Vlasov Theory Research Articles

Proton Temperature Anisotropy Constraint Associated With Alpha Beam Instability in the Solar Wind

AbstractSolar wind observations in the space of the proton temperature anisotropy versus parallel proton beta always show a distorted rhomboid‐like pattern, where the boundaries of this plot are associated with several instabilities. However, the specific mechanism on the constraint of the proton temperature anisotropy is unclear in the low‐beta plasma. In this work, we study the kinetic instabilities driven by proton temperature anisotropy and alpha beam with the Vlasov theory and investigate the nonlinear evolution of these instabilities with hybrid simulations. We also compare the theoretical and simulation results with Wind observations. The alpha beam with drift velocity leads to a new kind of Alfvén/ion‐cyclotron instability (A/IC instability) in the region of . In the region of and , the nonlinear wave‐particle interaction through the A/IC instability can decelerate alpha beams to and result in the anti‐correlation between and . In the region of and , the nonlinear wave‐particle interaction through the A/IC instability further leads to slight increase of . The present results provide a potential mechanism for regulating the proton temperature anisotropy in the low‐beta plasma with in the solar wind.

Soil-tunnel interaction under the effect associated with the longitudinal direction

Longitudinal structural analysis of shield tunnels is studied by considering the tunnel as a Timoshenko beam based on elastic foundations under Vlasov's theory. This article presents the structural behavior of a Timoshenko-Vlasov beam jointly, using a single differential system that brings together all the equilibrium and deformation equations. Solving the presented formulation, shield tunnels can be covered under any type of action, ground pressure and support or boundary conditions. This new compacted notation facilitates the possibility of expressing the interaction between soil and structure by introducing the variability of longitudinal support conditions on the elastic springs. This differential system is resolved with the numeric procedure named Finite Transfer Method. To verify the effectiveness of the procedure followed, the structure of the subway tunnel in Shanghai has been modeled and the results obtained have been compared with those offered in the literature. The proposed model has been exemplified by introducing the variability of longitudinal elastic spring conditions representing actual soil behavior.

A new buckling model for thin-walled micro-beams based on modified gradient elasticity: Coupling effect and size effect

More and more thin-walled micro-beams, including the I-section thin-walled micro-beams, are applied in MEMS. The buckling of a thin-walled micro-beam exhibits some different phenomena from that of a large scale thin-walled beam, such as the size effect, and the coupling effects caused by strain gradient. To investigate the influence of the coupling effects and size effect on the buckling of thin-walled micro-beams, a new buckling model is proposed. Based on modified gradient elasticity (MGE), the governing equations and boundary conditions of the new model are derived from the variational principle. The governing equations and boundary conditions can be simplified to the classical Vlasov's theory, the MGE Bernoulli–Euler beam model, MGE Bernoulli–Euler beam model considering bi-directional flexure respectively. Solved by the feature expansion method, the size effects of the thin-walled micro-beam can be captured by the new model, that is the buckling load increases with internal length scales. Besides the classical flexural–torsional coupling effect, two new coupling effects, such as the higher-order flexure–flexure coupling and the higher-order flexural–torsional coupling, are also captured by the new model. These two new high-order coupling effects of the new model reduce the critical buckling load of thin-walled micro-beams. The higher-order flexural–flexural coupling effect is far greater than the other two coupling effects, and the higher-order flexural–flexural coupling effect is on the same order of magnitude as the effect of MGE (size effect), with the former being negative and the latter positive. The size effect and higher-order coupling effect cannot be ignored and need to be considered carefully for the buckling of thin-walled micro-beams.

