Gradient Scale Length Research Articles

Growth of Rayleigh instability in a Hall thruster channel having dust in exit region

In a Hall thruster, the axial movement of ions towards the exit of the chamber and the divergence of ion beam sputter the wall and thus the maximum erosion of wall of the exit region takes place. This way the dust is generated, which is largely confined near the exit of the channel. By considering dust particles with negative charge, we derive and solve the dispersion equation for the growth of Rayleigh instability in the thruster plasma and examine its growth rate for different density, charge, mass and temperature of the dust particles. We also talk about the role of density gradient scale length on the growth of the instability.

Double layers in an inhomogeneous magnetized bi‐ion plasma

A simpler analytical approach is employed to obtain energy integral equation for a pseudo‐particle in a pseudo‐potential, which admits double layer (DL) solutions for the non‐linear low‐frequency electrostatic perturbations in non‐uniform plasma consisting of electrons and two kinds of ions. One of the ion species has field‐aligned shear flow and electrons are superthermal kappa distributed. This theoretical model is applied to the upper ionospheric oxygen‐dominated plasma that has small concentration of protons along with upward flow of oxygen ions. Under suitable boundary conditions, both rarefactive (density dip) and compressive (density hump) DLs are obtained solving energy integral equation using the plasma parameters of ionosphere around altitude of 800 km. The amplitude and width of the DLs depend upon the scale lengths of density and temperature gradients as well as on the ratio of equilibrium densities of oxygen and hydrogen.

Resonance behavior of functionally graded polymer composite nanoplates reinforced with graphene nanoplatelets

For the first time, forced resonance vibration of Graphene Nano-Platelets (GNPs) reinforced Functionally Graded Polymer Composite (FG-PC) nanoplates is studied. The effective Young's modulus is determined using the Halpin–Tsai model while the rule of mixture is used to compute the effective Poisson's ratio and mass density. The governing equations, classical and non-classical boundary conditions are obtained through Hamilton's principle for nonlocal strain gradient Kirchhoff plate theory. Employing Navier solution procedure, a closed form solution is introduced for forced resonance vibration of the nanoplate. The influences of the GNPs distribution schemes, nonlocal and strain gradient length scale parameters, weight fraction and the total number of layers of GNPs as well as geometrically parameters are discussed in detail. The results show that the impact of layer's number on the resonance position depends on the reinforcement patterns.

Characteristics of elastic waves in radial direction of anisotropic solid sphere, a new closed-form solution

This study aims to study the radial wave dispersion of anisotropic solid sphere. In order to include the size-dependent effects, the nonlocal strain gradient theory in conjunction with three dimensional (3D) elasticity theory is used. This model presents both stiffness softening and hardening mechanisms of nanostructures using a nonlocal stress field parameter and also a strain gradient length scale parameter. The nanoparticles are made of different crystallinity (i.e. gold, beryllium oxide, rutile, and hematite), while the equations of wave motion are obtained based on mentioned theories in which an analytic technique is utilized to find the wave frequency, phase velocity and group velocity as a function of wave number. The reliability of present approach is verified for radial vibration by comparing obtained results with those provided in literature. It is demonstrated that, radius of nanoparticle, small-scale parameters, wave number, and crystallinity of nanoparticle have a remarkable effect on radial wave characteristics of anisotropic solid sphere. In addition, the possibility of replacing an anisotropic model with isotropic ones is studied.

Few-cycle laser wakefield acceleration on solid targets with controlled plasma scale length

We measure the emission of energetic electrons from the interaction between relativistic-intensity ultrashort laser pulses and a solid density plasma with a tunable density gradient scale length. We detect an electron beam that only appears with few-cycle pulses (<10 fs) and large plasma scale lengths (L > λ0). Numerical simulations, in agreement with the experiments, reveal that these electrons are accelerated by a laser wakefield. Plasma waves are indeed resonantly excited by the few-cycle laser pulses in the near-critical density region of the plasma. Electrons are then injected by ionization into the plasma waves and accelerated to relativistic energies. In this laser wakefield acceleration regime, the plasma waves are rotated by the plasma density gradient, which results in the electrons not being emitted in the same direction as the driving laser pulse.

