Thin Flux Tube Research Articles

Multiple Scattering of Seismic Waves from Ensembles of Upwardly Lossy Thin Flux Tubes

Our previous semi-analytic treatment of f- and p-mode multiple scattering from ensembles of thin flux tubes (Hanson and Cally, Astrophys. J. 781, 125; 791, 129, 2014) is extended by allowing both sausage and kink waves to freely escape at the top of the model using a radiative boundary condition there. As expected, this additional avenue of escape, supplementing downward loss into the deep solar interior, results in substantially greater absorption of incident f- and p-modes. However, less intuitively, it also yields mildly to substantially smaller phase shifts in waves emerging from the ensemble. This may have implications for the interpretation of seismic data for solar plage regions, and in particular their small measured phase shifts.

Effects of Radiative Diffusion on Thin Flux Tubes in Turbulent Solar-like Convection

We study the combined effects of convection and radiative diffusion on the evolution of thin magnetic flux tubes in the solar interior. Radiative diffusion is the primary supplier of heat to convective motions in the lower convection zone, and it results in a heat input per unit volume of magnetic flux tubes that has been ignored by many previous thin flux tube studies. We use a thin flux tube model subject to convection taken from a rotating spherical shell of turbulent, solar-like convection as described by Weber, Fan, and Miesch (2011, Astrophys. J., 741, 11; 2013, Solar Phys., 287, 239), now taking into account the influence of radiative heating on flux tubes of large-scale active regions. Our simulations show that flux tubes of less than or equal to 60 kG subject to solar-like convective flows do not anchor in the overshoot region, but rather drift upward due to the increased buoyancy of the flux tube earlier in its evolution as a result of the inclusion of radiative diffusion. Flux tubes of magnetic field strengths ranging from 15 kG to 100 kG have rise times of less than or equal to 0.2 years, and exhibit a Joy's Law tilt-angle trend. Our results suggest that radiative heating is an effective mechanism by which flux tubes can escape from the stably stratified overshoot region, and that flux tubes do not necessarily need to be anchored in the overshoot region to produce emergence properties similar to those of active regions on the Sun.

Small scale alpha-squared effect in the solar convection zone

Phenomena of self-organization of macroscopic structures in nature caused by the negative turbulent viscosity, which is basically the transfer of energy of the turbulent helical motion from small to large eddies (so-called inverse energy cascade), are reviewed. The effect of the helical negative turbulent diffusion (small-scale α2 effect by Kraichnan) that occurs in the strong helical turbulent highly conducting fluid in the large-scale, nonuniform magnetic field is considered. This effect has the property of “pulling” continuous magnetic fields together in discrete structures. Our calculations for the two models of the solar convection zone have shown that, in the deep layers, there are favorable conditions for the emergence of the effect of the helical negative turbulent diffusion. Under the assumption that the lifetime of helical structures τ2 equals the lifetime of convective cells τ1, the calculated values of the coefficient of the helical negative magnetic turbulent diffusion η T α reach two-thirds of the coefficient of the positive turbulent diffusion ηT. This should greatly facilitate the self-organization of originally weak continuous magnetic fields in isolated thin magnetic flux tubes.

Nonlinear slow magnetoacoustic waves in coronal plasma structures

Context. There is abundant observational evidence of longitudinal waves in the plasma structures of the solar corona. These essentially compressive waves are confidently interpreted as slow magnetoacoustic waves. The use of the slow waves in plasma diagnostics and estimating their possible contribution to plasma heating and acceleration require detailed theoretical modelling. Aims. We investigate the role of obliqueness and magnetic effects in the evolution of slow magnetoacoustic waves, also called tube waves, in field-aligned plasma structures. Special attention is paid to the wave damping caused by nonlinear steepening. Methods. We considered an untwisted straight axisymmetric field-aligned plasma cylinder and analysed the behaviour of the slow magnetoacoustic waves that are guided by this plasma structure. We adopted a thin flux tube approximation. We took into account dissipation caused by viscosity, resistivity and thermal conduction, and nonlinearity. Effects of stratification and dispersion caused by the finite radius of the flux tube were neglected. Results. We derive the Burgers-type evolutionary equation for tube waves in a uniform plasma cylinder. Compared with a plane acoustic wave, the formation of shock fronts in tube waves is found to occur at a larger distance from the source. In addition, tube waves experience stronger damping. These effects are most pronounced in plasmas with the parameter β at about or greater than unity. In a low-β plasma, the evolution of tube waves can satisfactorily be described with the Burgers equation for plane acoustic waves.

