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.     

Waves in Twisted Magnetic Flux Tubes

The modes of oscillation of a twisted magnetic flux tube in an incompressible medium are investigated analytically. An exact dispersion relation for the case of uniform twist is obtained. In contrast to the case of an untwisted incompressible tube, body, surface, and hybrid (surface-body) modes arise.     

On Analysis of Dual Spacecraft Stereoscopic Observations to Determine the Three-Dimensional Morphology and Plasma Properties of Solar Coronal Flux Tubes

By using two spacecraft equipped with multi-bandpass X-ray telescopes, it is possible to obtain direct 3-dimensional morphology of coronal structures which is essential for understanding the energetics and dynamics of the solar atmosphere. X-ray observations taken only in orbit about the Earth are inadequate to fully resolve the 3-dimensional nature of the solar corona. These Earth-orbit observations produce 2-dimensional images and an appropriate model must be included to derive the 3-dimensional structures from the line-of-sight information. Stereoscopic observations from space will remove this limitation and are needed if we are to improve our knowledge of the 3-dimensional morphology of the corona.Several important points regarding a stereoscopic mission are investigated and illustrated using model coronal flux tubes and image-rendering techniques. Synthesized images are formed by integrating the emission from volume elements along the line-of-sight path through a 3-dimensional volume in which a set of model flux tubes are located. The flux tubes are defined by (1) a plasma model defining the emissivity for a specific density, temperature, and pressure distribution, and (2) a magnetic field model from which a set of field lines are selected to define the geometry of the flux tubes. The field lines are used to define the flux-tube volume by assuming an initial base radius and conservation of flux. An effective instrumental spectral-response function is folded into the integration. Analysis of pairs of these synthesized images with various angular perspectives are used to investigate the effect of angular separation on mission objectives. The resulting images and analysis provide guidelines for developing a stereoscopic mission.Our study produced four important results, namely: (1) An angular separation of 30 degrees maximizes the scientific return by direct triangulation analysis because of the tradeoff between increased line-of-sight resolution of position and decreased recognition of individual loop structures arising from the overlapping of multiple loops with increasing angular separation. (2) The analysis benefits from the use of time-differential images to select flux tubes from the collection of numerous overlapping systems by selecting only recently heated or cooled flux tubes. (3) An analysis needs to be developed for algebraic reconstruction techniques applying a priori information, specific to the solar coronal structures, i.e., flux-tube continuity, maximum emission strength, non-negative emission, previous history, and maximum gradients of emission. (4) An analysis strategy combining triangulation, modeling techniques, and algebraic restoration is necessary to derive a complete understanding of the 3-dimensional morphology of the magnetic field. In the same way that helioseismology is classical viewing of the Sun with a tailored set of analysis tools for probing the interior of the Sun, heliostereoscopy is classical viewing of the X-ray emitting corona and requires a tailored set of analysis tools to deduce the true 3-dimensional structure of the corona.     

A Fixed-point Scheme for the Numerical Construction of Magnetohydrostatic Atmospheres in Three Dimensions

Magnetohydrostatic models of the solar atmosphere are often based on idealized analytic solutions because the underlying equations are too difficult to solve in full generality. Numerical approaches, too, are often limited in scope and have tended to focus on the two-dimensional problem. In this article we develop a numerical method for solving the nonlinear magnetohydrostatic equations in three dimensions. Our method is a fixed-point iteration scheme that extends the method of Grad and Rubin (Proc. 2nd Int. Conf. on Peaceful Uses of Atomic Energy 31, 190, 1958) to include a finite gravity force. We apply the method to a test case to demonstrate the method in general and our implementation in code in particular.     

Observational Evidence for Rising Penumbral Flux Tubes?

On 13 May 2000 parts of a penumbra were observed in an active region NOAA 8990 with the La Palma Stokes Polarimeter attached to the Swedish Vacuum Solar Telescope. The stratification over the solar atmosphere of different physical parameters is retrieved from these data by using the Stokes inversion based on response functions. The results confirm the previous findings of the penumbral structure. In general, the magnetic field becomes weaker and more horizontal with increasing distance from the umbra and the line-of-sight velocities are increasing towards the outer boundary of the penumbra. The results also suggest the existence of the unresolved fine structure of the penumbra. The stratifications of the temperature and of the magnetic field strength indicate the presence of rising flux tubes, which were predicted theoretically by Schlichenmaier, Jahn and Schmidt (1998, Astron. Astrophys. 337, 897).     