Fluttering and divergence instability of functionally graded viscoelastic nanotubes conveying fluid based on nonlocal strain gradient theory.

In this paper, a size-dependent viscoelastic pipe model is developed to investigate the size effects on flutter and divergence instability of functionally graded viscoelastic nanotubes conveying fluid. The nonlocal strain gradient theory and the Kelvin-Voigt model are used to consider the significance of nonlocal field, strain gradient field, and viscoelastic damping effects. The dimensionless equation of transverse motion and related classical and non-classical boundary conditions are derived using the variational approach. The partial differential equations are discretized to a system of ordinary differential equations by the use of Galerkin's method. The frequency equation is obtained as a function of dimensionless flow velocity, small-scale parameters, damping coefficient, and power-law parameter. Numerical results are presented to study the dynamical behavior of the system and are compared with experimental and theoretical results reported by other researchers. Coupled and single mode fluttering related to higher vibration modes of fluid-conveying nanotubes supported at both ends are studied for the first time. It is found that coupled mode fluttering can be seen for different vibration modes by increasing the flow velocity in the absence of structural damping. Structural damping changes the dynamical behavior of the system, in which by increasing the flow velocity, single mode fluttering occurs instead of coupled mode fluttering. In addition, the presence of structural damping increases the critical flow velocity and, as a result, increases the stability of the system. The results also show that increasing the nonlocal parameter will have a stiffness-softening effect, while increasing the strain gradient length scale has an opposing effect.

Mathematical modeling and stochastic stability analysis of viscoelastic nanobeams using higher-order nonlocal strain gradient theory

This paper analyzes stochastic vibrations of a viscoelastic nanobeam under axial loadings. Based on the higher-order nonlocal strain gradient theory and the Liapunov functional method, bounds of the almost sure asymptotic stability of a nanobeam are obtained as a function of retardation time, variance of the stochastic force, higher-order and lower-order scale coefficients, strain gradient length scale, and intensity of the deterministic component of axial loading. Analytical results from this study are first compared with those obtained from the Monte Carlo simulation. Numerical calculations are performed for the Gaussian and harmonic non-white processes as models of axial forces.

Statistical properties of the plasma fluctuations and turbulent cross-field fluxes in the outboard mid-plane scrape-off layer of Alcator C-Mod

The Alcator C-Mod mirror Langmuir probe diagnostic is used to characterize plasma fluctuations in the outboard mid-plane scrape-off layer (SOL). Ohmically heated L-mode plasmas with Greenwald fractions from 0.1 to 0.45 are analyzed. In plasmas with a low line-averaged core density, the radial profiles of the average electron density and temperature feature a two-scale structure with a strong gradient in the near-SOL and a weak gradient in the far-SOL. The relative fluctuation levels of the electron density are below 25% throughout the entire SOL and the fluctuation driven heat flux is dominated by plasma density fluctuations. In the plasma with the highest line-averaged core density, the radial profiles of the electron density and temperature feature only a single weak gradient length scale across the SOL. The plasma density at the limiter radius is approximately 10 times larger than in low-density discharges. The average electron temperature at the separatrix falls to approximately 30 eV while the magnitude of the relative temperature fluctuations increases with the line-averaged core plasma density. With increasing plasma density, the fluctuation-driven heat fluxes increase by more than one order of magnitude while retaining a relative fluctuation level of approximately 3. Even though the SOL is on average cooler than in low density discharges, large electron temperature fluctuations govern the turbulence driven heat flux towards the wall while contributions from density fluctuations and triple correlations contribute approximately 35% and 15%, respectively.

Wave Propagation of Porous Nanoshells.

This study aims at investigating the wave propagation of porous nanoshells. The Bi-Helmholtz non-local strain gradient theory is employed in conjunction with a higher-order shear deformation shell theory, in order to include the size-dependent effects. The nanoshells are made of a porous functionally graded material (P-FGM), whose properties vary continuously along the thickness direction. A variational approach is here applied to handle the governing equations of the problem, which are solved analytically to compute the wave frequencies and phase velocities as function of the wave numbers. The sensitivity of the wave response is analyzed for a varying porosity volume fraction, material properties, non-local parameters, strain gradient length scales, temperature, humidity, and wave numbers. Based on the results, it is verified that the size-dependence of the response is almost the same to the one of plates, beams and tubes.

