exhibiting the enhancement of the growth rate. This originates from the coupling property of the new zonal flow eigenmode equation system. We refer to this state as a coupled zonal flows eigenmode, which leads to a spatial modulation of zonal flows affected by the turbulence structure.

MHD Stability in Flowing Plasmas: Connection between Fusion Plasma and Astrophysics Research

Axisymmetric magneto-rotational instability (MRI) is studied in comparison with interchange instability (IntI) in a rotating cylindrical plasma. MRI is driven by the shear of plasma rotation, and the IntI by the density gradient with effective gravity due to the plasma rotation. The eigenmode equation for the MRI has the same form as that for the IntI. The local stability criterion is also summarized in a similar statement as “the spatial gradient of centrifugal force greater than the square of Alfven frequency causes instability.” However, the MRI is essentially different from the IntI because of the non-Hermitian property. The Keplerian rotation generates irregular singularity at the center of the disk, which yields a continuum of eigenvalues with non-orthogonal and square-integrable eigenfunctions.

M = 1 ideal internal kink modes in a line-tied screw pinch

It is well known that the radial displacement of the m=1 internal kink mode in a periodic screw pinch has a steep jump at the resonant surface where k∙B=0 [Rosenbluth, Dagazian, and Rutherford, Phys. Fluids 16, 1894 (1973)]. In a line-tied system, relevant to solar and astrophysical plasmas, the resonant surface is no longer a valid concept. It is then of interest to see how line-tying alters the aforementioned result for a periodic system. If the line-tied kink also produces a steep gradient, corresponding to a thin current layer, it may lead to strong resistive effects even with weak dissipation. Numerical solution of the eigenmode equations shows that the fastest growing kink mode in a line-tied system still possesses a jump in the radial displacement at the location coincident with the resonant surface of the fastest growing mode in the periodic counterpart. However, line-tying thickens the inner layer and slows down the growth rate. As the system length L approaches infinity, both the inner layer thickness and the growth rate approach the periodic values. In the limit of small ϵ∼Bϕ∕Bz, the critical length for instability Lc∼ϵ−3. The relative increase in the inner layer thickness due to line-tying scales as ϵ−1(Lc∕L)2.5.

Short Wavelength Ion Temperature Gradient Driven Instabilityin Noncircular Flux Surface Plasmas with Finite Aspect Ratio

By employing the local equilibrium of shaped tokamak plasmas, agyrokinetic model with integral eigenmode equations is developed toinvestigate effects of the finite aspect ratio and noncircular fluxsurface on short wavelength ion temperature gradient (SWITG) drivenmodes. It is found that when nonadiabatic electron and trapped particleeffects are not considered, the SWITG mode can be stabilized by finiteaspect ratio A, elongation κ and triangularity δ, andcan be destabilized by the Shafranov shift gradient∂R0/∂r.

Stability of Tokamak Equilibrium with Internal Transport Barrieragainst High-n MHD Ballooning Mode

A new eigen-mode equation for the tokamak high-n (the toroidalmode number) ideal magnetohydrodynamic (MHD) ballooning mode in tokamakplasmas is derived to include the toroidal effects that are significantfor stability of configurations with internal transport barriers(ITBs). For tokamak equilibria of shift circular flux surfaces, thesetoroidal effects basically are the finite inverse aspect ratio and theShafranov shift. The former yields the averaged favourable curvaturestabilization while the latter further strengthens this effect, leadingto a low shear stable channel connecting the first and second stabilityregions, and to the shrinkage of unstable region in the (s,α)diagram. The dependence of the critical shear, below which the plasma isstable, on these effects is given. These results are important forunderstanding the ITB physics in some regards.