Magneto-Acoustic Waves in Compressible Magnetically Twisted Flux Tubes

The oscillatory modes of a magnetically twisted compressible flux tube embedded in a compressible magnetic environment are investigated in cylindrical geometry. Solutions to the governing equations to linear wave perturbations are derived in terms of Whittaker’s functions. A general dispersion equation is obtained in terms of Kummer’s functions for the approximation of weak and uniform internal twist, which is a good initial working model for flux tubes in solar applications. The sausage, kink and fluting modes are examined by means of the derived exact dispersion equation. The solutions of this general dispersion equation are found numerically under plasma conditions representative of the solar photosphere and corona. Solutions for the phase speed of the allowed eigenmodes are obtained for a range of wavenumbers and varying magnetic twist. Our results generalise previous classical and widely applied studies of MHD waves and oscillations in magnetic loops without a magnetic twist. Potential applications to solar magneto-seismology are discussed.     

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。     

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.     

The Importance of Photospheric Intense Flux Tubes for Coronal Heating

Dissipation of magnetic energy in the corona requires the creation of very fine scale-lengths because of the high magnetic Reynolds number of the plasma. The formation of current sheets is a natural possible solution to this problem and it is now known that a magnetic field that is stressed by continous photospheric motions through a series of equilibria can easily form such sheets. Furthermore, in a large class of 3D magnetic fields without null points there are locations, called quasi-separatrix layers (QSLs), where the field-line linkage changes drastically. They are the relevant generalisation of normal separatrices to configurations without nulls: along them concentrated electric currents are formed by smooth boundary motions and 3D magnetic reconnection takes place when the layers are thin enough. With a homogenous normal magnetic field component at the boundaries, the existence of thin enough QSL to dissipate magnetic energy rapidly requires that the field is formed by flux tubes that are twisted by a few turns. However, the photospheric field is not homogeneous but is fragmented into a large number of thin flux tubes. We show that such thin tubes imply the presence of a large number of very thin QSLs in the corona. The main parameter on which their presence depends is the ratio between the magnetic flux located outside the flux tubes to the flux inside. The thickness of the QSLs is approximately given by the distance between neighbouring flux tubes multiplied by the ratio of fluxes to a power between two and three (depending on the density of flux tubes). Because most of the photospheric magnetic flux is confined in thin flux tubes, very thin QSLs are present in the corona with a thickness much smaller than the flux tube size. We suggest that a turbulent resistivity is triggered in a QSL, which then rapidly evolves into a dynamic current sheet that releases energy by fast reconnection at a rate that we estimate to be sufficient to heat the corona. We conclude that the fragmentation of the photospheric magnetic field stimulates the dissipation of magnetic energy in the corona.     

Tubes of Magnetic Flux and Electric Current in Space Physics

The singularities of an irrotational magnetic field are lines of electric current. This property derives from the relationship between vector fields and the topology of the underlying three-space and allows for a definition of cosmic field flux tubes and flux ropes as cores (in the sense of the physics of defects) of helical singularities. When applied to force-free flux ropes, and assuming current conservation, an interesting feature is the quantization of the radii, pitches, and helicities. One expects similar quantization effects in the general case. In the special case when the total electric current vanishes, a force-free rope embedded in a medium devoid of magnetic field is nonetheless topologically stable, because it is the core of a singularity of the vector potential. Magnetic merging is also briefly discussed in the same framework.     

Non-equilibrium of Magnetic Flux Tubes emerging into the Solar Corona

A lack of equilibrium of twisted magnetic flux tubes emerging from the photosphere into the corona is considered. Assuming mass and flux conservation in the tube and an isothermal tube that is in thermal equilibrium with the surrounding plasma, it is shown that a sufficently rapid temperature increase through the transition zone may lead to the loss of magnetohydrostatic equilibrium of the emerging flux tube due to the enhancement of the plasma pressure inside the tube. The non-equilibrium leads to a rapid expansion of the tube to reach a new equilibrium state. The rise and expansion of the tube before and after the non-equilibrium are accompanied by an increase in the twist of the magnetic field. This may lead to the field exceeding the threshold for the onset of the kink instability and a subsequent explosive release of magnetic energy.     

