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DOE OSTI2012

Selective Plasmonic Gas Sensing: H2, NO2, and CO Spectral Discrimination by a Single Au-CeO2 Nanocomposite Film

A Au-CeO2 nanocomposite film has been investigated as a potential sensing element for high-temperature plasmonic sensing of H2, CO, and NO2 in an oxygen containing environment. The CeO2 thin film was deposited by molecular beam epitaxy (MBE) and Au was implanted into the as-grown film at an elevated temperature followed by high temperature annealing to form well-defined Au nanoclusters. The Au-CeO2 nanocomposite film was characterized by x-ray diffraction (XRD) and Rutherford backscattering spectrometry (RBS). For the gas sensing experiments, separate exposures to varying concentrations of H2, CO, and NO2 were performed at a temperature of 500°C in oxygen backgrounds of 5.0, 10, and ~21% O2. Changes in the localized surface plasmon resonance (LSPR) absorption peak were monitored during gas exposures and are believed to be the result of oxidation-reduction processes that fill or create oxygen vacancies in the CeO2. This process affects the LSPR peak position either by charge exchange with the Au nanoparticles or by changes in the dielectric constant surrounding the particles. Multivariate analysis was used to gauge the inherent selectivity of the film between the separate analytes. From principal component analysis (PCA), unique and identifiable responses were seen for each of the analytes. Linear discriminant analysis (LDA) was also used and showed separation between analytes as well as trends in gas concentration. Results indicate that the Au-CeO2 thin film is selective to O2, H2, CO, and NO2 in separate exposures. Combined with the observed stability over long exposure periods, the Au-CeO2 film shows good potential as an optical sensing element for harsh environmental conditions.

36 MATERIALS SCIENCE↗

DOE OSTI2013

Simulation of linear and nonlinear Landau damping of lower hybrid waves

The linear physics of lower hybrid waves (LHWs) and their nonlinear interaction with particles through Landau damping are studied with the gyrokinetic electron and fully kinetic ion (GeFi) particle simulation model in the electrostatic limit. Unlike most other wave modes, the LHWs can resonantly interact with both electrons and ions, with the former being highly magnetized and latter nearly unmagnetized around the lower hybrid frequency. Direct interactions of LHWs with electrons and/or ions are investigated for cases with various k{sub ∥}/k,T{sub i}/T{sub e}, and wave amplitudes. In the linear electron Landau damping (ELD), the dispersion relation and the linear damping rate obtained from our simulation agree well with the analytical linear theory. As the wave amplitude increases, the nonlinear Landau effects are present, and a transition from strong decay at smaller amplitudes to weak decay at larger amplitudes is observed. In the nonlinear stage, the LHWs in the long time evolution finally exhibit a steady Bernstein-Greene-Kruskal mode, in which the wave amplitude is saturated above the noise level. While the resonant electrons are trapped in the wave field in the nonlinear ELD, the resonant ions are untrapped in the LHW time scales. The ion Landau damping is thus predominantly in a linear fashion, leading to a wave saturation level significantly lower than that in the ELD. On the long time scales, however, the ions are still weakly trapped. The results show a coupling between the LHW frequency and the ion cyclotron frequency during the long-time LHW evolution.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1991

Approximations for inclusion of rotor lag dynamics in helicopter flight dynamics models

Approximate forms are suggested for augmenting linear rotor/body response models to include rotor lag dynamics. Use of an analytically linearized rotor/body model has shown that the primary effect comes from the additional angular rate contributions of the lag inertial response. Addition of lag dynamics may be made assuming these dynamics are represented by an isolated rotor with no shaft motion. Implications of such an approximation are indicated through comparison with flight test data and sensitivity of stability levels with body rate feedback.

