Wave propagation in steady stratified one-dimensional cylindrical waveguides

Aims. This paper studies the propagation of longitudinal magnetic tube waves in a stratified isothermal flux tube with an internal equilibrium background flow. Methods. The governing differential equation is solved by means of Laplace transforms and temporal and spatial solutions are developed, with boundary conditions given by various footpoint drivers, namely a monochromatic source, a delta function pulse, and a sinusoidal pulse. The effect of the background flow is to introduce an increase in amplitude of the wave perturbation and changes in phase shift when compared with the corresponding static case. Results. Results are presented and applied to conditions in the solar atmosphere. When the source is driven continuously, the forced atmospheric oscillations are shown to have large percentage differences when compared to the corresponding static case. For the free atmospheric oscillations, percentage increases in amplitude merely a few percent are found and vary greatly in height but are practically unaltered in time. Phase shifts up to a radian are introduced and weakly depend on both height and time. Conclusions. The results presented in this paper may have interesting observational consequences, especially when using the tools of magnetic seismology of solar atmospheric wave guides (i.e. flux tubes from photosphere to corona) in light of the present and near-future high spatial and temporal resolution space missions, e.g. Hinode, Solar Dynamics Observatory, or Solar Orbiter

Kink oscillations in magnetic tubes with twisted annulus

Aims.We study kink waves in a magnetic flux tube modelled as a straight core surrounded by a magnetically twisted annulus, both embedded in a straight ambient external field, and derive the dispersion relation for this configuration. Methods.The existence and behaviour of the kink modes are examined with specific attention to the effect that the addition of magnetic twist has on phase speeds and periods. Analytic expansions to the short and long wavelength approximations are also considered. Results.The magnetic twist is found to introduce of an infinite set of body modes into solutions of the dispersion relation not present in the untwisted case. Moreover, for the kink modes, the width of interval of this infinite set, generally found to occupy phase speeds around the annulus' longitudinal Alfvén speed, increases for longer wavelengths. Two surface modes are also present in the solution, one at each surface: the internal and the external edges of the annulus. The magnetic twist is found to increase or decrease the phase speeds of these surface modes that are depending on the ratio of internal and external Alfvén speeds in the flux tube. Conclusions.The magnetic twist of the annulus region of a flux tube is found to have a marked effect on the phase speeds of occurring modes. A straight annulus layer increased (or decreased) the periods of the surface modes for a tube modelled as a density (magnetic) enhancement. The addition of twist reduces the periods of the modes in both cases

The effect of elliptic shape on the period ratio P-1/P-2 of emerging coronal loops

Aims. We determine the effect of an elliptical shape on the period ratio for the standing transversal oscillations of a longitudinally stratified coronal loop throughout its emergence from the low solar atmosphere into the ubiquitously magnetised corona. Methods. Under the assumption that elliptical curvature has a negligible effect on eigenfrequencies, the equation that describes the projection of a density profile onto a magnetic flux tube with elliptical shape is obtained in a gravitationally stratified atmosphere. The effect of the elliptical shape on the period ratio of the fundamental mode to the first harmonic (P-1/P-2) at various stages of emergence is determined, assuming that the oscillation periods are much shorter than the characteristic time scale of loop emergence. Results. We find that there are two separate cases of elliptical shape that occur, the minor ellipse and the major ellipse. It is then shown how the period ratio P-1/P-2 is dependent upon the ellipticity (epsilon), the parameter characterising the stage of emergence (lambda) and the density scale height (H). Ellipticity is found to make an important contribution to P-1/P-2 for the minor ellipse when compared to its counterpart of standing oscillations of stratified loops with semi-circle or circle-arc shape. The major ellipse was found to have a lesser effect on the period ratio of standing oscillations. We also find the value of P-1/P-2 is dependent upon the stage of emergence of the loop, where the greatest contribution from emergence to the ratio of P-1/P-2 is when the loop is almost fully emerged. The important implication for magneto-seismological interpretations of the observations of oscillating coronal loops is that measurements of ellipticity and stage of emergence should supplement observations of oscillation periods and should be considered when applying observed frequencies of the fundamental mode and first harmonic to determine the diagnostic properties of these oscillating loops, e. g. the density scale height or strength of magnetic field. Neglecting the determination of ellipticity and stage of emergence may result in a 35% error in estimating density scale height