THE SCATTERING OFf- ANDp-MODES FROM ENSEMBLES OF THIN MAGNETIC FLUX TUBES: AN ANALYTICAL APPROACH

Motivated by the observational results of Braun (1995), we extend the model of Hanson & Cally (2014) to address the effect of multiple scattering of f and p-modes by an ensemble of thin vertical magnetic flux tubes in the surface layers of the Sun. As in observational Hankel analysis we measure the scatter and phase shift from an incident cylindrical wave in a coordinate system roughly centred in the core of the ensemble. It is demonstrated that, although thin flux tubes are unable to interact with high order fluting modes individually, they can indirectly absorb energy from these waves through the scatters of kink and sausage components. It is also shown how the distribution of absorption and phase shift across the azimuthal order m depends strongly on the tube position, as well as on the individual tube characteristics. This is the first analytical study into an ensembles multiple scattering regime, that is embedded within a stratified atmosphere.

THE HISTORY OF A QUIET-SUN MAGNETIC ELEMENT REVEALED BY IMaX/SUNRISE

Isolated flux tubes are considered to be fundamental magnetic building blocks of the solar photosphere. Their formation is usually attributed to the concentration of magnetic field to kG strengths by the convective collapse mechanism. However, the small size of the magnetic elements in quiet-Sun areas has prevented this scenario from being studied in fully resolved structures. Here we report on the formation and subsequent evolution of one such photospheric magnetic flux tube, observed in the quiet Sun with unprecedented spatial resolution (0\farcs 15 - 0\farcs 18) and high temporal cadence (33 s). The observations were acquired by the Imaging Magnetograph Experiment (IMaX) aboard the \textsc{Sunrise} balloon-borne solar observatory. The equipartition field strength magnetic element is the result of the merging of several same polarity magnetic flux patches, including a footpoint of a previously emerged loop. The magnetic structure is then further intensified to kG field strengths by convective collapse. The fine structure found within the flux concentration reveals that the scenario is more complex than can be described by a thin flux tube model with bright points and downflow plumes being established near the edges of the kG magnetic feature. We also observe a daisy-like alignment of surrounding granules and a long-lived inflow towards the magnetic feature. After a subsequent weakening process, the field is again intensified to kG strengths. The area of the magnetic feature is seen to change in anti-phase with the field strength, while the brightness of the bright points and the speed of the downflows varies in phase. We also find a relation between the brightness of the bright point and the presence of upflows within it.

Fast magnetohydrodynamic oscillation of longitudinally inhomogeneous prominence threads: an analogue with quantum harmonic oscillator

Previous works indicate that the frequency ratio of second and first harmonics of kink oscillations has tendency towards 3 in the case of prominence threads. We aim to study the magnetohydrodynamic oscillations of longitudinally inhomogeneous prominence threads and to shed light on the problem of frequency ratio. Classical Sturm--Liouville problem is used for the threads with longitudinally inhomogeneous plasma density. We show that the spatial variation of total pressure perturbations along the thread is governed by the stationary Schr\"{o}dinger equation, where the longitudinal inhomogeneity of plasma density stands for the potential energy. Consequently, the equation has bounded solutions in terms of Hermite polynomials. Boundary conditions at the thread surface lead to transcendental dispersion equation with Bessel functions. Thin flux tube approximation of the dispersion equation shows that the frequency of kink waves is proportional to the expression \alpha(2n+1), where \alpha is the density inhomogeneity parameter and n is the longitudinal mode number. Consequently, the ratio of the frequencies of second and first harmonics tends to 3 in prominence threads. Numerical solution of the dispersion equation shows that the ratio only slightly decreases for thicker tubes in the case of smaller longitudinal inhomogeneity of external density, therefore the thin flux tube limit is a good approximation for prominence oscillations. However, stronger longitudinal inhomogeneity of external density may lead to the significant shift of frequency ratio for wider tubes and therefore the thin tube approximation may fail. The tendency of frequency ratio of second and first harmonics towards 3 in prominence threads is explained by the analogy of the oscillations with quantum harmonic oscillator, where the density inhomogeneity of the threads plays a role of potential energy.

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.

AN ANALYTICAL APPROACH TO SCATTERING BETWEEN TWO THIN MAGNETIC FLUX TUBES IN A STRATIFIED ATMOSPHERE

We expand on recent studies to analytically model the behavior of two thin flux tubes interacting through the near- and acoustic far-field. The multiple scattering that occurs between the pair alters the absorption and phase of the outgoing wave, when compared to non-interacting tubes. We have included both the sausage and kink scatter produced by the pair. It is shown that the sausage mode's contribution to the scattered wave field is significant, and plays an equally important role in the multiple scattering regime. A disparity between recent numerical results and analytical studies, in particular the lack of symmetry between the two kink modes, is addressed. This symmetry break is found to be caused by an incorrect solution for the near-field modes.