Indentation of sandwich beams: Comparison of Vlasov, Winkler, and shear theories with composite surfaces reinforced by CNTs and ANN model

The use of sandwich structures has spread to a wide range of fields today. As a result of their thin top and low stiffness, sandwich beams are very susceptible to local loads. One practical solution to increase the sandwich structure's indentation resistance is to add bending strength to the top. The structural properties of carbon nanotubes make them well-suited for use in polymer matrixes; their addition improves the attributes of polymers. Experimental and theoretical investigations of the sandwich structure's indentation behavior are presented in this article. As well, simulations of sandwich structures are conducted using Abaqus software. A tensile test of composite samples and Mori Tanaka's theory were used to determine the elastic modulus of epoxy resin reinforced with carbon nanotubes at different mass percentages. The elastic modulus of composites can be improved by adding carbon nanotubes up to 0.3 % by mass while the mechanical properties are reduced when more than 0.3 % of carbon nanotubes were added. Finally, the analytical solution and Abaqus modeling results for the indentation of the sandwich structure were compared with the results obtained from the experimental tests. A machine learning model is proposed for prediction of sandwich beams deflection. The results of the predictions were phenomenal, with MAE of less than 2 %. The computational costs have been compared with the experimental results, and the machine learning method is very efficient.

Study on the mechanism and response law of fracture movement on the super-high position hard-thick strata

In this paper, a thick plate structural mechanical model was established for the hard-thick rock strata in the Ordos region, which was characterized by the occurrence of high-energy strong earthquakes caused by the fracture of hard-thick rock strata. Subsequently, based on Vlasov's theory, the evolution process of hard-thick rock strata was analyzed. And the paper validated the analysis results using high-energy mine earthquake and surface subsidence data. The following conclusions were drawn: (1) The hard-thick strata in the cretaceous system will not be broken during the advancing and mining process of the test panel of the Shilawusu coal mine. (2) When the test panel is mined to a distance of two panel widths, no fracture occurred in the lower part of the hard-thick strata, because no separated space was formed. (3) When the test panel was advanced to about 856 m, the hard-thick strata have fractured in a vertical direction. (4) No high-energy mine earthquake event has occurred during mining at test panel, and the amount of surface subsidence is approximately 200 mm. (5) In the mining at test panel, two high-energy mining earthquakes occurred at 837 m, 1153 m away from the initial position of the panel, respectively, and the maximum amount of surface subsidence increased to 1397 mm, which accords with the results of the first and periodic breaks calculated by theory. The research results of this paper are of guiding significance for the study of the breaking law of hard-thick strata under similar engineering geological conditions and disaster pre-control.

Bending theory of composite pressure vessels: A closed-form analytical approach

Composite pressure vessels are strategic for the present and future of the aerospace, transportation, and energy industries. These shell structures are utilized to store and transport liquids and gases and are principally composed of a cylindrical shell and two elliptical heads. The current rules for the structural design of composite pressure vessels make use of the membrane theory that excludes the bending of the structure. Nevertheless, the cylindrical shell and the elliptical heads present different curvature radii. These changes generate a localized bending deformation that determines an increase in the stress state as an effect of a local bending moment and shear force distributions that arise in the geometric transition zone. The stresses related to the bending effects die out rapidly, moving away from the junction plane, but they are sensibly higher than the membrane stress and, therefore, cannot be neglected in the mechanical assessment. Here, a closed-form analytical solution for the bending theory of composite shells is proposed. It can assist in the design of composite pressure vessels and drive the design towards reliable configurations for structural integrity. Using the Donnell–Mustari–Vlasov theory for thin-walled composite shells, the governing fourth-order differential equation is deduced and solved to obtain the displacements and stresses acting in the junction. The analytical results are successfully compared with those of reference finite element models for validation; their accuracy is proved considering different thicknesses, lay-ups, and flattening factors of the vessel

Asymmetric Folded Plate with Parallel Edges in Validation of their Static Behavior by Combining Vlasov Torsion Theory with Bernoulli Bending Theory