Mckillip, Robert, Jr.↗

NASA NTRS1986

Toward comparing experiment and theory for corroborative research on hingeless rotor stability in forward flight (an experimental and analytical investigation of isolated rotor-flap-lag stability in forward flight)

For flap-lag stability of isolated rotors, experimental and analytical investigations are conducted in hover and forward flight on the adequacy of a linear quasisteady aerodynamics theory with dynamic inflow. Forward flight effects on lag regressing mode are emphasized. Accordingly, a soft inplane hingeless rotor with three blades is tested at advance ratios as high as 0.55 and at shaft angles as high as 20 degrees. The 1.62 m model rotor is untrimmed with an essentially unrestricted tilt of the tip path plane. In combination with lag natural frequencies, collective pitch settings and flap-lag coupling parameters, the data base comprises nearly 1200 test points (damping and frequency) in forward flight and 200 test points in hover. By computerized symbolic manipulation, a linear analytical model is developed in substall to predict stability margins with mode identificaton. To help explain the correlation between theory and data it also predicts substall and stall regions of the rotor disk from equilibrium values. The correlation shows both the strengthts and weaknesses of the theory in substall.

Gaonkar, G.↗

NASA NTRS2008

Production of Magnetic Turbulence by Cosmic Rays Drifting Upstream of Supernova Remnant Shocks

I will present results of our recent two- and three-dimensional Particle-In-Cell simulations of magnetic-turbulence production by cosmic-ray ions drifting upstream of supernova remnant shocks. These studies' aim is twofold: test recent predictions of strong amplification in short wavelength, non-resonant wave modes, and study the subsequent evolution of the magnetic turbulence, including its backreaction on cosmic-ray trajectories. We confirm that the drifting cosmic rays give rise to a turbulent magnetic field, but show that an oblique filamentary mode grows more rapidly than the non-resonant parallel modes found in analytical theory. The field perturbations grow more slowly than estimated using a quasi-linear analytical approach for the parallel plane-wave mode, and saturate in amplitude at deltaB/B approximately equal to 1. The backreaction of the magnetic turbulence on the particles leads to an alignment of the bulk-flow velocities of the cosmic rays and the background medium. This is an essential characteristic of cosmic ray-modified shocks: the upstream flow speed is continuously changed by the cosmic rays. The reduction of relative drift between cosmic rays and background medium accounts for the saturation of the instability at only moderate magnetic-field amplitudes. It is possible that the prolonged magnetic field growth observed in recent MHD simulations results from a cosmic-ray current assumed to be constant and thus immune to the backreaction from the turbulent field. We speculate that the parallel plane-wave mode found in analytical treatments very quickly leads co filamentation, which we observe in our PIC modeling and is also apparent in the MHD simulations.

Stroman, Thomas↗

DOE OSTI2024

Permutation-adapted complete and independent basis for atomic cluster expansion descriptors

Atomic cluster expansion (ACE) methods provide a systematic way to describe particle local environments of arbitrary body order. For practical applications it is often required that the basis of cluster functions be symmetrized with respect to rotations and permutations. Existing methodologies yield sets of symmetrized functions that are over-complete. These methodologies thus require an additional numerical procedure, such as singular value decomposition (SVD), to eliminate redundant functions. In this work, it is shown that analytical linear relationships for subsets of cluster functions may be derived using recursion and permutation properties of generalized Wigner symbols. From these relationships, subsets (blocks) of cluster functions can be selected such that, within each block, functions are guaranteed to be linearly independent. It is conjectured that this block-wise independent set of permutation-adapted rotation and permutation invariant (PA-RPI) functions forms a complete, independent basis for ACE. Along with the first analytical proofs of block-wise linear dependence of ACE cluster functions and other theoretical arguments, numerical results are offered to demonstrate this. The utility of the method is demonstrated in the development of an ACE interatomic potential for tantalum. Using the new basis functions in combination with Bayesian compressive sensing sparse regression, some high degree descriptors are observed to persist and help achieve high-accuracy models.

Angular momentum↗

DOE OSTI2014

An analytic model for buoyancy resonances in protoplanetary disks

Zhu et al. found in three-dimensional shearing box simulations a new form of planet-disk interaction that they attributed to a vertical buoyancy resonance in the disk. We describe an analytic linear model for this interaction. We adopt a simplified model involving azimuthal forcing that produces the resonance and permits an analytic description of its structure. We derive an analytic expression for the buoyancy torque and show that the vertical torque distribution agrees well with the results of the Athena simulations and a Fourier method for linear numerical calculations carried out with the same forcing. The buoyancy resonance differs from the classic Lindblad and corotation resonances in that the resonance lies along tilted planes. Its width depends on damping effects and is independent of the gas sound speed. The resonance does not excite propagating waves. At a given large azimuthal wavenumber k{sub y} > h {sup –1} (for disk thickness h), the buoyancy resonance exerts a torque over a region that lies radially closer to the corotation radius than the Lindblad resonance. Because the torque is localized to the region of excitation, it is potentially subject to the effects of nonlinear saturation. In addition, the torque can be reduced by the effects of radiative heat transfer between the resonant region and its surroundings. For each azimuthal wavenumber, the resonance establishes a large scale density wave pattern in a plane within the disk.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2018