Characterization of a Knudsen force based vacuum sensor for N2[sbnd]H2O gas mixtures

Radiometric phenomena can arise in gas flows in which the local molecular mean free path is in the same order as the temperature gradient length scale. The forces exerted on immersed bodies by such rarefied flows are known as Knudsen (or thermophoretic) forces and can play a major role in nonisothermal microflows. In this work, the direct simulation Monte Carlo (DSMC) method is used to numerically characterize the force and thermal mechanisms on a second generation Microscale In-Plane Knudsen Radiometric Actuator (MIKRA) designed for gas pressure and composition sensing. First, a brief verification analysis of the present DSMC solver in reproducing the thermal conductivity for the gas species of interest – nitrogen and water vapor – is carried out. The geometric impact of the gap between MIKRA beams is then investigated for pure N2 environments. In addition, the beam heat transfer coefficients for N2H2O binary mixtures are extracted from the DSMC data and a dimensionless correlation is proposed for predicting the corresponding Knudsen forces. These results are intended to assist in the operation and design of the MIKRA device.

Propagation of input parameter uncertainties in transport models

The many sources of uncertainty in validation studies of plasma turbulence in magnetically confined fusion devices are well-known. In this paper, we investigate how to efficiently transform uncertainties in experimentally derived transport model inputs into model prediction uncertainties, using the quasilinear trapped-gyro-Landau-fluid (TGLF) turbulent transport model [Staebler et al., Phys. Plasmas 14, 055909 (2007)]. We use the rapidly converging and computationally inexpensive non-intrusive probabilistic collocation method (PCM) to propagate input parameter uncertainty probability distribution functions (PDFs) through TGLF, yielding PDFs of predicted transport fluxes. We observe in many cases that the flux PDFs exhibit significant non-normal features such as strong skewness, even when the input distributions were normal. To illustrate the utility of the PCM approach, we apply this methodology to transport predictions for a DIII-D ITER baseline plasma [Grierson et al., Phys. Plasmas 25, 022509 (2018)] in which the mix of neutral beam injection (NBI) and electron cyclotron heating (ECH) was varied. The model predictions show clear changes in the parametric dependencies and sensitivities of the turbulence between the two heating mixes. Specifically, when only NBI heating was used, the transport fluxes responded significantly only to the ion temperature gradient scale length. However, when both NBI and ECH were applied, the electron transport channels demonstrate a strong sensitivity to the electron temperature and density gradients not observed in the NBI-only case. Additional context for the PCM approach is provided by comparing its predictions with those obtained via a local flux-matching approach. A new set of validation metrics based on the Wasserstein distance is proposed for PDF-based comparisons.

Scaling trends of the critical E × B shear for edge harmonic oscillation onset in DIII-D quiescent H-mode plasmas

Quiescent H-mode (QH-mode) has been identified as an attractive stationary operational regime in tokamaks due to its lack of edge localized modes (ELMs), along with good particle and impurity control aided by the presence of magnetohydrodynamic modes such as the edge harmonic oscillation (EHO) or edge turbulence. Experiments on the DIII-D tokamak explore local access conditions for QH-mode through measurements of the critical edge rotational shear necessary for the transition from a QH-mode with a coherent EHO to a typical ELMy H-mode. The critical E × B shear and EHO frequency are predicted by a nonlinear phase-dynamics model relating the pressure and velocity perturbations in the edge pedestal region. The reduced theoretical model predicts a linear relationship between critical shearing rate and , where is the ion acoustic velocity, Lp the pressure gradient scale length, and the radial width of the mode. This scaling of the critical shearing rate agrees with the experimental trend, although the absolute magnitude of the shearing rate threshold is over-predicted by the model. Through a normalized predicted scaling, the model demonstrates the dynamic transition into and out of QH-mode qualitatively, within a single plasma discharge. The experimental comparison lends insight into improving the theoretical model by including more accurate geometry and toroidal mode number physics for more accurate quantitative predictions.