Aims: A new hybrid procedure that combines the Vlasov torsion theory with the Bernoulli bending theory is presented herein, to demonstrate qualitatively and quantitatively the operation of asymmetric folded plates with parallel edges, which are loaded with gravity static loads. Background: A recently proposed technique based on Vlasov torsion theory is used for the exact calculation of the Principal Elastic Reference System in a reinforced concrete folded plate having an asymmetric thin-walled open cross-section with parallel edges. Moreover, the warping moment (or bi-moment) concept of the Vlasov theory is combined with the pure-bending around two axes, according to the Bernoulli bending theory, to determine the normal stresses along the longitudinal dimension of the folded plate. Methods: Τhe warping properties of a thin-walled open cross-section are determined by calculating: (a) the elastic characteristics (elastic center, principal axes) of the section, (b) the principal start point, the sectorial coordinates, the wrapping moment of inertia and the wrapping stiffness of the section. Finally, the normal stresses along the longitudinal dimension are calculated considering the bi-axes flexure with the bi-moment phenomenon. Results: Τhe exact solution of normal stresses at the middle section of an examined folded plate along the longitudinal dimension is found by combining the Bernoulli bending theory for prismatic beams and the Vlasov torsion theory for thin-walled open sections. Conclusion: The current procedure can be used as a benchmark analysis method of asymmetric folded plates in order to evaluate the reliability of the results of various analysis F.E.M. software, covering an open issue of the structural analysis of special structures.

The Wigner-Vlasov formalism for time-dependent quantum oscillator

This paper presents a comprehensive investigation of the problem of a harmonic oscillator with time-depending frequencies in the framework of the Vlasov theory and the Wigner function apparatus for quantum systems in the phase space. A new method is proposed to find an exact solution of this problem using a relation of the Vlasov equation chain with the Schrödinger equation and with the Moyal equation for the Wigner function. A method of averaging the energy function over the Wigner function in the phase space can be used to obtain time-dependent energy spectrum for a quantum system. The Vlasov equation solution can be represented in the form of characteristics satisfying the Hill equation. A particular case of the Hill equation, namely the Mathieu equation with unstable solutions, has been considered in details. An analysis of the dynamics of an unstable quantum system shows that the phase space square bounded with the Wigner function level line conserves in time, but the phase space square bounded with the energy function line increases. In this case the Vlasov equation characteristic is situated on the crosspoint of the Wigner function level line and the energy function line. This crosspoint moves in time with a trajectory that represents the unstable system dynamics. Each such trajectory has its own energy, and the averaging of these energies by the Wigner function results in time-dependent discreet energy spectrum for the whole system. An explicit expression has been obtained for the Wigner function of the 4th rank in the generalized phase space

Instabilities Driven by Proton Temperature Anisotropy in the Presence of Alpha Particles: Implications for Proton-temperature-anisotropy Constraint in the Solar Wind

In situ measurements reveal that proton temperature anisotropy is ubiquitous in the solar wind. Various plasma instabilities have been proposed to regulate the distribution of the proton temperature anisotropy in the solar wind; detailed constraint processes are still unclear. In this paper, we study the effects of alpha beams on both the forward and backward proton temperature anisotropy instabilities at parallel and oblique propagation with the Vlasov theory, and compare the theoretical results with the Wind observation. As the alpha-beam drift velocity v α /v A increases, the growth rates of forward Alfvén/ion-cyclotron (FA/IC) and backward magnetosonic/whistler (BM/W) instabilities increase, those of backward Alfvén/ion-cyclotron (BA/IC) and forward magnetosonic/whistler (FM/W) instabilities decrease, and those of the mirror and forward Alfvén wave (FAW) instabilities are nearly constant. In particular, there are different constraining mechanisms on the distribution of proton temperature anisotropy for different values of the alpha-beam drift velocity. The proton temperature anisotropy instability together with the alpha beam can provide a potential explanation for the distribution of the proton temperature anisotropy in the solar wind.

Distribution of alpha temperature anisotropy in the slow and fast solar wind: WIND observations and Vlasov theory

Context. Alpha particles in the solar wind frequently show temperature anisotropies, which can be constrained by various kinetic instabilities. However, the detailed constraint mechanism remains unclear, especially in different types of solar wind. Aims. Using data obtained by the Wind spacecraft between 2005 and 2015, we investigate the distribution of alpha temperature anisotropy and the correlation with background plasma parameters in the slow and fast solar wind. Methods. We present a comparison between observations of the alpha temperature anisotropy by the Wind spacecraft and theoretical results obtained with the linear Vlasov theory. Results. The results show that the distribution of the alpha temperature anisotropy depends not only on the parallel temperature ratio of alpha to proton Tα∥/Tp∥ and the proton temperature anisotropy Tp⊥/Tp∥, but also on the drift velocity of alpha to proton vα/vA and the solar wind velocity vsw. In particular, there are clearly different constraining mechanisms on the distribution of alpha temperature anisotropy in the slow and fast solar wind. Conclusions. The alpha temperature anisotropy instability together with different types of free energy can be effective for regulating the alpha temperature anisotropy in the slow and fast solar wind.

