Comment on “Note on the Initial Value Problem for Coronal Loop Kink Waves” By P. S. Cally

In our comment we argue that the results obtained by Cally (2006) are based on a wrong solution of the initial value problem.     

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.     

On Coronal Loop Heating by Torsional Alfvén Waves

Heating of coronal loops by linear resonant Alfvén waves, excited by the footpoints motions in the photosphere, has been studied. The analysis of single-layer heating is extended to multilayer heating, in semiempirical treatment. Heating and nonthermal velocities in different layers of loops in X-ray bright points, active regions, and large-scale structures are estimated. The average value of velocity is found to be in agreement with the observations.     

Numerical Simulations of Impulsively Generated Magnetosonic Waves in a Coronal Loop

We consider impulsively excited magnetosonic waves in a highly magnetized coronal loop that is approximated by a straight plasma slab of enhanced mass density. Numerical results reveal that wavelet spectra of time signatures of these waves possess characteristic shapes that depend on the position of the initial pulse: in the case of a pulse launched inside the slab, these spectra are of a tadpole shape, while for a pulse excited in the ambient medium these spectra display more complex structures with branches of long and short-period waves. These short period oscillations correspond to waves that are trapped inside the slab, and the long-period oscillations are associated with waves that propagate through the ambient medium and reach the detection point. These findings are compatible with recent theoretical studies and observations by the solar eclipse coronal imaging system (SECIS).     

冕环振荡的物理特征研究

太阳动力学观测站(Solar Dynamics Observatory,SDO)上的太阳大气成像仪(Atmospheric Imaging Assembly,AIA)利用谱线Fe IX 171 (A)在2010年10月16日对整个日面进行了连续的高分辨率观测,获得了高质量的数据.这些高质量的数据提供了仔细研究冕环振荡的样本.通过分析这份数据,发现活动区NOAA 1112在此期间爆发了一个M2.9级的耀斑.该耀斑触发了太阳表面的多个冕环产生强烈振荡.其中最为明显的两个振荡冕环呈现出截然不同的振荡特征.位于西492 Mm,南170 Mm(简称W492/S170,后面出现的坐标位置均采取这种标注)处的冕环做周期为pA0 =385 s的简谐振荡,其振荡方程为x=2.2sin[2π55(t-768)],其中t为时间,单位为s;而位于W559/S142处的冕环则是一种典型的阻尼振荡,其阻尼振荡周期为Pf =449 s,相应的振荡方程可表示为x=24.8e-2π/342tsin[2π/449(t-1128)].     

A Research on the Physical Features of Coronal Loop Oscillations

Observations of the solar full-disk were carried out by the Atmo- spheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO) with the Fe IX 171 Å line on 16th October 2010. The obtained high-quality data permit us to elaborate on the coronal loop oscillations. It is found that a major flare of GOES (Geostationary Operational Environmental Satellite) class M_2.9 occurred in the active region NOAA 1112 during this period, which triggered a number of coronal loops on the solar surface to oscillate. Among them, there are two coronal loops exhibiting oscillations with different physical features. The oscillation of the coronal loop located at W₄₉₂/S₁₇₀ is a simple harmonic oscillation with a period of 385_s, which abides by the oscillating equation of x = 2.2 sin[2π/385(t–768)], while the other located at W559/S142 is a damping oscillation with a period of 449s, and the oscillating equation is expressed by x = 24.8e - 2π/343 t sin[2π/449(t–1128)], where t is the observational time in units of second.     

The Cosmological Problem as Initial Value Problem on the Observer's Past Light Cone: Observations

In a previous paper it was shown how cosmological models can be characterized by few initial data on the observer's past light cone. Part of the inner geometry of the past cone as well as some matter variables taken on the cone (in the simplest case the velocity and density of a cosmological fluid) determine a world model uniquely within the past light cone. Since the initial data to some degree enter the relations between observable quantities such as redshift, luminosity, angular diameter, number counts, distortion parameters and background radiation intensity, it might be possible to determine the required initial data from observations. This problem is discussed in the present article for some observable relatios involving the quantities just mentioned.     

