Numerical experiments on the Helmholtz equation derived from the solar radiation transfer equation in three-dimensional space

A numerical scheme using the finite-difference approach to solve the modified Helmholtz partial differential equation derived from the solar radiative transfer equation is developed and tested along with the method of evaluating the slant-path optical depth. For overhead solar incidence, we obtain good agreement between the finite-difference approach and the semianalytical solution in terms of the local intensity, local flux, average intensity, and average flux. For face-parallel oblique solar incidence in which the semianalytical method is not applicable, we compare the results with those of previous studies utilizing the Monte Carlo method and the approximate semianalytical method. We show that the present numerical scheme can be applied to any incident solar angle which the approximate semianalytical method is incapable of. Comparisons with results from Monte Carlo method reveal reasonable agreement for the averaged intensity and flux density.     

He3 nuclei in the low energy primary cosmic radiation

The flux of He³ nuclei and the ratio He³/(He³+He⁴) in the low energy primary cosmic radiation have been determined using a stack of nuclear emulsions exposed at 3·1 g. cm.^?2 of atmospheric depth from Fort Churchill, Canada, in June 1963. The grain-densityversus residual range method was used to determine the masses of the helium nuclei. Using a sample of 146 helium nuclei whose masses could be identified, the ratio He³/(He³+He⁴) is obtained as 0·14±0·04 for the kinetic energy interval 115–210 MeV per nucleon and 0·43±0·11 for the rigidity interval 0·85–1·05 BV. The differential fluxes of He³ nuclei are determined as 0·017±0·006, 0·045±0·015, and 0·054±0·017 particles/M². Sr. Sec. MeV/nucleon, in the kinetic energy intervals of 117–183, 183–217, and 217–250 MeV/nucleon respectively. These results are compared with those of other investigators. From the results of the present work the amount of matter traversed in space by the primary cosmic ray helium nuclei of energy 115–210 MeV/nucleon is obtained as 4·7±1·8 gm. cm.^?2 of hydrogen.     

Simulation of turbulent magnetic reconnection in the small-scale solar wind

Some observational examples for the possible occurrence of the turbulent magnetic reconnection in the solar wind are found by analysing Helios spacecraft's high resolution data. The phenomena of turbulent magnetic reconnections in small scale solar wind are simulated by introducing a third order accuracy upwind compact difference scheme to the compressible two_dimensional MHD flow. Numerical results verify that the turbulent magnetic reconnection process could occur in small scale interplanetary solar wind, which is a basic feature characterizing the magnetic reconnection in high_magnetic Reynolds number (RM=2 000-10 000) solar wind. The configurations of the magnetic reconnection could evolve from a single X_line to a multiple X-line reconnection, exhibiting a complex picture of the formation, merging and evolution of magnetic islands, and finally the magnetic reconnection would evolve into a low_energy state. Its life_span of evolution is about one hour order of magnitude. Various magnetic and flow signatures are recorded in the numerical test for different evolution stages and along different crossing paths, which could in principle explain and confirm the observational samples from the Helios spacecraft. These results are helpful for revealing the basic physical processes in the solar wind turbulence.     

Modulation space estimates for the fractional integral operators

We obtain the boundedness for the fractional integral operators from the modulation Hardy space μp,q to the modulation Hardy space μr,q for all 0 < p < ∞. The result is an extension of the known result for the case 1 < p < ∞ and it contains a larger range of r than those in the classical result of the Lp → Lr boundedness in the Lebesgue spaces. We also obtain some estimates on the modulation spaces for the bilinear fractional operators.     

Isothermal flow behind a shock wave propagating in the solar wind

A Parker-type blast wave, which is headed by a strong shock, driven out by a propelling contact surface, moving into an ambient solar wind having a strictly inverse square law radial decay in density, is studied. Assuming the self-similar flow behind the shock to be isothermal, approximate analytical and exact numerical solutions are obtained. There is a good agreement between the approximate analytical and exact numerical solutions. It is observed that the mathematical singularity in density at the contact surface is removed for the isothermal flow.     

Magnetic diffusion and the scaling laws for solar wind simulation

Zusammenfassung Es werden Theorien über die Position und die Dicke der Grenze zwischen dem Magnetfeld der Erde und dem Sonnenwindplasma diskutiert. Die Eigenschaften eines im Laboratorium erzeugten Plasmas werden angegeben, welche diese Zusammenwirkung simulieren kann. Es wird ein Experiment beschrieben, welches die korrekten Modellgesetze erfüllt, und die Störung des Magnetfeldes im Modell wird gezeigt. Es wurde gefunden, dass die Eigenschaften des Grenzgebietes mit den theoretischen Voraussagen nicht übereinstimmen. Der Unterschied kann aber durch die Diffusion des Magnetfeldes in den Plasmastrom erklärt werden, wenn ein genauerer Ausdruck für die elektrische Leitfähigkeit benützt wird. Die Bedingungen für die exakte Modellähnlichkeit sind dann nicht mehr erfüllt. Es wird vorgeschlagen, dass eine höhere Elektrontemperatur benötigt wird, um die elektrische Leitfähigkeit genügend zu crhöhen.     

Investigations on multiply charged nuclei of the primary cosmic radiation near the geomagnetic equator using nuclear emulsions

The charge spectrum obtained from measurements made on tracks of 1329 nuclei of the cosmic radiation of Z?6 and E?7·5 GeV/n has been studied in detail using emulsion stacks exposed over Hyderabad, India (cut off rigidity=16·8 GV) at a depth of 6·8 g/cm² of air for 7¼ hrs. This spectrum has been compared with that available for nuclei of E≥1·5 GeV/n. Within the limits of experimental errors, no dependence of the flux values or ratios of intensities with energy are noticed. This result has important implications on the origin and history of the cosmic radiation. Integral energy spectra of M and H-nuclei have been derived making use of the flux values at the two locations; the spectra thus derived have the least errors associated with these.     