Upgrade of the ECE diagnostic on EAST.

The electron cyclotron emission (ECE) diagnostic on the experimental advanced superconducting tokamak (EAST) was upgraded recently to provide electron temperature profile measurement with wider radial coverage and better precision. The lower limit of the ECE detection frequency band was extended from 104 GHz to 97 GHz by adding a new 8-channel heterodyne radiometer, which ensures capability for the measurement of the second harmonic ECE with toroidal magnetic field down to 1.75 T. Also, the existing 32-channel heterodyne radiometer has been upgraded, with the frequency interval for the lower frequency range up to 120 GHz reduced from 2 GHz to 1 GHz by introducing eight channels in the intermediate frequency part. In addition, a plan is presented to incorporate tunable yttrium iron garnet filters into the existing heterodyne radiometer to obtain detailed measurements of the electron temperature gradient scale length as well as finer spatial pinpointing of magnetohydrodynamic modes. Examples from DIII-D are provided where similar ECE diagnostic allowed precise measurement of the center and width of neoclassical tearing modes.

Whistler modes in highly nonuniform magnetic fields. I. Propagation in two-dimensions

The propagation of whistler wave packets is studied in a large laboratory plasma with nonuniform ambient magnetic fields. The wave packets are excited by magnetic loop antennas and, for uniform fields, have the topology of helicon modes in unbounded plasmas. The waves propagate across the separatrix of X-type nulls and across circular field lines of magnetic islands. Classical wave refraction is not observed when the magnetic gradient scale length becomes shorter than the wavelength. Since the ambient magnetic field is generated by a line current, it is two dimensional and the wave magnetic field is measured and displayed in transverse planes. Using wave bursts, the space-time evolution of wave packets is observed. The group velocity has been measured. Waves can cross a separatrix and refract into nearly parallel whistler modes. The phase rotation of helicons is lost on a strongly curved field. In highly nonuniform magnetic fields, the phase velocity is determined from multipoint magnetic hodograms, the group velocity from the Poynting vector fields. The energy flow is highly field aligned while the phase flow can be highly oblique to the ambient magnetic field without encountering cyclotron resonance absorption. In the limiting case of circular field lines, counter propagating waves create azimuthal standing waves while still propagating radially. Standard concepts of plane wave refraction do not apply to whistler modes in highly nonuniform magnetic fields. Implications of these observations on research in space plasmas and helicon devices will be pointed out.

A four-field gyrofluid model with neoclassical effects for the study of the rotation velocity of magnetic islands in tokamaks

A four-field system of equations which includes the neoclassical flow damping effects and the lowest-order finite-Larmor-radius (FLR) corrections is deduced from a system of gyrofluid equations. The FLR corrections to the poloidal flow damping are calculated by solving a simplified version of the gyrokinetic equation. This system of equations is applied to the study of a chain of freely rotating magnetic islands in a tokamak, resulting from the nonlinear evolution of a resistive tearing mode, to determine the island rotation velocity consistently with the fields' radial profiles close to the resonant surface. The island rotation velocity is determined by imposing the torque balance condition. The equations thus deduced are applied to the study of two different collisionality regimes, namely the weak-damping regime and the intermediate-damping regime. The equations reduce, in the weak-damping regime, to a form already obtained in previous works, while an additional term, containing the lowest order FLR corrections to the poloidal flow damping, appears in the intermediate-damping regime. The numerical integration of the final system of equations allows the determination of the dependence of the island rotation velocity on the plasma collisionality and the island width compared to the ion Larmor radius. The results show that, in the intermediate-damping regime, the island rotation velocity is almost completely determined by the neoclassical effects, with the island width playing a minor role. The parameter ηi=Ln/LT, where Ln and LT are the density and temperature gradient length scales, plays an important role in determining the island rotation velocity.