Effects of Field-Aligned Flows on Standing Kink and Sausage Modes Supported by Coronal Loops

Fundamental standing modes and their overtones play an important role in coronal seismology. We examine the effects of a significant field-aligned flow on standing modes that are supported by coronal loops, which are modeled here as cold magnetic slabs. Of particular interest are the period ratios of the fundamental to its (n?1)th overtone [P ₁/nP _n ] for kink and sausage modes, and the threshold half-width-to-length ratio for sausage modes. For standing kink modes, the flow significantly reduces P ₁/nP _n in general, the effect being particularly strong for higher n and weaker density contrast [ $\rho_{0}/\rho_{\rm e}$ ] between loops and their surroundings. That said, even when $\rho_{0}/\rho_{\rm e}$ approaches infinity, this effect is still substantial, reducing the minimal P ₁/nP _n by up to 13.7?% (24.5?%) for n=2 (n=4) relative to the static case, when the Alfvén Mach number [M _A] reaches 0.8, where M _A measures the loop flow speed in units of the internal Alfvén speed. Although it is not negligible for standing sausage modes, the flow effect in reducing P ₁/nP _n is not as strong. However, the threshold half-width-to-length ratio is considerably higher in the flowing case than in its static counterpart. For $\rho_{0}/\rho_{\rm e}$ in the range [9,1024] and M _A in the range [0,0.5], an exhaustive parameter study yields that this threshold is well fitted by $(d/L)_{\rm cutoff, fit} = \frac{1}{2}\sqrt{\frac{1}{\rho_{0}/\rho_{\rm e}-1}} \exp (3.7 M_{\mathrm{A}}^{2} )$ , which involves the two parameters in a simple way. This allows one to analytically constrain the combination $(\rho_{0}/\rho_{\rm e}, M_{\mathrm {A}})$ for a loop with a known width-to-length ratio when a standing sausage oscillation is identified. It also allows one to examine the idea of partial sausage modes in more detail, and the flow is found to significantly reduce the spatial extent where partial modes are allowed.     

On Solving the Coronal Heating Problem

The question of what heats the solar corona remains one of the most important problems in astrophysics. Finding a definitive solution involves a number of challenging steps, beginning with an identification of the energy source and ending with a prediction of observable quantities that can be compared directly with actual observations. Critical intermediate steps include realistic modeling of both the energy release process (the conversion of magnetic stress energy or wave energy into heat) and the response of the plasma to the heating. A variety of difficult issues must be addressed: highly disparate spatial scales, physical connections between the corona and lower atmosphere, complex microphysics, and variability and dynamics. Nearly all of the coronal heating mechanisms that have been proposed produce heating that is impulsive from the perspective of elemental magnetic flux strands. It is this perspective that must be adopted to understand how the plasma responds and radiates. In our opinion, the most promising explanation offered so far is Parker's idea of nanoflares occurring in magnetic fields that become tangled by turbulent convection. Exciting new developments include the identification of the “secondary instability” as the likely mechanism of energy release and the demonstration that impulsive heating in sub-resolution strands can explain certain observed properties of coronal loops that are otherwise very difficult to understand. Whatever the detailed mechanism of energy release, it is clear that some form of magnetic reconnection must be occurring at significant altitudes in the corona (above the magnetic carpet), so that the tangling does not increase indefinitely. This article outlines the key elements of a comprehensive strategy for solving the coronal heating problem and warns of obstacles that must be overcome along the way.     

The Effect of Magnetic and Density Differences on the Fast Kink Oscillations of Neighboring Coronal Loops

It is reasonable that neighboring coronal loops may obtain similar momentum during a flare. The fast kink oscillations (FKOs) between them are thus mainly influenced by their physical differences. We discuss the dependencies of FKO on the physical properties of coronal loops in a low-\(\beta \) thin-tube approximation. From the analysis, we obtain the analytic relationship between the density [\(\rho _{\mathrm{i}}\)] and magnetic field [\(B\)] of loops and the corresponding period [\(\tau \)] and amplitude [\(A\)] of FKO, which may provide us with a powerful tool to diagnose the physical differences between neighboring loops.     