Magnetohydrodynamic waves in a compressible magnetic flux tube with elliptical cross-section

Aims. The propagation of magnetohydrodynamic (MHD) waves in a finite, compressible magnetic flux tube with an elliptical cross-section embedded in a magnetic environment is investigated. Methods. We present the derivation of the general dispersion relation of linear magneto-acoustic wave propagation for a compressible magnetic flux tube with elliptical cross-section in a plasma with finite beta. The wave modes of propagation for the n = 0 (symmetric) sausage and n = 1 (anti-symmetric) kink oscillations are then examined within the limit of the thin flux tube approximation. Results. It is shown that a compressible magnetic tube with elliptical cross-section supports slow and fast magneto-acoustic waves. In the thin tube approximation, the slow sausage mode and the slow and fast kink modes are found in analogue to a circular cross-section. However, the kink modes propagate with different phase speeds depending on whether the axial displacement takes place along the major or minor axis of the ellipse. This feature is present in both the slow and the fast bands, providing two infinite sets of slow kink modes and two infinite sets of fast kink modes, i.e. each corresponding cylindrical mode splits into two sets of modes due to the ellipticity. The difference between the phase speeds along the different axis is dependent on the ratio of the lengths of the two axes. Analytical expressions for the phase speeds are found. We show that the sausage modes do not split due to the introduced ellipticity and only the phase speed is modified when compared to the appropriate cylindrical counterpart. The percentage difference between the periods of the circular and elliptical cross-sections is also calculated, which reaches up to 21% for oscillations along the major axis. The level of difference in period could be very important in magneto-seismological applications, when observed periods are inverted into diagnostic properties (e. g. magnetic field strength, gravitational scale height, tube expansion parameter). Also shown is the perturbation of focal points of the elliptical cross-section for different modes. It is found that the focal points are unperturbed for the sausage mode, but are perturbed for all higher modes

Periodic Recurrence Patterns In X-Ray Solar Flare Appearances

The temporal recurrence of micro-flare events is studied for a time interval before and after of major solar flares. Our sample is based on the X-ray flare observations by the Geostationary Operational Environmental Satellite (GOES) and Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI). The analyzed data contain 1330/301 M-class and X-class GOES/RHESSI energetic solar flares and 4062/4119 GOES/RHESSI micro-flares covering the period elapse since 2002. The temporal analysis of recurrence, by Fast Fourier Transform, of the micro-flares, shows multiple significant periods. Based on the GOES and RHESSI data, the temporal analysis also demonstrates that multiple periods manifest simultaneously in both statistical samples without any significant shift over time. In the GOES sample, the detected significant periods are: 11.33, 5.61, 3.75, 2.80, and 2.24 minutes. The RHESSI data show similar significant periods at 8.54, 5.28, 3.66, 2.88, and 2.19 minutes. The periods are interpreted as signatures of standing oscillations, with the longest period (P 1) being the fundamental and others being higher harmonic modes. The period ratio of the fundamental and higher harmonics (P 1/P N ) is also analyzed. The standing modes may be signatures of global oscillations of the entire solar atmosphere encompassing magnetized plasma from the photosphere to the corona in active regions

Late-time expansion in the semiclassical theory of the Hawking radiation

We give a detailed treatment of the back-reaction effects on the Hawking spectrum in the late-time expansion within the semiclassical approach to the Hawking radiation. We find that the boundary value problem defining the action of the modes which are regular at the horizon admits in general the presence of caustics. We show that for radii less that a certain critical value $r_c$ no caustic occurs for all values of the wave number and time and we give a rigorous lower bound on such a critical value. We solve the exact system of non linear equations defining the motion, by an iterative procedure rigorously convergent at late times. The first two terms of such an expansion give the $O(\omega/M)$ correction to the Hawking spectrum.Comment: 17 pages, 1 figure, LaTex, typos corrected, one intermediate formula adde