Magnetohydrostatic Model of a Bald-Patch Flare

On 18 May, 1994, a subflare was observed in AR 7722 in X-rays by Yohkoh/SXT and in H at National Astronomical Observatory of Japan. The associated brightenings are due to small-scale magnetic energy release, triggered by parasitic fluxes emerging and moving at the edge of leading sunspots. Using the magnetohydrostatic equations derived by Low (1992), we model the magnetic field configuration by extrapolation of the Kitt Peak photospheric field, taking into account the effects of pressure and gravity. H flare kernels are shown to be located at computed separatrices associated with field lines which are tangent to the photosphere, namely 'bald patches (BPs). This is evidence that BPs can be involved in flares, and that current sheets can be dissipated in low levels of the solar atmosphere. The presence of dense plasma which is supported against gravity in the magnetic dips above BPs is correlated to dark elongated features observed in H. Mass flows in these flat fibrils are discussed in the context of energy release in the BP separatrices. The effect of the plasma on the computed magnetic configuration is shown to be of secondary importance with respect to the topology of the field.     

Linear MHD Sausage Waves in Compressible Magnetically Twisted Flux Tubes

Oscillations of magnetic flux tubes are of great importance as they contain information about the geometry and fine structure of the flux tubes. Here we derive and analytically solve in terms of Kummer’s functions the linear governing equations of wave propagation for sausage surface and body modes (m=0) of a magnetically twisted compressible flux tube embedded in a compressible uniformly magnetized plasma environment in cylindrical geometry. A general dispersion relation is obtained for such flux tubes. Numerical solutions for the phase velocity are obtained for a wide range of wavenumbers and for varying magnetic twist. The effect of magnetic twist on the period of oscillations of sausage surface modes for different values of the wavenumber and vertical magnetic field strength is calculated for representative photospheric and coronal conditions. These results generalize and extend previous studies of MHD waves obtained for incompressible or for compressible but nontwisted flux tubes. It is found that magnetic twist may change the period of sausage surface waves of the order of a few percent when compared to counterparts in straight nontwisted flux tubes. This information will be most relevant when high-resolution observations are used for diagnostic exploration of MHD wave guides in analogy to solar-interior studies by means of global eigenoscillations in helioseismology.     

Conditions for Propagation of Torsional Waves in Solar Magnetic Flux Tubes

Propagation of torsional waves along isothermal and initially-untwisted magnetic-flux tubes embedded in the solar atmosphere is studied analytically. Conditions for wave propagation along thin and wide magnetic-flux tubes are determined, and it is shown that the propagation along thin tubes is cutoff free; however, for wide tubes the propagation is affected by a cutoff frequency. A method to determine the cutoff frequency is presented and applied to a specific model of solar magnetic flux tubes. An interesting result is that the cutoff frequency is a local quantity in the model and that its value at a given height determines the frequency that torsional tube waves must have to propagate at this height.     

Non-ideal MHD Properties of Magnetic Flux Tubes in the Solar Photosphere

Magnetic flux tubes reaching from the solar convectivezone into the chromosphere have to pass through the relatively cool, and therefore non-ideal (i.e. resistive) photospheric region enclosed between the highly ideal sub-photospheric and chromospheric plasma. It is shown that stationary MHD equilibria of magnetic flux tubes which pass through this region require an inflow of photospheric material into the flux tube and a deviation from iso-rotation along the tube axis. This means that there is a difference in angular velocity of the plasma flow inside the tube below and above the non-ideal region. Both effects increase with decreasing cross section of the tube. Although for characteristic parameters of thick flux tubes the effect is negligible, a scaling law indicates its importance for small-scale structures. The relevance of this `inflow effect' for the expansion of flux tubes above the photosphere is discussed.     

Nonlinear Evolution of Axisymmetric Twisted Flux Tubes in the Solar Tachocline

We study the periodicity of twisting motions in sunspot penumbral filaments, which were recently discovered from space (Hinode) and ground-based (SST) observations. A sunspot was well observed for 97 minutes by Hinode/SOT in the G-band (4305 Å) on 12 November 2006. By the use of the time?–?space gradient applied to intensity space?–?time plots, twisting structures can be identified in the penumbral filaments. Consistent with previous findings, we find that the twisting is oriented from the solar limb to disk center. Some of them show a periodicity. The typical period is about ≈?four minutes, and the twisting velocity is roughly 6 km s^?1. However, the penumbral filaments do not always show periodic twisting motions during the time interval of the observations. Such behavior seems to start and stop randomly with various penumbral filaments displaying periodic twisting during different intervals. The maximum number of periodic twists is 20 in our observations. Studying this periodicity can help us to understand the physical nature of the twisting motions. The present results enable us to determine observational constraints on the twisting mechanism.     