The Effect of a Twisted Magnetic Field on the Nature of Kink MHD Waves

We consider a pressureless plasma in a thin magnetic-flux tube with a twisted magnetic field. We study the effect of twisted magnetic field on the nature of propagating kink waves. To do this, the restoring forces of oscillations in the linear ideal magnetohydrodynamics (MHD) were obtained. In the presence of a twisted magnetic field, the ratio of the magnetic-tension force to the gradient of the magnetic pressure increases for the mode with negative azimuthal wave number, but it decreases for the mode with positive azimuthal wave number. For the kink mode with positive azimuthal mode number, the ratio of the forces is more affected by the twisted magnetic field in dense loops. For the kink mode with negative azimuthal mode number, the perturbed magnetic pressure is negligible under some conditions. The magnetic twist increases (diminishes) the damping of the kink waves with positive (negative) azimuthal mode number due to resonant absorption. Our conclusion is that introducing a twisted magnetic field breaks the symmetry between the nature of the kink waves with positive and negative azimuthal wave number, and the wave can be a purely Alfvénic wave in the entire loop.     

Effect of Variable Background on an Oscillating Hot Coronal Loop

We investigate the effect of a variable, i.e. time-dependent, background on the standing acoustic (i.e. longitudinal) modes generated in a hot coronal loop. A theoretical model of 1D geometry describing the coronal loop is applied. The background temperature is allowed to change as a function of time and undergoes an exponential decay with characteristic cooling times typical for coronal loops. The magnetic field is assumed to be uniform. Thermal conduction is assumed to be the dominant mechanism for damping hot coronal oscillations in the presence of a physically unspecified thermodynamic source that maintains the initial equilibrium. The influence of the rapidly cooling background plasma on the behaviour of standing acoustic (longitudinal) waves is investigated analytically. The temporally evolving dispersion relation and wave amplitude are derived by using the Wenzel–Kramers–Brillouin theory. An analytic solution for the time-dependent amplitude that describes the influence of thermal conduction on the standing longitudinal (acoustic) wave is obtained by exploiting the properties of Sturm–Liouville problems. Next, numerical evaluations further illustrate the behaviour of the standing acoustic waves in a system with a variable, time-dependent background. The results are applied to a number of detected loop oscillations. We find a remarkable agreement between the theoretical predictions and the observations. Despite the emergence of the cooling background plasma in the medium, thermal conduction is found to cause a strong damping for the slow standing magneto–acoustic waves in hot coronal loops in general. In addition to this, the increase in the value of thermal conductivity leads to a strong decay in the amplitude of the longitudinal standing slow MHD waves.     

Observations of the 24 September 1997 Coronal Flare Waves

We report coincident observations of coronal and chromospheric flare wave transients in association with a flare, large-scale coronal dimming, metric radio activity and a coronal mass ejection. The two separate eruptions occurring on 24 September 1997 originate in the same active region and display similar morphological features. The first wave transient was observed in EUV and H data, corresponding to a wave disturbance in both the chromosphere and the solar corona, ranging from 250 to approaching 1000 km s^–1 at different times and locations along the wavefront. The sharp wavefront had a similar extent and location in both the EUV and H data. The data did not show clear evidence of a driver, however. Both events display a coronal EUV dimming which is typically used as an indicator of a coronal mass ejection in the inner corona. White-light coronagraph observations indicate that the first event was accompanied by an observable coronal mass ejection while the second event did not have clear evidence of a CME. Both eruptions were accompanied by metric type II radio bursts propagating at speeds in the range of 500–750 km s^–1, and neither had accompanying interplanetary type II activity. The timing and location of the flare waves appear to indicate an origin with the flaring region, but several signatures associated with coronal mass ejections indicate that the development of the CME may occur in concert with the development of the flare wave.     

Homotopy Continuation Method of Arbitrary Order of Convergence for the Two-Body Universal Initial Value Problem

In this paper, an efficient iterative method of arbitrary positive integer order of convergence 2 will be established for the two-body universal initial value problem. The method is of dynamic nature in the sense that, on going from one iterative scheme to the subsequent one, only additional instruction is needed. Moreover, which is the most important, the method does not need any a priori knowledge of the initial guess. A property which avoids the critical situations between divergent to very slow convergent solutions, that may exist in other numerical methods which depend on initial guess. Some applications of the method are also given.     

