Propagation of acoustic waves in the non‐isothermal solar atmosphere

The acoustic cutoff frequency was originally introduced by Lamb in the study of the propagation of acoustic waves in a stratified, isothermal medium. In this paper, we use a new method to generalize Lamb's result for a stratified, non‐isothermal medium and obtain the local acoustic cutoff frequency, which describes the propagation of acoustic waves in such a medium. The main result is that the cutoff frequency is a local quantity and that its value at a given atmospheric height determines the frequency acoustic waves must have in order to propagate at this height. Application of this result to specific physical problems like the solar atmosphere is discussed. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cutoff frequency and the reflection of vertically propagating magnetoacoustic gravity waves in the solar atmosphere

Based on a plane-parallel isothermal model solar atmosphere stratified in the field of gravity, we investigate the main patterns of vertical propagation of magnetoacoustic gravity waves (MAGWs) in the approximation of a horizontal potential magnetic field. We have established that the cutoff frequency for MAGWs below which they cannot propagate does not depend on the magnetic field strength and is equal to that for acoustic gravity waves, the Lamb frequency. The cutoff frequency is shown to be unaffected by the linear interaction between counterpropagating MAGWs that results from a nonuniform height distribution of the Alfvén velocity and that causes the reflection of propagating waves at relatively large heights.     

The origin of cutoff frequencies for torsional tube waves propagating in the solar atmosphere

Torsional waves supported by magnetic flux tubes have long been thought to bear a high potential for supplying energy and momentum to the upper solar atmosphere, thereby contributing to its heating and to the driving of dynamic events like spicules. This hope rested on the belief that their propagation is not impeded by cutoff restrictions, unlike longitudinal and kink waves. We point out that this applies only to thin, isothermal tubes. When they widen in the chromosphere, and as a result of temperature gradients, cutoff restrictions arise. We compare them to recent observational reports of such waves and of vortex motions and find that their long period components are already affected by cutoff restrictions. An observational strategy is proposed that should permit the derivation of better information on vortex flows from off‐center observations with next generation telescopes (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global Acoustic Resonance in a Stratified Solar Atmosphere

The upward propagation of linear acoustic waves in a gravitationally stratified solar atmosphere is studied. The wave motion is governed by the Klein?–?Gordon equation, which contains a cutoff frequency introduced by stratification. The acoustic cutoff may act as a potential barrier when the temperature decreases with height. It is shown that waves trapped below the barrier could be subject to a resonance that extends into the entire unbounded atmosphere of the Sun. The parameter space characterizing the resonance is explored.     

Kink Wave Propagation in Thin Isothermal Magnetic Flux Tubes

We investigated the propagation of kink waves in thin and isothermal expanding flux tubes in cylindrical geometry. By using the method of radial expansion for fluctuating variables we obtained a new kink wave equation. We show that including the radial component of the tube magnetic field leads to cutoff-free propagation of kink waves along thin flux tubes.     

Sausage Mode Propagation in a Thick Magnetic Flux Tube

The aim of this paper is to model the propagation of slow magnetohydrodynamic (MHD) sausage waves in a thick expanding magnetic flux tube in the context of the quiescent (VAL-C) solar atmosphere. The propagation of these waves is found to be described by the Klein–Gordon equation. Using the governing MHD equations and the VAL-C atmosphere model we study the variation of the cut-off frequency along and across the magnetic tube guiding the waves. Due to the radial variation of the cut-off frequency the flux tubes act as low frequency filters for the waves.     

Numerical Simulations of Torsional Alfvén Waves in Axisymmetric Solar Magnetic Flux Tubes

We numerically investigate Alfvén waves propagating along an axisymmetric and non-isothermal solar flux tube embedded in the solar atmosphere. The tube magnetic field is current-free and diverges with height, and the waves are excited by a periodic driver along the tube magnetic field lines. The main results are that the two wave variables, the velocity and magnetic field perturbations in the azimuthal direction, behave differently as a result of gradients of the physical parameters along the tube. To explain these differences in the wave behavior, the time evolution of the wave variables and the resulting cutoff period for each wave variable are calculated and used to determine regions in the solar chromosphere where strong wave reflection may occur.     