Parton showers as sources of energy-momentum deposition in the QGP and their implication for shockwave formation at RHIC and at the LHC

We derive the distribution of energy and momentum transmitted from a primary fast parton and its medium-induced bremsstrahlung gluons to a thermalized quark-gluon plasma. Our calculation takes into account the important and thus far neglected effects of quantum interference between the resulting color currents. We use our result to obtain the rate at which energy is absorbed by the medium as a function of time and find that the rate is modified by the quantum interference between the primary parton and secondary gluons. This Landau-Pomeranchuk-Migdal type interference persists for time scales relevant to heavy ion phenomenology. We further couple the newly derived source of energy and momentum deposition to linearized hydrodynamics to obtain the bulk medium response to realistic parton propagation and splitting in the quark-gluon plasma. We find that because of the characteristic large angle in-medium gluon emission and the multiple sources of energy deposition in a parton shower, formation of well defined Mach cones by energetic jets in heavy ion reactions is not likely.Comment: 8 pages, 4 figure

Dynamic Behavior of Spicules Inferred from Perpendicular Velocity Components

Understanding the dynamic behavior of spicules, e.g., in terms of magnetohydrodynamic (MHD) wave mode(s), is key to unveiling their role in energy and mass transfer from the photosphere to corona. The transverse, torsional, and field-aligned motions of spicules have previously been observed in imaging spectroscopy and analyzed separately for embedded wave-mode identification. Similarities in the Doppler signatures of spicular structures for both kink and torsional Alfvén wave modes have led to the misinterpretation of the dominant wave mode in these structures and is a subject of debate. Here, we aim to combine line- of-sight (LOS) and plane-of-sky (POS) velocity components using the high spatial/temporal resolution Hα imaging-spectroscopy data from the CRisp Imaging SpectroPolarimeter based at the Swedish Solar Telescope to achieve better insight into the underlying nature of these motions as a whole. The resultant three-dimensional velocity vectors and the other derived quantities (e.g., magnetic pressure perturbations) are used to identify the MHD wave mode(s) responsible for the observed spicule motion. We find a number of independent examples where the bulk transverse motion of the spicule is dominant either in the POS or along the LOS. It is shown that the counterstreaming action of the displaced external plasma due to spicular bulk transverse motion has a similar Doppler profile to that of the m = 0 torsional Alfvén wave when this motion is predominantly perpendicular to the LOS. Furthermore, the inferred magnetic pressure perturbations support the kink wave interpretation of observed spicular bulk transverse motion rather than any purely incompressible MHD wave mode, e.g., the m = 0 torsional Alfvén wav

Integro-differential diffusion equation for continuous time random walk

In this paper we present an integro-differential diffusion equation for continuous time random walk that is valid for a generic waiting time probability density function. Using this equation we also study diffusion behaviors for a couple of specific waiting time probability density functions such as exponential, and a combination of power law and generalized Mittag-Leffler function. We show that for the case of the exponential waiting time probability density function a normal diffusion is generated and the probability density function is Gaussian distribution. In the case of the combination of a power-law and generalized Mittag-Leffler waiting probability density function we obtain the subdiffusive behavior for all the time regions from small to large times, and probability density function is non-Gaussian distribution.Comment: 12 page

New solutions of Heun general equation

We show that in four particular cases the derivative of the solution of Heun general equation can be expressed in terms of a solution to another Heun equation. Starting from this property, we use the Gauss hypergeometric functions to construct series solutions to Heun equation for the mentioned cases. Each of the hypergeometric functions involved has correct singular behavior at only one of the singular points of the equation; the sum, however, has correct behavior