Verification of 5D continuum gyrokinetic code COGENT: Studies of kinetic drift wave instability

COGENT (Continuum Gyrokinetic Edge New Technology) is a kinetic plasma simulation code that is being developed by the edge simulation laboratory (ESL) collaboration. The original version of the code has been developed in a 4D phase space (2D configuration space and 2D velocity space) to address kinetic plasma phenomena in a complex magnetic field geometry including the core, magnetic separatrix, and scrape-off layer regions. This paper is focused on extending the original 4D phase space to a 5D phase space (3D2V) to address full kinetic turbulences in a tokamak edge region. Here, we report on the current status of 5D COGENT, which presently operates in a shear-less, simple slab geometry. As a verification study, we use the problems of collision-less drift wave instability (universal instability) including its modification in the presence of collisional effects. The simulation model includes the gyrokinetic equations for ion and electron species coupled to the long-wavelength limit of the gyro-Poisson equation. Collisional effects are represented by the Krook collision model. Linear analytical results for the drift mode growth rate and real frequency are recovered, and the non-linear stage is modelled and analysed as well. In addition, extensive 5D runs have been performed to address the effects of the drift wave instability on blob/filamentary structures characteristic of a tokamak edge. Finally, a helical-shaped potential perturbation is observed to grow exponentially in time while spinning around the filament axis with electron drift frequency.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS2015

Description, Usage, and Validation of the MVL-15 Modified Vortex Lattice Analysis Capability

MVL-15 is the most recent version of the Modified Vortex-Lattice (MVL) code developed within the Aerodynamics Systems Analysis Branch (ASAB) at NASA LaRC. The term "modified" refers to the primary modification of the core vortex-lattice methodology: inclusion of viscous aerodynamics tables that are linked to the linear solution via iterative processes. The inclusion of the viscous aerodynamics inherently converts the MVL-15 from a purely analytic linearized method to a semi-empirical blend which retains the rapid execution speed of the linearized method while empirically characterizing the section aerodynamics at all spanwise lattice points. The modification provides a means to assess non-linear effects on lift that occur at angles of attack near stall, and provides a means to determine the drag associated with the application of design strategies for lift augmentation such as the use of flaps or blowing. The MVL-15 code is applicable to the analyses of aircraft aerodynamics during cruise, but it is most advantageously applied to the analysis of aircraft operating in various high-lift configurations. The MVL methodology has been previously conceived and implemented; the initial concept version was delivered to the ASAB in 2001 (van Dam, C.), subsequently revised (Gelhausen, P. and Ozoroski, T. 2002 / AVID Inc., Gelhausen, P., and Roberts, M. 2004), and then overhauled (Ozoroski, T., Hahn, A. 2008). The latest version, MVL-15 has been refined to provide analysis transparency and enhanced to meet the analysis requirements of the Environmentally Responsible Aviation (ERA) Project. Each revision has been implemented with reasonable success. Separate applications of the methodology are in use, including a similar in-house capability, developed by Olson, E. that is tailored for structural and acoustics analyses. A central premise of the methodology is that viscous aerodynamic data can be associated with analytic inviscid aerodynamic results at each spanwise wing section, thereby providing a pathway to map viscous data to the inviscid results. However, a number of factors can sidetrack the analysis consistency during various stages of this process. For example, it should be expected that the final airplane lift curve and drag polar results depend strongly on the geometry and aerodynamics of the airfoil section; however, flap deflections and flap chord extensions change the local reference geometry of the input airfoil, the airplane wing, the tabulated non-dimensional viscous aerodynamics, and the spanwise links between the linear and the viscous aerodynamics. These changes also affect the bound circulation and therefore, calculation and integration of the induced angle of attack and induced drag. MVL-15 is configured to ensure these types of challenges are properly addressed. This report is a comprehensive manual describing the theory, use, and validation of the MVL-15 analysis tool. Section 3 summarizes theoretical, procedural, and characteristic features of MVL-15, and includes a list of the files required to setup, execute, and summarize an analysis. Section 4, Section 5, Section 6, and Section 7 combine to comprise the User's Guide portions of this report. The MVL-15 input and output files are described in Section 4 and Section 5, respectively; the descriptions are supplemented with example files and information about the file formats, parameter definitions, and typical parameter values. Section 6 describes the Wing Geometry Setup Utility and the 2d-Variants Utility files that simplify and assist setting up a consistent set of MVL-15 geometry and aerodynamics input parameters and input files. Section 7 describes the use of the 3d-Results Presentation Utility file that can be used to automatically create summary tables and charts from the MVL-15 output files. Section 8 documents the Validation Results of an extensive and varied validation test matrix, including results of an airplane analysis representative of the ERA Program. A start-to-finish example of the airplane analysis procedure is described in Section 7.