On the Connection Between Mean Field Dynamo Theory and Flux Tubes

Mean field dynamo theory deals with various mean quantities and does not directly throw any light on the question of existence of flux tubes. We can, however, draw important conclusions about flux tubes in the interior of the Sun by combining additional arguments with the insights gained from solar dynamo solutions. The polar magnetic field of the Sun is of order 10 G, whereas the toroidal magnetic field at the bottom of the convection zone has been estimated to be 100000 G. Simple order-of-magnitude estimates show that the shear in the tachocline is not sufficient to stretch a 10 G mean radial field into a 100000 G mean toroidal field. We argue that the polar field of the Sun must get concentrated into intermittent flux tubes before it is advected to the tachocline. We estimate the strengths and filling factors of these flux tubes. Stretching by shear in the tachocline is then expected to produce a highly intermittent magnetic configuration at the bottom of the convection zone. The meridional flow at the bottom of the convection zone should be able to carry this intermittent magnetic field equatorward, as suggested recently by Nandy and Choudhuri (2002). When a flux tube from the bottom of the convection zone rises to a region of pre-existing poloidal field at the surface, we point out that it picks up a twist in accordance with the observations of current helicities at the solar surface.     

Sausage MHD Waves in Incompressible Flux Tubes with Twisted Magnetic Fields

Twisted magnetic flux tubes are of considerable interest because of their natural occurrence from the Sun’s interior, throughout the solar atmosphere and interplanetary space up to a wide range of applicabilities to astrophysical plasmas. The aim of the present work is to obtain analytically a dispersion equation of linear wave propagation in twisted incompressible cylindrical magnetic waveguides and find appropriate solutions for surface, body and pseudobody sausage modes (i.e. m = 0) of a twisted magnetic flux tube embedded in an incompressible but also magnetically twisted plasma. Asymptotic solutions are derived in long- and short-wavelength approximations. General solutions of the dispersion equation for intermediate wavelengths are obtained numerically. We found, that in case of a constant, but non-zero azimuthal component of the equilibrium magnetic field outside the flux tube the index ν of Bessel functions in the dispersion relation is not integer any more in general. This gives rise to a rich mode-structure of degenerated magneto-acoustic waves in solar flux tubes. In a particular case of a uniform magnetic twist the total pressure is found to be constant across the boundary of the flux tube. Finally, the effect of magnetic twist on oscillation periods is estimated under solar atmospheric conditions. It was found that a magnetic twist will increase, in general, the periods of waves approximately by a few percent when compared to their untwisted counterparts.     

How to Reach Superequipartition Field Strengths in Solar Magnetic Flux Tubes

A number of independent arguments indicate that the toroidal flux system responsible for the sunspot cycle is stored at the base of the convection zone in the form of flux tubes with field strength close to 10⁵ G. Although the evidence for such strong fields is quite compelling, how such field strength can be reached is still a topic of debate. Flux expulsion by convection should lead to about the equipartition field strength, but the magnetic energy density of a 10⁵-G field is two orders of magnitude larger than the mean kinetic energy density of convective motions. Line stretching by differential rotation (i.e., the “Ω effect” in the classical mean-field dynamo approach) probably plays an important role, but arguments based on energy considerations show that it does not seem feasible that a 10⁵-G field can be produced in this way. An alternative scenario for the intensification of the toroidal flux system in the overshoot layer is related to the explosion of rising, buoyantly unstable magnetic flux tubes, which opens a complementary mechanism for magnetic-field intensification. A parallelism is pointed out with the mechanism of “convective collapse” for the intensification of photospheric magnetic flux tubes up to field strengths well above equipartition; both mechanisms, which are fundamentally thermal processes, are reviewed.     

On Three-Dimensional Magnetic Skeleton Elements Due to Discrete Flux Sources

The magnetic field in the solar corona plays an important role in coronal heating, flaring activity and many other phenomena studied on the Sun. Magnetic topology is frequently used to understand complicated coronal magnetic fields. By calculating the skeleton of a field, it is possible to build up a sophisticated representation of the key elements of a field’s configuration. This paper determines a simple relation between the numbers of separators (X), coronal null points (N_c), flux domains (D) and flux sources (S) in such a configuration: D=X+S−N_c−1. This equation is used to explain the behaviour of some of the bifurcations found in Magnetic Charge Topology, and to show that a one-to-one relationship exists between the number of circuits in the domain graph and the augmented null graph. Finally, it is shown that in quiet-Sun regions, the number of separators is approximately proportional to the number of flux sources.