Modeling Coronal Loop Abundances: Effects of Parameter Variations on Observables

Coronal abundance variations arise from and affect solar atmospheric processes such as coronal heating and structural dynamics. Lenz (1999) presented initial results of a numerical theoretical study of abundances and ion heating rates in static, steady-state coronal loops. We present here a closer investigation of two fundamental aspects of the physics of coronal abundances, relating modeling parameters to observables: (1) the effect of varying the abundances on the electron temperature and (2) the effect of varying the ion heating rate on abundances.     

Propagation of the Slow Magnetoacoustic Waves in Coronal Loops Above Sunspots

The observations from the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO) have revealed the weak dis- turbances (WDs) propagating in the fan-like coronal loops of the active region (AR 11092) at 171 ?A, 193 ?A, and 211 ?A. These WDs seem to be a common phenomenon in this part of the active region. The disturbances originate from the bright loop foot, and propagate along the loops. The observed propagation speed decreases with the increasing temperature, and varies between 40 km/s and 121 km/s, close to and less than the sound speed in coronal loops. Consid- ering the projection effect and the different angles of the loops with respect to the line of sight, this is exactly what the slow-wave model expects. The wavelet analysis shows that the periods of the WDs observed in different wavebands have no signi?cant difference, the two distinct periods, 3 min and more than 10 min, are all detected in the three EUV wavebands. Not only the coronal loops but also the sunspot region in the chromosphere exhibit intensity oscillations with a period of the order of 3 min. This result suggests that the sunspot oscillations can propagate into the corona through the chromosphere and transition region.     

慢磁声波在黑子上方冕环中的传播

太阳动力学观测站(Solar Dynamics Observatory,SDO)装载的太阳大气成像系统(Atmospheric Imaging Assembly,AIA)通过3个极紫外波段(171 (A)、193 (A)和211 (A))的观测资料展现了太阳活动区(AR11092)扇形冕环微弱的扰动传播,这种波动现象普遍存在于活动区冕环.该扰动传播起源于明亮的环足处,其传播沿冕环方向.冕环中观测到的扰动传播速度随温度升高而减小,速度范围是40～121 km/s,其值接近且小于声速.考虑投影效应以及环与视线方向的倾角大小,这正好是慢波模型所预期的.冕环的扰动周期在不同波段(3个极紫外波段)没有明显的区别,都存在3 min以及10多分钟的周期.此外,不仅仅冕环扰动表现出3 min左右的周期,色球层的黑子区域同样存在相同的振荡周期.这个结果表明黑子振荡是可以穿过色球以及过渡区到达日冕的.     

The Parker Problem and the Theory of Coronal Heating

To illustrate his theory of coronal heating, Parker initially considers the problem of disturbing a homogeneous vertical magnetic field that is line-tied across two infinite horizontal surfaces. It is argued that, in the absence of resistive effects, any perturbed equilibrium must be independent of z. As a result random footpoint perturbations give rise to magnetic singularities, which generate strong Ohmic heating in the case of resistive plasmas. More recently these ideas have been formalized in terms of a magneto-static theorem but no formal proof has been provided. In this paper we investigate the Parker hypothesis by formulating the problem in terms of the fluid displacement. We find that, contrary to Parker's assertion, well-defined solutions for arbitrary compressibility can be constructed which possess non-trivial z-dependence. In particular, an analytic treatment shows that small-amplitude Fourier disturbances violate the symmetry ∂_z = 0 for both compact and non-compact regions of the (x, y) plane. Magnetic relaxation experiments at various levels of gas pressure confirm the existence and stability of the Fourier mode solutions. More general footpoint displacements that include appreciable shear and twist are also shown to relax to well-defined non-singular equilibria. The implications for Parker's theory of coronal heating are discussed.     

Effect of Flow on Resonant Absorption of Slow MHD Waves in Coronal Arcades

Resonant absorption of slow MHD waves is studied numerically by using the SGH method and is applied to a model of a coronal arcade in the presence of equilibrium plasma flows. The arcade is approximated by a 1D horizontal magnetic slab that is non-uniform along the vertical direction and which is surrounded by two homogeneous media. While propagating from the photosphere upwards into the corona, the magneto-acoustic waves can be resonantly absorbed in the inhomogeneous region of the arcade. Computational results show that the resonant absorption of the impinging waves strongly depends on the equilibrium model and on the characteristics of the driving wave. The results also indicate that the presence of an equilibrium plasma flow along the magnetic field of the arcade reduces the resonant absorption for the flow speed parameters considered.