Ozoroski, Thomas A.↗

DOE OSTI2012

Plasmonics Based Harsh Environment Compatible Chemical Sensors

Au-YSZ, Au-TiO 2 and Au-CeO 2 nanocomposite films have been investigated as a potential sensing element for high-temperature plasmonic sensing of H 2 , CO, and NO 2 in an oxygen containing environment. The Au-YSZ and Au-TiO 2 films were deposited using PVD methods, while the CeO 2 thin film was deposited by molecular beam epitaxy (MBE) and Au was implanted into the as-grown film at an elevated temperature followed by high temperature annealing to form well-defined Au nanoclusters. Each of the films were characterized by x-ray diffraction (XRD) and Rutherford backscattering spectrometry (RBS). For the gas sensing experiments, separate exposures to varying concentrations of H 2 , CO, and NO 2 were performed at a temperature of 500°C in oxygen backgrounds of 5.0, 10, and ~21% O 2 . Changes in the localized surface plasmon resonance (LSPR) absorption peak were monitored during gas exposures and are believed to be the result of oxidation-reduction processes that fill or create oxygen vacancies in the respective metal oxides. This process affects the LSPR peak position either by charge exchange with the Au nanoparticles or by changes in the dielectric constant surrounding the particles. Hyperspectral multivariate analysis was used to gauge the inherent selectivity of the film between the separate analytes. From principal component analysis (PCA), unique and identifiable responses were seen for each of the analytes. Linear discriminant analysis (LDA) was also used on the Au-CeO 2 results and showed separation between analytes as well as trends in gas concentration. Results indicate that each of the films are is selective towards O 2 , H 2 , CO, and NO 2 in separate exposures. However, when the films were analyzed in a sensor array based experiment, ie simultaneous exposures to the target gases, PCA analysis of the combined response showed an even greater selective character towards the target gases. Combined with the observed stability over long exposure periods, each of these Au-metal oxide films shows good potential as an optical sensing element for harsh environmental conditions.

47 OTHER INSTRUMENTATION↗

NASA NTRS1979

Attenuation of sound in a low Mach number nozzle flow

The energy conversion mechanisms which govern the emission of low frequency sound from an axisymmetric jet pipe of arbitrary nozzle contraction ratio in the case of low Mach number nozzle flow are discussed. The energy of the incident sound which flows through the nozzle is used to maintain two distinct characteristic disturbances in the exterior fluid. First, there is the emitted radiation which has the directivity equivalent to that of a monopole-dipole combination. Second, essentially incompressible vortex waves are induced on the jet by vortex shedding from the lip of the nozzle and may involve the excitation of instability modes. Two linearized analytical models are examined to determine the partition of the emitted energy between the radiation field and the vortex waves. One of these is an exact linear theory in which the jet boundary is treated as a vortex sheet. The second model assumes that the width of the mean shear layer of the jet cannot be neglected. The results are discussed with reference to recent nozzle attenuation measurements.

Howe, M. S.↗

NASA NTRS1987

Design sensitivity analysis of nonlinear structural response

A unified theory is described of design sensitivity analysis of linear and nonlinear structures for shape, nonshape and material selection problems. The concepts of reference volume and adjoint structure are used to develop the unified viewpoint. A general formula for design sensitivity analysis is derived. Simple analytical linear and nonlinear examples are used to interpret various terms of the formula and demonstrate its use.