Non-field-aligned Proton Beams and Their Roles in the Growth of Fast Magnetosonic/Whistler Waves: Solar Orbiter Observations

The proton beam is an important population of the non-Maxwellian proton velocity distribution in the solar wind, but its role in wave activity remains unclear. In particular, the velocity vector of the proton beam and its influence on wave growth/damping have not been addressed before. Here we explore the origin and the associated particle dynamics of a kinetic wave event in the solar wind by analyzing measurements from Solar Orbiter and comparing them with theoretical predictions from linear Vlasov theory. We identify the waves as outward-propagating circularly polarized fast magnetosonic/whistler (FM/W) waves. The proton’s velocity distribution functions can destabilize FM/W waves. According to linear Vlasov theory, the velocity fluctuations of the core and the beam associated with FM/W waves render the original field-aligned background drift velocity non-field-aligned. This non-field-aligned drift velocity carrying the information of the velocity fluctuations of the core and the beam is responsible for the wave growth/damping. Specifically, for the FM/W waves we analyze, the non-field-aligned fluctuating velocity of the beam population is responsible for the growth of these unstable waves in the presence of a proton beam. In contrast, the core population plays the opposite role, partially suppressing the wave growth. Remarkably, the observed drift velocity vector between the core and the beam is not field aligned during an entire wave period. This result contrasts the traditional expectation that the proton beam is field aligned.

The effect of heavy ions on the dispersion properties of kinetic Alfvén waves in astrophysical plasmas

Context.Spacecraft measurements have shown Kinetic Alfvén Waves propagating in the terrestrial magnetosphere at lower wave-normal angles than predicted by linear Vlasov theory of electron-proton plasmas. To explain these observations, it has been suggested that the abundant heavy ion populations in this region may have strong, non-trivial effects that allow Alfvénic waves to acquire right-handed polarization at lower angles with respect to the background magnetic field, as in the case of typical electron-proton plasma.Aims.We study the dispersion properties of Alfvénic waves in plasmas with stationary phase-space distribution functions with different heavy ion populations. Our extensive numerical analysis has allowed us to quantify the role of the heavy ion components on the transition from the left-hand polarized electromagnetic ion-cyclotron (EMIC) mode to the right-hand polarized kinetic Alfvén wave (KAW) mode.Methods.We used linear Vlasov-Maxwell theory to obtain the dispersion relation for oblique electromagnetic waves. The dispersion relation of Alfvén waves was obtained numerically by considering four different oxygen ion concentrations ranging between 0.0 and 0.2 for all propagation angles, as a function of both the wavenumber and the plasma beta parameter.Results.The inclusion of the heavy O+ions is found to considerably reduce the transition angle from EMIC to KAW both as a function of the wave number and plasma beta. With increasing O+concentrations, waves become more damped in specific wavenumber regions. However, the inclusion of oxygen ions may allow weakly damped KAW to effectively propagate at smaller wave-normal angles than in the electron-proton case, as suggested by observations.

Longitudinal mode-coupling instabilities of proton bunches in the CERN Super Proton Synchrotron

In this paper, we study single-bunch instabilities observed in the CERN Super Proton Synchrotron (SPS). According to the linearized Vlasov theory, radial or azimuthal mode-coupling instabilities result from a coupling of bunch-oscillation modes, which belong to either the same or adjacent azimuthal modes, respectively. We show that both instability mechanisms exist in the SPS by applying the Oide-Yokoya approach to compute van Kampen modes for the realistic longitudinal impedance model of the SPS. The results agree with macroparticle simulations and are consistent with beam measurements. In particular, we see that the uncontrolled longitudinal emittance blowup of single bunches observed before the recent impedance reduction campaign (2018--2021) is due to the radial mode-coupling instability. Unexpectedly, this instability is as strong as the azimuthal mode-coupling instability, which is possible in the SPS for other combinations of bunch length and intensity. We also demonstrate the significant role of rf nonlinearity and potential-well distortion in determining these instability thresholds. Finally, we discuss the effect of the recent impedance reduction campaign on beam stability in single- and double-rf configurations.