Non-reflective Propagation of Kink Waves in Coronal Magnetic Loops

Propagating kink waves are ubiquitously observed in solar magnetic wave guides. We consider the possibility that these waves propagate without reflection although there is some inhomogeneity. We briefly describe the general theory of non-reflective, one-dimensional wave propagation in inhomogeneous media. This theory is then applied to kink-wave propagation in coronal loops. We consider a coronal loop of half-circle shape embedded in an isothermal atmosphere, and assume that the plasma temperature is the same inside and outside the loop. We show that non-reflective kink-wave propagation is possible for a particular dependence of the loop radius on the distance along the loop. A viable assumption that the loop radius increases from the loop footpoint to the apex imposes a lower limit on the loop expansion factor, which is the ratio of the loop radii at the apex and footpoints. This lower limit increases with the loop height; however, even for a loop that is twice as high as the atmospheric scale height, it is small enough to satisfy observational constraints. Hence, we conclude that non-reflective propagation of kink waves is possible in a fairly realistic model of coronal loops.     

Nonlinear Numerical Simulations of Magneto-Acoustic Wave Propagation in Small-Scale Flux Tubes

We present results of nonlinear, two-dimensional, numerical simulations of magneto-acoustic wave propagation in the photosphere and chromosphere of small-scale flux tubes with internal structure. Waves with realistic periods of three to five minutes are studied, after horizontal and vertical oscillatory perturbations are applied to the equilibrium model. Spurious reflections of shock waves from the upper boundary are minimized by a special boundary condition. This has allowed us to increase the duration of the simulations and to make it long enough to perform a statistical analysis of oscillations. The simulations show that deep horizontal motions of the flux tube generate a slow (magnetic) mode and a surface mode. These modes are efficiently transformed into a slow (acoustic) mode in the v _A<c _S atmosphere. The slow (acoustic) mode propagates vertically along the field lines, forms shocks, and remains always within the flux tube. It might effectively deposit the energy of the driver into the chromosphere. When the driver oscillates with a high frequency, above the cutoff, nonlinear wave propagation occurs with the same dominant driver period at all heights. At low frequencies, below the cutoff, the dominant period of oscillations changes with height from that of the driver in the photosphere to its first harmonic (half period) in the chromosphere. Depending on the period and on the type of the driver, different shock patterns are observed.     

Parametric generation of acoustic-gravity waves in the solar atmosphere

Based on a plane isothermal solar-atmosphere model, we investigate the parametric generation of acoustic-gravity waves (AGWs) in the approximation of a fixed field for vertically propagating disturbances. Both nonpropagating and propagating AGWs are shown to be generated at the difference frequency via the nonlinear interaction of primary waves in the frequency range “forbidden” for the propagation of AGWs during their linear generation. An acoustic wind has been found to be formed in the solar atmosphere at zero difference frequency; its velocity increases with height in inverse proportion to the decreasing ambient density. We study the nonlinear generation of AGWs at the second harmonic during the interaction of disturbances from the forbidden frequency range.     

细磁通量管运动的特征

从细磁通量管运动方程组导出了特征速度、特征线及其相容关系。这些结果是用特征线法研究细磁通量管运动的基础，也是正确提出定解问题的依据。表征细磁通量管横向波传播的特征速度与管内流动的ＡｌｆｖｅｎＭａｃｈ数有关。当管内流速超过ｋｅｌｖｉｎ－Ｈｅｌｍｈｏｌｔｚ不稳定性临界值时，不存在横向波模式。     

Least-Squares Fitting Methods for Estimating the Winding Rate in Twisted Magnetic-Flux Tubes

We investigate least-squares fitting methods for estimating the winding rate of field lines about the axis of twisted magnetic-flux tubes. These methods estimate the winding rate by finding the values for a set of parameters that correspond to the minimum of the discrepancy between vector magnetic-field measurements and predictions from a twisted flux-tube model. For the flux-tube model used in the fitting, we assume that the magnetic field is static, axisymmetric, and does not vary in the vertical direction. Using error-free, synthetic vector magnetic-field data constructed with models for twisted magnetic-flux tubes, we test the efficacy of fitting methods at recovering the true winding rate. Furthermore, we demonstrate how assumptions built into the flux-tube models used for the fitting influence the accuracy of the winding-rate estimates. We identify the radial variation of the winding rate within the flux tube as one assumption that can have a significant impact on the winding-rate estimates. We show that the errors caused by making a fixed, incorrect assumption about the radial variation of the winding rate can be largely avoided by fitting directly for the radial variation of the winding rate. Other assumptions that we investigate include the lack of variation of the field in the azimuthal and vertical directions in the magnetic-flux tube model used for the fitting, and the inclination, curvature, and location of the flux-tube axis. When the observed magnetic field deviates substantially from the flux-tube model used for the fitting, we find that the winding-rate estimates can be unreliable. We conclude that the magnetic-flux tube models used in this investigation are probably too simple to yield reliable estimates for the winding rate of the field lines in solar magnetic structures in general, unless additional information is available to justify the choice of flux-tube model used for the fitting.     