CONDITIONS FOR TRANSVERSE WAVES PROPAGATION ALONG THIN MAGNETIC FLUX TUBES ON THE SUN

The propagation of kink waves in the thin gravity stratified flux tubes with a generalized magnetic field distribution model is considered in cylindrical geometry. The new kink wave equations for both wave variables are obtained. It is shown that the inclusion of the radial component of an unperturbed tube magnetic field sufficiently transforms the conditions for the propagation of transverse waves. It is demonstrated that, for the models of isothermal and polytropic atmosphere in the tube and its environment, the propagation of kink waves along thin magnetic flux tubes is cutoff-free.

Probing the structure of local magnetic field of solar features with helioseismology

AbstractMotivated by the problem of local solar subsurface magnetic structure, we have used numerical simulations to investigate the propagation of waves through monolithic magnetic flux tubes of different sizes. A cluster model can be a good approximation to simulate sunspots as well as solar plage regions which are composed of an ensemble of compactly packed thin flux tubes. Simulations of this type are powerful tools to probe the structure and the dynamics of various solar features which are directly related to solar magnetic field activity.

A THEORY ON THE CONVECTIVE ORIGINS OF ACTIVE LONGITUDES ON SOLAR-LIKE STARS

Using a thin flux tube model in a rotating spherical shell of turbulent, solar-like convective flows, we find that the distribution of emerging flux tubes in our simulation is inhomogeneous in longitude, with properties similar to those of active longitudes on the Sun and other solar-like stars. The large-scale pattern of flux emergence our simulations produce exhibits preferred longitudinal modes of low order, drift with respect to a fixed reference system, and alignment across the Equator at low latitudes between 15 degrees. We suggest that these active-longitude-like emergence patterns are the result of columnar, rotationally aligned giant cells present in our convection simulation at low latitudes. If giant convecting cells exist in the bulk of the solar convection zone, this phenomenon, along with differential rotation, could in part provide an explanation for the behavior of active longitudes.

SYNOPTIC MAPPING OF CHROMOSPHERIC MAGNETIC FLUX

We used daily full-disk Ca II 854.2?nm magnetograms from the Synoptic Optical Long Term Investigations of the Sun (SOLIS) facility to study the chromospheric magnetic field from 2006 April through 2009 November. We determined and corrected previously unidentified zero offsets in the SOLIS magnetograms. By tracking the disk passages of stable unipolar regions, the measured net flux densities were found to systematically decrease from the disk center to the limb by a factor of about two. This decrease was modeled using a thin flux tube model with a difference in signal formation height between the center and limb sides. Comparison of photospheric and chromospheric observations shows that their differences are largely due to horizontal spreading of magnetic flux with increasing height. The north polar magnetic field decreased nearly linearly with time during our study period while the south polar field was nearly constant. We used the annual change in the viewing angle of the polar regions to estimate the radial and meridional components of the polar fields and found that the south polar fields were tilted away from the pole. Synoptic maps of the chromospheric radial flux density distribution were used as boundary conditions for extrapolation of the field from the chromosphere into the corona. A comparison of modeled and observed coronal hole boundaries and coronal streamer positions showed better agreement when using the chromospheric rather than the photospheric synoptic maps.

GLOBAL AND LOCAL CUTOFF FREQUENCIES FOR TRANSVERSE WAVES PROPAGATING ALONG SOLAR MAGNETIC FLUX TUBES

The propagation of linear transverse waves along a thin isothermal magnetic flux tube is affected by a global cutoff frequency that separates propagating and non-propagating waves. In this paper, wave propagation along a thin but non-isothermal flux tube is considered and a local cutoff frequency is derived. The effects of different temperature profiles on this local cutoff frequency are studied by considering different power-law temperature distributions as well as the semi-empirical VAL C model of the solar atmosphere. The results show that the conditions for wave propagation strongly depend on the temperature gradients. Moreover, the local cutoff frequency calculated for the VAL C model gives constraints on the range of wave frequencies that are propagating in different parts of the solar atmosphere. These theoretically predicted constraints are compared to observational data and are used to discuss the role played by transverse tube waves in the atmospheric heating and dynamics, and in the excitation of solar atmospheric oscillations.