Flexural analysis of I-section beams functionally graded materials

This paper aims to present the flexural behavior of thin functionally graded (FG) I-beams. The characteristics of metal-ceramic materials are defined using a power-law function dependent on volume fraction. The bending-torsion equations for this problem are derived based on Vlasov's theory for thin-walled beams and the principle of minimum total potential energy. All geometric properties are expressed according to the functional graded power index law. A nonlinear algebraic system is obtained, and the deflection of the structure is numerically derived. To confirm the precision and effectiveness of the suggested method, a standard benchmark test case is implemented, that concerns the flexural analysis of an FGM I-section beam according to power index and aspect ratio parameters.

Alpha Particle Temperature Anisotropy in Earth’s Magnetosheath

In magnetized plasmas, temperature anisotropy manifests as distinct temperatures (T ⊥j , T ∥j ). Numerous prior studies have demonstrated that as plasma beta (β ∥j ) increases, the range of temperature anisotropy (R j = T ⊥j /T ∥j ) narrows. This limiting effect is conventionally taken as evidence that kinetic microinstabilities are active in the plasma, and has been previously observed for protons in the magnetosheath. This study is the first to use data from the Magnetic Multiscale Mission to investigate these instability-driven limits on alpha particle (ionized helium) anisotropy in Earth’s magnetosheath. The distribution of data over the (β ∥j , R j ) plane was plotted and shows the characteristic narrowing in the range of R j -values as β ∥j increases. The contours of the data distribution align well with the contours of the constant growth rate for the ion cyclotron, mirror, parallel firehose, and oblique firehose instabilities, which were calculated using linear Vlasov theory.

Kinetic Simulations of Proton Mirror Instability: Phase Relations and Thermodynamics

Mirror-mode waves driven by the large temperature anisotropy of T ⊥/T ∣∣ > 1 have been widely observed in the solar wind, planetary magnetosheaths, heliosheath, etc. Recent studies have shown that the phase relations and thermodynamics of the mirror waves observed in the terrestrial magnetosheath may well be interpreted by the linear mixed kinetic–MHD theory of proton mirror instability. In particular, the energy laws possess the form of double-polytropic closures with the thermodynamic exponents being functions of β ⊥,∣∣ = p ⊥,∣∣/(B 2/2μ 0). In this study, we examine the time evolution of proton mirror instability based on the hybrid particle simulations for twenty sets of β ⊥,∣∣ values. Quantitative comparisons between the kinetic simulations, linear Vlasov theory, and observations are made in terms of the growth rates, phase relations, thermodynamic conditions, etc., which show high agreements. In particular, the dependences of various compressibility and thermodynamic exponents on β ⊥,∣∣ are confirmed by the kinetic simulations, which show that the polytropic exponents are in the ranges of γ ⊥ = 0.64 ± 0.21, and γ ∣∣ = 1.07 ± 0.12 consistent with the theoretical predictions and mirror observations of γ ⊥ < 1 and γ ∣∣ ≳ 1. It is shown that the observed features, including the various perturbations and wavelengths, may indeed be reproduced by the nonlinear simulations. The saturated temperature anisotropy β ⊥/β ∣∣ and plasma β ∣∣ show an anticorrelation, which may well be fitted by the modified mirror instability threshold of with γ ⊥ ≈ 0.8, γ ∣∣ ≈ 1.3, and the saturated magnetic field of δ B/B ≈ 0.26 ∼ 0.97 increases with increasing values of β ≈ 1.6 ∼ 8.3.