Physical properties of the quiet solar chromosphere–corona transition region

The physical properties of the quiet solar chromosphere–corona transition region are studied. Here the structure of the solar atmosphere is governed by the interaction of magnetic fields above the photosphere. Magnetic fields are concentrated into thin tubes inside which the field strength is great. We have studied how the plasma temperature, density, and velocity distributions change along a magnetic tube with one end in the chromosphere and the other one in the corona, depend on the plasma velocity at the chromospheric boundary of the transition region. Two limiting cases are considered: horizontally and vertically oriented magnetic tubes. For various plasma densities we have determined the ranges of plasma velocities at the chromospheric boundary of the transition region for which no shock waves arise in the transition region. The downward plasma flows at the base of the transition region are shown to be most favorable for the excitation of shock waves in it. For all the considered variants of the transition region we show that the thermal energy transfer along magnetic tubes can be well described in the approximation of classical collisional electron heat conduction up to very high velocities at its base. The calculated extreme ultraviolet (EUV) emission agrees well with the present-day space observations of the Sun.     

Terrestrial low-frequency bursts: Escape paths of radio waves through the bow shock

From the analysis of 119 low-frequency (LF) burst spectra observed onboard the Wind spacecraft, we propose an interpretation of the frequency-time characteristics including the low frequency cutoff of the LF burst spectra, and we use these characteristics to sound the bow shock structure at large tailward distances from Earth. When observed from within the solar wind, LF bursts appear to be made of two spectral components. The high frequency one is bursty and observed above twice the solar wind plasma frequency f_psw. The low frequency one is diffuse (ITKR) and its spectrum extends from about 2f_psw to a cutoff frequency f_c not much higher than f_psw; its onset time δt(f) increases as the frequency f decreases. For each of the 119 events observed from near the Lagrange point L1, the solar wind density variations were measured and the variations of the density jump across the shock calculated from plasma data all along a shock model over more than 2000R_E. But, except for a few events, neither the solar wind overdensities nor the shock density barrier can prevent waves with frequencies below f_c from reaching the spacecraft. Scattering on plasma density inhomogeneities was then introduced to account for the propagation of the LF burst waves in the magnetosheath, from near Earth to their escape point through the bow shock at a frequency-dependent distance |X_esc(f)| (GSE), and then in the solar wind to the spacecraft. In such media, at frequencies between 2f_psw and f_psw, the bulk speed of the scattered waves decreases rapidly as f decreases, and this accounts for the observed variations of the onset time δt(f). Angular scattering can also account for the observed cutoff at f_c if the distance |X_esc(f)| increases exponentially when f/f_psw decreases. As the shock model we used meets that requirement, we consider that this model is valid, which implies that the bow shock still exists beyond 1000R_E from the Earth. The observed decrease of the average spectral intensity of the LF burst between about 1.5f_psw and 2f_psw can also be explained by the scattering in the solar wind if we take into account the angular distribution of the rays when they leave the bow shock.     

Resonantly Damped Surface and Body MHD Waves in a Solar Coronal Slab with Oblique Propagation

The theory of magnetohydrodynamic (MHD) waves in solar coronal slabs in a zero-β configuration and for parallel propagation of waves does not allow the existence of surface waves. When oblique propagation of perturbations is considered, both surface and body waves are able to propagate. When the perpendicular wavenumber is larger than a certain value, the body kink mode becomes a surface wave. In addition, a sausage surface mode is found below the internal cutoff frequency. When nonuniformity in the equilibrium is included, surface and body modes are damped by resonant absorption. In this paper, first, a normal-mode analysis is performed and the period, the damping rate, and the spatial structure of the eigenfunctions are obtained. Then, the time-dependent problem is solved, and the conditions under which one or the other type of mode is excited are investigated.     