Scaling of spoke rotation frequency within a Penning discharge

A rotating plasma spoke is shown to develop in two-dimensional full-sized kinetic simulations of a Penning discharge cross-section. Electron cross-field transport within the discharge is highly anomalous and correlates strongly with the spoke phase. Similarity between collisional and collisionless simulations demonstrates that ionization is not necessary for spoke formation. Parameter scans with discharge current Id, applied magnetic field strength B, and ion mass mi show that the spoke frequency scales with eErLn/mi, where Er is the radial electric field, Ln is the gradient length scale, and e is the fundamental charge. This scaling suggests that the spoke may develop as a non-linear phase of the collisionless Simon-Hoh instability.

Random vibrations of functionally graded nanobeams based on unified nonlocal strain gradient theory

In present study the dynamic response of a functionally graded nanobeam supported by visco-elastic foundation to a stationary random excitation, based on non-local strain gradient theory has been investigated for the first time. The governing equations of motion are derived in framework of a unified beam theory (which includes different beam theories such as Euler–Bernoulli, Timoshenko and Reddy beam theories as special case), and Hamilton’s principle. The results of present study for some limit cases have been compared with results available in open literature and excellent agreement is observed. Effects of various parameters (i.e. non-local parameter, strain gradient length scale, power-low index of FGM model, different beam theories etc.) have been investigated comprehensively and illustrated graphically.

Nonlinear ballooning modes in tokamaks: stability and saturation

The nonlinear dynamics of magneto-hydrodynamic ballooning mode perturbations is conjectured to be characterised by the motion of isolated elliptical flux tubes. The theory of stability, dynamics and saturation of such tubes in tokamaks is developed using a generalised Archimedes’ principle. The equation of motion for a tube moving against a drag force in a general axisymmetric equilibrium is derived and then applied to a simplified ‘s–α’ equilibrium. The perturbed nonlinear tube equilibrium (saturated) states are investigated in an ‘s–α’ equilibrium with specific pressure and magnetic shear profiles. The energy of these nonlinear (ballooning) saturated states is calculated. In some cases, particularly at low magnetic shear, these finitely displaced states can have a lower energy than the equilibrium state even if the profile is linearly stable to ballooning modes (infinitesimal tube displacements) at all radii. Thus nonlinear ballooning modes can be metastable. The amplitude of the saturated tube displacement in such cases can be as large as the pressure gradient scale length. We conjecture that triggering a transition into these filamentary states can lead to hard instability limits. A short survey of different pressure profiles is presented to illustrate the variety of behaviour of perturbed elliptical flux tubes.

The study of micro-inextensible piezoelectric cantilever plate

In this paper, a micro-inextensible piezoelectric cantilever plate is analyzed and its nonlinear dynamic behaviour is studied. The nonlinear oscillation differential equation is established by using Hamilton’s principle with the application of strain gradient theory to consider the size effect, and inextensible theory to consider the large deformation and rotation effect of cantilever plate. Based on MATLAB software, using the Runge-Kuta method, we can obtain the response of the nonlinear oscillation differential equation. The influences of the strain gradient length scale parameter and voltage on the dynamic response of micro piezoelectric cantilever plate are investigated separately. The results confirmed an increase of the stiffness of the system by using the strain gradient theory and the amplitude of the vibration is reduced. The vibration of the system can be controlled by applying an active voltage. The effect of external excitation frequency on nonlinear dynamic behaviour is considered by using Poincare surface of section and diagrams of waveforms, phase and bifurcation.

Compressive and rarefactive double layers in non-uniform plasma with q $q$ -nonextensive distributed electrons

Electrostatic solitary waves and double layers (DLs) formed by the coupled ion acoustic (IA) and drift waves have been investigated in non-uniform plasma using \(q\)-nonextensive distribution function for the electrons and assuming ions to be cold \(T_{i}< T_{e}\). It is found that both compressive and rarefactive nonlinear structures (solitary waves and DLs) are possible in such a system. The steeper gradients are supportive for compressive solitary (and double layers) and destructive for rarefactive ones. The \(q\)-nonextensivity parameter \(q\) and the magnitudes of gradient scale lengths of density and temperature have significant effects on the amplitude of the double layers (and double layers) as well as on the speed of these structures. This theoretical model is general which has been applied here to the \(F\)-region ionosphere for illustration.