Time Evolution of the Macroscopic Characteristics of a Thin Current Sheet in the Course of Its Formation in the Earth’s Magnetotail
A numerical model is developed that allows tracing the time evolution of a current sheet from a relatively thick current configuration with isotropic distributions of the pressure and temperature in an extremely thin current sheet, which plays a key role in geomagnetic processes. Such a configuration is observed in the Earth’s magnetotail in the stage preceding a large-scale geomagnetic disturbance (substorm). Thin current sheets are reservoirs of the free energy released during geomagnetic disturbances. The time evolution of the components of the pressure tensor caused by changes in the structure of the current sheet is investigated. It is shown that the pressure tensor in the current sheet evolves in two stages. In the first stage, a current sheet with a thickness of eight to ten proton Larmor radii forms. This stage is characterized by the plasma drift toward the current sheet and the Earth and can be described in terms of the Chu–Goldberger–Low approximation. In the second stage, an extremely thin current sheet with an anisotropic plasma pressure tensor forms, due to which the system is maintained in an equilibrium state. Estimates of the characteristic time of the system evolution agree with available experimental data.
The Earth’s magnetosphere is an open dynamic system permanently interacting with the solar wind, i.e., the plasma flow from the Sun. Some plasma processes in the magnetosphere are of spontaneous explosive character, while others develop rather slowly as compared to the characteristic times of plasma particle motion in it. The large-scale current sheet in the magnetotail can be in an almost equilibrium state both in quiet periods and during geomagnetic perturbations, and its variations can be considered quasistatic. Thus, under some conditions, the magnetotail current sheet can be described as an equilibrium plasma system. Its state depends on various parameters, in particular, on those determining the dynamics of charged particles. Knowing the main governing parameters, one can study the structure and properties of the current sheet equilibrium. This work is devoted to the self-consistent modeling of the equilibrium thin current sheet (TCS) of the Earth’s magnetotail, the thickness of which is comparable with the ion gyroradius. The main objective of this work is to examine how the TCS structure depends on the parameters characterizing the particle dynamics and magnetic field geometry. A numerical hybrid self-consistent TCS model in which the tension of magnetic field lines is counterbalanced by the inertia of ions moving through the sheet is constructed. The ion dynamics is considered in the quasi-adiabatic approximation, while the electron motion, in the conductive fluid approximation. Depending on the values of the adiabaticity parameter κ (which determines the character of plasma particle motion) and the dimensionless normal component of the magnetic field , the following two scenarios are considered: (A) the adiabaticity parameter is proportional to the particle energy and = const and (B) the particle energy is fixed and the adiabaticity parameter is proportional to . The structure of the current sheet and particle dynamics in it are studied as functions of the parameters κ and . It is shown that, in scenario A, the current sheet thickness decreases with increasing adiabaticity parameter due to a decrease in the ion gyroradius. Accordingly, the radius of curvature of magnetic field lines decreases, which leads to an increase in the contribution of electron drift currents near the neutral plane z = 0. Numerical simulations demonstrate that current equilibria can exist if the adiabaticity parameter lies in the range . At κ ~ 0.7, the contribution of electron drift currents to the total current density is much larger than the contribution of ions and the ion motion becomes chaotic. At larger values of the adiabaticity parameter, no equilibrium solutions were found in the framework of the given one-dimensional model. Therefore, the value κ = 0.7 corresponds to the upper applicability limit of the quasi-adiabatic model of the current sheet. In scenario B, an increase in the parameter κ leads to the appearance of a large number of quasi-trapped ions in the current sheet, due to which the current sheet thickens and the amplitude of the current density decreases. As a result, equilibrium solutions exist in a much narrower range of the adiabaticity parameter, . Consequences of the existence of parametric boundaries of equilibrium solutions for the TCS under actual geomagnetic conditions are discussed.
Electron dynamics and acceleration in an electromagnetic field configuration modeling the current sheet configuration of the Earth’s magnetotail region is investigated. A focus is made on the role of the dawn−dusk magnetic field component By in the convection electron heating by an electric field Ey. For numerical integration of a large number of test particle trajectories over long time intervals, the equations of motion written in the guiding center approximation are used. It is shown that the presence of a By ≠ 0 magnetic field significantly changes the electron heating and allows electrons with small pitch angles to gain energy much more efficiently than the equatorial electrons. As a result, the convection heating in the current sheet with By ≠ 0 leads to the formation of an accelerated anisotropic population of particles with energies higher than a few hundred electronvolts. The obtained results and spacecraft observations in the Earth’s magnetotail are compared, and possible limitations in the proposed model approaches are discussed.
This work is devoted to the investigation of particle acceleration during magnetospheric dipolarizations. A numerical model is presented taking into account the four scenarios of plasma acceleration that can be realized: (A) total dipolarization with characteristic time scales of 3 min; (B) single peak value of the normal magnetic component Bz occurring on the time scale of less than 1 min; (C) a sequence of rapid jumps of Bz interpreted as the passage of a chain of multiple dipolarization fronts (DFs); and (D) the simultaneous action of mechanism (C) followed by the consequent enhancement of electric and magnetic fluctuations with the small characteristic time scale 1 s. In a frame of the model, we have obtained and analyzed the energy spectra of four plasma populations: electrons e, protons Hþ, helium Heþ, and oxygen Oþ ions, accelerated by the above-mentioned processes (A)–(D). It is shown that Oþ ions can be accelerated mainly due to the mechanism (A); Hþ and Heþ ions (and to some extent electrons) can be more effectively accelerated due to the mechanism (C) than the single dipolarization (B). It is found that high-frequency electric and magnetic fluctuations accompanying multiple DFs (D) can strongly accelerate electrons e and really weakly influence other populations of plasma. The results of modeling demonstrated clearly the distinguishable spatial and temporal resonance character of particle acceleration processes. The maximum particle energies depending on the scale of the magnetic acceleration region and the value of the magnetic field are estimated. The shapes of energy spectra are discussed.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
Event logs collected by modern information and technical systems usually contain enough data for automated process models discovery. A variety of algorithms was developed for process models discovery, conformance checking, log to model alignment, comparison of process models, etc., nevertheless a quick analysis of ad-hoc selected parts of a journal still have not get a full-fledged implementation. This paper describes an ROLAP-based method of multidimensional event logs storage for process mining. The result of the analysis of the journal is visualized as directed graph representing the union of all possible event sequences, ranked by their occurrence probability. Our implementation allows the analyst to discover process models for sublogs defined by ad-hoc selection of criteria and value of occurrence probability
The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.
Let G be a semisimple algebraic group whose decomposition into the product of simple components does not contain simple groups of type A, and P⊆G be a parabolic subgroup. Extending the results of Popov , we enumerate all triples (G, P, n) such that (a) there exists an open G-orbit on the multiple flag variety G/P × G/P × . . . × G/P (n factors), (b) the number of G-orbits on the multiple flag variety is finite.
I give the explicit formula for the (set-theoretical) system of Resultants of m+1 homogeneous polynomials in n+1 variables