Bounded strict solar property of strict suns in the space C(Q)

The intersection of an arbitrary strict sun M in C(Q) with a closed span Π ? C(Q) (in particular, with a closed ball) is shown to be a strict protosun, provided that the natural condition M ∩ int Π ≠ Ø is satisfied. This property is shown to characterize closed spans in C(Q).     

Investigations on the multiply charged nuclei of the primary cosmic radiation near the geomagnetic equator using nuclear emulsion

A nuclear emulsion stack was exposed over Hyderabad, India (geomagnetic latitudeλ=7·6° N) under a mean atmospheric depth of 6·8 g./cm.², on 24 March 1960. The relative and absolute intensities of Be and B nuclei and nuclei of charge Z?6 (the S-nuclei) have been determined in this stack. The intensities were extrapolated to the top of the atmosphere using measured fragmentation parameters involved in collisions of cosmic ray nuclei in graphite. The flux values of Be, B and S nuclei at the top of the atmosphere have been determined to be 0·025±0·011, 0·101±0·023 and 1·12±0·10 particles/m.² sec. sr. respectively. The ratios of intensities B/S, Be/S and Be/B have been found to be 0·09±0·02, 0·022±0·008 and 0·25±0·11 respectively. These ratios of intensities have been used (a) to compute the amount of matter traversed by the radiation before reaching the vicinity of the earth as 2·6±0·6 g./cm.², (b) to show that the Fermi mechanism of acceleration of particles is not efficient at these energies (1–10 GeV/n.) in interstellar space and (c) to show that preferential acceleration of H₁-nuclei (Z?20) is not of overwhelming importance. The derived source composition of nuclei of energy ?7·5 GeV/n suggests that N, O, F and H₃-nuclei (Z=10–15) have similar abundances in the universe and in cosmic rays, and that C and H₁-nuclei are overabundant in cosmic rays.     

Investigations on the multiply charged nuclei of the primary cosmic radiation near the geomagnetic equator using nuclear emulsions

The interaction mean-free paths and fragmentation parameters associated with collisions of nuclei of Z?6 and E?7·5 Gev/n have been measured using emulsion-graphite sandwich stacks exposed to the primary cosmic radiation at Hyderabad, India (Geomagnetic latitudeλ=7·6° N). The results are based on a total of 213 nuclear interactions observed in graphite. Since the atomic number of graphite is very close to the mean atomic number of air, the parameters derived for graphite medium are directly applicable to air. A comparative study has been made of the results thus obtained with similar data available for nuclei of E?1·5 Gev/n.     

Multiscale behavior and fractional kinetics from the data of solar wind–magnetosphere coupling

Multiscale phenomena are ubiquitous in nature as well as in laboratories. A broad range of interacting space and time scales determines the dynamics of many systems which are inherently multiscale. In many systems multiscale phenomena are not only prominent, but also they often play the dominant role. In the solar wind–magnetosphere interaction, multiscale features coexist along with the global or coherent features. Underlying these phenomena are the mathematical and theoretical approaches such as phase transitions, turbulence, self-organization, fractional kinetics, percolation, etc. The fractional kinetic equations provide a suitable mathematical framework for multiscale behavior. In the fractional kinetic equations the multiscale nature is described through fractional derivatives and the solutions of these equations yield infinite moments, showing strong multiscale behavior. Using a Lévy flights approach, we analyze the correlated data of the solar wind–magnetosphere coupling. Based on this analysis a model of the multiscale features is proposed and compared with the solutions of diffusion-type equations. The equation with fractional spatial derivative shows strong multiscale behavior with infinite moments. On the other hand, the equation with space dependent diffusion coefficients yield finite moments, indicating Gaussian type solutions and absence of long tails typically associated with multiscale behavior.     

On capture and acceleration of heavy ions (a-particle) in high-speed solar wind

The α-particles and other heavy ions, as well es a few protons are observed to be faster than the main part of protons by about the local Alfven speed in the high-speed solar wind. It is suggested that when the velocity of the solar wind is equal to the local Alfven velocity, another Iow-frequency kinetic Alfvan wave will be excitated, and trap ail the α-particles and a few protons, so these ions have a local Alfven velocity faster than the other parts of the solar wind. The undamping kinetic Alfven waves change into Iow-frequency Alfven solitons in the solar wind. This model can explain the observation and give the conditions of wave excitated and ions trapped.     

Representation of the space spanned by a sequence in a Banach space

Summary Let (x _n ) be a general sequence of a Banach space, there is given a particular representation of the elements of the space spanned by (x _n ), by means of the elements of (x _n ). Some properties of this representation are shown, also if (x _n is minimal or normingM-basis. Moreover some open questions are solved.     

On capture and acceleration of heavy ions (α-particle) in high-speed solar wind

The α-particles and other heavy ions, as well as a few protons are observed to be faster than the main part of protons by about the local Alfven speed in the high-speed solar wind. It is suggested that when the velocity of the solar wind is equal to the local Alfven velocity, another low-frequency kinetic Alfven wave will be excitated, and trap all the α-particles and a few protons, so these ions have a local Alfven velocity faster than the other parts of the solar wind. The undamping kinetic Alfven waves change into low-frequency Alfven solitons in the solar wind. This model can explain the observation and give the conditions of wave excitated and ions trapped.