Cardoso, J. B.↗

DOE OSTI2014

Relation between energetic and standard geodesic acoustic modes

Geodesic Acoustic Modes (GAMs) are electrostatic, axisymmetric modes which are non-linearly excited by turbulence. They can also be excited linearly by fast-particles; they are then called Energetic-particle-driven GAMs (EGAMs). Do GAMs and EGAMs belong to the same mode branch? Through a linear, analytical model, in which the fast particles are represented by a Maxwellian bump-on-tail distribution function, we find that the answer depends on several parameters. For low values of the safety factor q and for high values of the fast ion energy, the EGAM originates from the GAM. On the contrary, for high values of q and for low values of the fast ion energy, the GAM is not the mode which becomes unstable when fast particles are added: the EGAM then originates from a distinct mode, which is strongly damped in the absence of fast particles. The impact of other parameters is further explored: ratio of the ion temperature to the electron temperature, width of the fast particle distribution, mass and charge of the fast ions. The ratio between the EGAM and the GAM frequencies was found in experiments (DIII-D) and in non-linear numerical simulations (code GYSELA) to be close to 1/2: the present analytical study allows one to recover this ratio.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2015

KINETIC ALFVÉN WAVE GENERATION BY LARGE-SCALE PHASE MIXING

One view of the solar wind turbulence is that the observed highly anisotropic fluctuations at spatial scales near the proton inertial length d{sub p} may be considered as kinetic Alfvén waves (KAWs). In the present paper, we show how phase mixing of large-scale parallel-propagating Alfvén waves is an efficient mechanism for the production of KAWs at wavelengths close to d{sub p} and at a large propagation angle with respect to the magnetic field. Magnetohydrodynamic (MHD), Hall magnetohydrodynamic (HMHD), and hybrid Vlasov–Maxwell (HVM) simulations modeling the propagation of Alfvén waves in inhomogeneous plasmas are performed. In the linear regime, the role of dispersive effects is singled out by comparing MHD and HMHD results. Fluctuations produced by phase mixing are identified as KAWs through a comparison of polarization of magnetic fluctuations and wave-group velocity with analytical linear predictions. In the nonlinear regime, a comparison of HMHD and HVM simulations allows us to point out the role of kinetic effects in shaping the proton-distribution function. We observe the generation of temperature anisotropy with respect to the local magnetic field and the production of field-aligned beams. The regions where the proton-distribution function highly departs from thermal equilibrium are located inside the shear layers, where the KAWs are excited, this suggesting that the distortions of the proton distribution are driven by a resonant interaction of protons with KAW fluctuations. Our results are relevant in configurations where magnetic-field inhomogeneities are present, as, for example, in the solar corona, where the presence of Alfvén waves has been ascertained.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS2002