Interplay of turbulence and proton-microinstability growth in space plasmas

Numerous prior studies have shown that as proton beta increases, a narrower range of proton temperature anisotropy values is observed. This effect has often been ascribed to the actions of kinetic microinstabilities because the distribution of observational data aligns with contours of constant instability growth rates in the beta-anisotropy plane. However, the linear Vlasov theory of instabilities assumes a uniform background in which perturbations grow. The established success of linear-microinstability theories suggests that the conditions in regions of extreme temperature anisotropy may remain uniform for a long enough time so that the instabilities have the chance to grow to sufficient amplitude. Turbulence, on the other hand, is intrinsically nonuniform and nonlinear. Thin current sheets and other coherent structures generated in a turbulent plasma may quickly destroy the uniformity. It is, therefore, not a-priori obvious whether the presence of intermittency and coherent structures favors or disfavors instabilities. To address this question, we examined the statistical distribution of growth rates associated with proton temperature-anisotropy driven microinstabilities and local nonlinear time scales in turbulent plasmas. Linear growth rates are, on average, substantially less than the local nonlinear rates. However, at the regions of extreme values of temperature anisotropy, near the “edges” of the populated part of the proton temperature anisotropy-parallel beta plane, the instability growth rates are comparable or faster than the turbulence time scales. These results provide a possible answer to the question as to why the linear theory appears to work in limiting plasma excursions in anisotropy and plasma beta.

Attenuation bands for flexural–torsional vibrations of locally resonant Vlasov beams

In this paper, a study on flexural–torsional vibrations of finite beams with a large number of resonators periodically attached along the length is presented. Emphasis is given in determining bandgaps, defined as frequency ranges free of resonances. The structural model is based on a modified Vlasov theory that incorporates the resonators effect by means of sectional inertial properties depending on the excitation frequency. Analyzing these last properties, the mechanism of weak and strong bandgap formation is enlightened. Analytical formulas for the edge frequencies of bandgaps are obtained. Also, an exact analytical solution for the free and forced vibration of simply supported beams is presented. This approach allows exploring, from another point of view, the location of attenuation bands.

Thermal buckling analysis of thin-walled closed section FG beam-type structures

In this paper, thermal buckling analysis of thin-walled, functionally graded (FG) beam-type structures with closed cross-sections and various boundary conditions is investigated. One dimensional finite element model, based on Euler–Bernoulli–Vlasov​ theory, is employed under the assumptions of large rotations and small strains. Two cases of temperature rise across the thickness of the cross-section walls are considered, which are uniform and linear. Stability analysis has been performed in eigenvalue manner and in load–deflection manner using Newton–Raphson method. Numerical results for thin-walled box beams and trapezoidal beams for two different FG materials configurations are presented to investigate the effects of temperature distribution, boundary conditions, and material properties on the critical buckling temperature and global buckling behaviour. The accuracy and reliability of the beam model is verified by comparing it with several benchmark examples. Good agreement was observed, showing possibility of further application of the proposed model which requires reduced computational time and provides greater modelling flexibility compared to commercial codes.

Mode Conversion of Kinetic Alfvén Waves in the Presence of Proton Beams and Its Role in Plasma Heating in the Solar Corona

The density inhomogeneity is a typical feature in various magnetoplasmas in the corona, where kinetic Alfvén waves (KAWs) are effectively generated and contribute greatly to the inhomogeneous heating of coronal plasmas. Proton beams exist widely in various space and solar plasma environments. In this paper, based on the kinetic Vlasov theory, we investigate the resonant mode conversion of Alfvén waves to KAWs in the presence of proton beams in an inhomogeneous plasma and the plasma heating of KAWs due to wave−particle interactions. It is found that the wave properties of excited KAWs are highly sensitive to the density of proton beams n bi /n 0, the drift velocity of proton beams v bi /v A, the proton-to-electron temperature ratio T i /T e , and the proton-beam-to-proton-temperature ratio T bi /T i . In addition, the maximum heating rate of KAWs Q m /Q 0 obviously increases with increasing n bi /n 0 and/or v bi /v A. As the electron beta β e increases, Q m /Q 0 decreases sharply in the region of 6 × 10−4 < β e < 10−2 and decreases slowly in the region of β e > 10−2. In addition, the applications of KAW dissipation to plasma heating are discussed in the corona and coronal loops. The KAWs associated with perturbed electric field E x ∼ a few V · m−1 are enough to supply the energy loss in the corona and coronal loops. The present results are of significant importance for comprehending the KAW excitation and the particle energization in the corona.