Linear MHD Wave Propagation in Time-Dependent Flux Tube

The propagation of magnetohydrodynamic (MHD) waves is an area that has been thoroughly studied for idealised static and steady state magnetised plasma systems applied to numerous solar structures. By applying the generalisation of a temporally varying background density to an open magnetic flux tube, mimicking the observed slow evolution of such waveguides in the solar atmosphere, further investigations into the propagation of both fast and slow MHD waves can take place. The assumption of a zero-beta plasma (no gas pressure) was applied in Williamson and Erdélyi (Solar Phys. 2013, doi: 10.1007/s11207-013-0366-9 , Paper I) is now relaxed for further analysis here. Firstly, the introduction of a finite thermal pressure to the magnetic flux tube equilibrium modifies the existence of fast MHD waves which are directly comparable to their counterparts found in Paper I. Further, as a direct consequence of the non-zero kinetic plasma pressure, a slow MHD wave now exists, and is investigated. Analysis of the slow wave shows that, similar to the fast MHD wave, wave amplitude amplification takes place in time and height. The evolution of the wave amplitude is determined here analytically. We conclude that for a temporally slowly decreasing background density both propagating magnetosonic wave modes are amplified for over-dense magnetic flux tubes. This information can be very practical and useful for future solar magneto-seismology applications in the study of the amplitude and frequency properties of MHD waveguides, e.g. for diagnostic purposes, present in the solar atmosphere.     

Nonlinear interaction between Alfvén and acoustic-gravity waves in the solar atmosphere

Based on a plane-parallel isothermal model solar atmosphere permeated by a uniform magnetic field directed against the action of gravity, we considered the nonlinear interaction between vertically propagating Alfvén and acoustic-gravity waves. We established that Alfvén waves are efficiently generated at the difference and sum frequencies. We ascertained that no acoustic-gravity waves are formed at the corresponding combination frequencies. A horizontal magnetohydrodynamic wind whose direction changes with height was found to be formed in the solar atmosphere at zero difference frequency.     

Oscillatory Response of the 3D Solar Atmosphere to the Leakage of Photospheric Motion

The direct propagation of acoustic waves, driven harmonically at the solar photosphere, into the three-dimensional solar atmosphere is examined numerically in the framework of ideal magnetohydrodynamics. It is of particular interest to study the leakage of 5-minute global solar acoustic oscillations into the upper, gravitationally stratified and magnetised atmosphere, where the modelled solar atmosphere possesses realistic temperature and density stratification. This work aims to complement and bring further into the 3D domain our previous efforts (by Erdélyi et al., 2007, Astron. Astrophys. 467, 1299) on the leakage of photospheric motions and running magnetic-field-aligned waves excited by these global oscillations. The constructed model atmosphere, most suitable perhaps for quiet Sun regions, is a VAL IIIC derivative in which a uniform magnetic field is embedded. The response of the atmosphere to a range of periodic velocity drivers is numerically investigated in the hydrodynamic and magnetohydrodynamic approximations. Among others the following results are discussed in detail: i) High-frequency waves are shown to propagate from the lower atmosphere across the transition region, experiencing relatively low reflection, and transmitting most of their energy into the corona; ii) the thin transition region becomes a wave guide for horizontally propagating surface waves for a wide range of driver periods, and particularly at those periods that support chromospheric standing waves; iii) the magnetic field acts as a waveguide for both high- and low-frequency waves originating from the photosphere and propagating through the transition region into the solar corona. Electronic Supplementary Material  The online version of this article () contains supplementary material, which is available to authorized users.     

Excess heating of corona and chromosphere above magnetic regions by non-linear Alfvén waves

Excess heating of the active region solar atmosphere is interpreted by the decay of MHD slow-mode waves produced in the corona through the non-linear coupling of Alfvén waves supplied from subphotospheric layers. It is stressed that the Alfvén-mode waves may be very efficiently generated directly in the convection layer under the photosphere in magnetic regions, and that such magnetic regions, at the same time, provide the ‘transparent windows’ for Alfvén waves in regard to the Joule and frictional dissipations in the photospheric and subphotospheric layers. Though the Alfvén waves suffer considerable reflection in the chromosphere and in the transition layer, a certain fraction of this large flux is propagated out to the corona, and a large velocity amplitude exceeding the local Alfvén velocity is attained during the propagation along the magnetic tubes of force into a region of lower density and weaker magnetic field. The otherwise divergence-free velocity field in Alfvén waves gets involved in such a case with a compressional component (slow-mode waves) which again is of considerable velocity amplitude relative to the local acoustic velocity when estimated by using the formulation for non-linear coupling between MHD wave modes derived by Kaburaki and Uchida (1971). Therefore, the compressional waves thus produced through the non-linear coupling of Alvén waves will eventually be thermalized to provide a heat source. The introduction of this non-linear coupling process and the subsequent thermalization of thus produced slow-mode waves may provide means of converting the otherwise dissipation-free Alfvén mode energy into heat in the corona. The liberated heat will readily be redistributed by conduction along the magnetic lines of force, with higher density as a consequence of increased scale height, and thus the loop-like structure of the coronal condensations (or probably also the thread-like feature of the general corona) may be explained in a natural fashion.