Structure and Dynamics of the Quiet Solar Chromosphere

The grant supported research on the structure of the quiet, nonmagnetic chromosphere and on wave excitation and propagation in both the nonmagnetic chromosphere and the magnetic network. The work on the structure of the chromosphere culminated in the recognition that between two competing views of the solar chromosphere, older models by Avrett and collaborators (referred to as VAL) and the newer, dynamical model by Carlsson & Stein (referred to as CS), the clear decision is in favor of the older models, and this in spite of the evident lack of physics, which does not include wave motion and oscillations. The contrast between the static VAL models and the dynamical CS model can be stated most succinctly by comparing the temperature variation implied by the VAL models and the temperature fluctuations of the CS model, which are, respectively, of the order of 10% for the VAL model (at heights where hydrogen is 50% ionized) and a factor of 10 (at the upper boundary of their chromospheric model). The huge fluctuations of the CS model have never been observed, whereas the smaller temperature variations of the VAL models are consistent with ground-based and space-based observations. While it should be obvious which model describes the Sun and which one fails, the case is far from settled in the minds of solar physicists. Thus, much educational work remains to be done and, of course, more research to develop arguments that make the case more convincing. The research on waves and oscillations has been based on a unified theory of excitation of acoustic waves in the field-free atmosphere and of transverse and longitudinal waves in magnetic flux tubes located in the magnetic network by noting, first, that impulsive excitation of all these waves in gravitationally stratified media leads to oscillations at the respective cutoff frequencies and, second, that the observed oscillation frequencies in the nonmagnetic and magnetic parts of the chromosphere match corresponding cutoff frequencies in the upper solar photosphere. The dynamical simulations by Carlsson and Stein have been most instructive and of fundamental importance for understanding wave propagation in a stratified medium by their "flaws", the most important of which is an intensity excess at the H2v emission peak in the H line of Ca II, which is surprising since the observed intensity should have been an upper limit to the simulated intensity. The only plausible explanation for a predicted intensity that is higher than observed is that energy is spread horizontally in upward propagation in the Sun, but not in the plane-wave modeling of CS (as well as by almost everyone else). Investigations of the horizontal size of the region disturbed by the upward-propagating shock in the acoustic-wave propagation implies that the waves in H2v bright-point oscillations emanate from a point source with a diameter corresponding to the width of an intergranular lane, about 100 km, and reach a size of about 4000 km in the upper layers of the chromosphere. Linear, analytic modeling of waves emanating from a point source in a stratified atmosphere shows that the upward-expanding propagation channel does not have sharp boundaries and that the shape of the wave front depends on the order of the wave behind the initial pulse. Otherwise, the behavior of the linear waves resembles that of the nonlinear shock waves observed in the Sun. Research that needs to be done to firm up the conclusions reached above concerns the numerical simulations of nonlinear waves and oscillations in a three-dimensional stratified atmosphere with impulsive excitation, and observations linking directly the horizontal size of the disturbed area in upward propagation to individual waves.

Kalkofen, Wolfgang↗

NASA NTRS2014

Impact of Cross-Axis Structural Dynamics on Validation of Linear Models for Space Launch System

A feasibility study was performed to examine the advisability of incorporating a set of Programmed Test Inputs (PTIs) during the Space Launch System (SLS) vehicle flight. The intent of these inputs is to provide validation to the preflight models for control system stability margins, aerodynamics, and structural dynamics. During October 2009, Ares I-X program was successful in carrying out a series of PTI maneuvers which provided a significant amount of valuable data for post-flight analysis. The resulting data comparisons showed excellent agreement with the preflight linear models across the frequency spectrum of interest. However unlike Ares I-X, the structural dynamics associated with the SLS boost phase configuration are far more complex and highly coupled in all three axes. This presents a challenge when implementing this similar system identification technique to SLS. Preliminary simulation results show noticeable mismatches between PTI validation and analytical linear models in the frequency range of the structural dynamics. An alternate approach was examined which demonstrates the potential for better overall characterization of the system frequency response as well as robustness of the control design.

Pei, Jing↗

NASA NTRS1993

Development of a computational testbed for numerical simulation of combustion instability

A synergistic hierarchy of analytical and computational fluid dynamic techniques is used to analyze three-dimensional combustion instabilities in liquid rocket engines. A mixed finite difference/spectral procedure is employed to study the effects of a distributed vaporization zone on standing and spinning instability modes within the chamber. Droplet atomization and vaporization are treated by a variety of classical models found in the literature. A multi-zone, linearized analytical solution is used to validate the accuracy of the numerical simulations at small amplitudes for a distributed vaporization region. This comparison indicates excellent amplitude and phase agreement under both stable and unstable operating conditions when amplitudes are small and proper grid resolution is used. As amplitudes get larger, expected nonlinearities are observed. The effect of liquid droplet temperature fluctuations was found to be of critical importance in driving the instabilities of the combustion chamber.

Grenda, Jeffrey↗

DOE OSTI2015

Gyrokinetic study of the impact of the electron to ion heating ratio on the turbulent diffusion of highly charged impurities

A gyrokinetic study based on numerical and analytical calculations is presented, which computes the dependence of the turbulent diffusion of highly charged impurities on the ratio of the electron to the ion heat flux of the plasma. Nonlinear simulations show that the size of the turbulent diffusion of heavy impurities can vary by one order of magnitude with fixed total heat flux and is an extremely sensitive function of the electron to ion heat flux ratio. Numerical linear calculations are found to reproduce the nonlinear results. Thereby, a quasi-linear analytical approach is used to explain the origin of this dependence.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