Torsional Alfvén waves in partially ionized solar plasma: effects of neutral helium and stratification

Ion-neutral collisions may lead to the damping of Alfven waves in chromospheric and prominence plasmas. Neutral helium atoms enhance the damping in certain temperature interval, where the ratio of neutral helium and neutral hydrogen atoms is increased. Therefore, the height-dependence of ionization degrees of hydrogen and helium may influence the damping rate of Alfven waves. We aim to study the effect of neutral helium in the damping of Alfven waves in stratified partially ionized plasma of the solar chromosphere. We consider a magnetic flux tube, which is expanded up to 1000 km height and then becomes vertical due to merging with neighboring tubes, and study the dynamics of linear torsional Alfven waves in the presence of neutral hydrogen and neutral helium atoms. We start with three-fluid description of plasma and consequently derive single-fluid magnetohydrodynamic (MHD) equations for torsional Alfven waves. Thin flux tube approximation allows to obtain the dispersion relation of the waves in the lower part of tubes, while the spatial dependence of steady-state Alfven waves is governed by Bessel type equation in the upper part of tubes. Consecutive derivation of single-fluid MHD equations results in a new Cowling diffusion coefficient in the presence of neutral helium which is different from previously used one. We found that shorter-period (< 5 s) torsional Alfven waves damp quickly in the chromospheric network due to ion-neutral collision. On the other hand, longer-period (> 5 s) waves do not reach the transition region as they become evanescent at lower heights in the network cores. Propagation of torsional Alfven waves through the chromosphere into the solar corona should be considered with caution: low-frequency waves are evanescent due to the stratification, while high-frequency waves are damped due to ion neutral collisions.

MAGNETIC FIELD INTENSIFICATION BY THE THREE-DIMENSIONAL “EXPLOSION” PROCESS

We investigate an intensification mechanism for the magnetic field near the base of the solar convection zone that does not rely on differential rotation. Such mechanism in addition to differential rotation has been suggested by studies of flux emergence, which typically require field strength in excess of those provided by differential rotation alone. We study here a process in which potential energy of the superadiabatically stratified convection zone is converted into magnetic energy. This mechanism, know as explosion of magnetic flux tubes, has been previously studied in the thin flux tube approximation as well as two-dimensional MHD simulations, we expand the investigation to three-dimensional MHD simulations. Our main result is that enough intensification can be achieved in a three-dimensional magnetic flux sheet as long as the spatial scale of the imposed perturbation normal to the magnetic field is sufficiently large. When this spatial scale is small, the flux sheet tends to rise toward the surface, resulting in a significant decrease of the magnetic field amplification.

THE EFFECT OF A TWISTED MAGNETIC FIELD ON THE PERIOD RATIOP1/P2OF NONAXISYMMETRIC MAGNETOHYDRODYNAMIC WAVES

The nonaxisymmetric magnetohydrodynamic (MHD) modes in a zero-beta cylindrical compressible thin magnetic flux tube modelled as a twisted core surrounded by a magnetically twisted annulus, both embedded in a straight ambient external field is considered. The dispersion relation is derived and solved analytically and numerically to obtain the frequencies of the nonaxisymmetric MHD waves. The main result is that the twisted magnetic annulus does affect the period ratio P_1/P_2 of the kink modes. For the kink modes, the magnetic twist in the annulus region can achieve deviations from P_1/P_2 = 2 of the same order of magnitude as in the observations. Furthermore, the effect of the internal twist on the fluting modes is investigated.

Comparing Simulations of Rising Flux Tubes Through the Solar Convection Zone with Observations of Solar Active Regions: Constraining the Dynamo Field Strength

We study how active-region-scale flux tubes rise buoyantly from the base of the convection zone to near the solar surface by embedding a thin flux tube model in a rotating spherical shell of solar-like turbulent convection. These toroidal flux tubes that we simulate range in magnetic field strength from 15 kG to 100 kG at initial latitudes of 1 degree to 40 degrees in both hemispheres. This article expands upon Weber, Fan, and Miesch (Astrophys. J., 741, 11, 2011) (Article 1) with the inclusion of tubes with magnetic flux of 10^20 Mx and 10^21 Mx, and more simulations of the previously investigated case of 10^22 Mx, sampling more convective flows than the previous article, greatly improving statistics. Observed properties of active regions are compared to properties of the simulated emerging flux tubes, including: the tilt of active regions in accordance with Joy's Law as in Article 1, and in addition the scatter of tilt angles about the Joy's Law trend, the most commonly occurring tilt angle, the rotation rate of the emerging loops with respect to the surrounding plasma, and the nature of the magnetic field at the flux tube apex. We discuss how these diagnostic properties constrain the initial field strength of the active region flux tubes at the bottom of the solar convection zone, and suggest that flux tubes of initial magnetic field strengths of \geq 40 kG are good candidates for the progenitors of large (10^21 Mx to 10^22 Mx) solar active regions, which agrees with the results from Article 1 for flux tubes of 10^22 Mx. With the addition of more magnetic flux values and more simulations, we find that for all magnetic field strengths, the emerging tubes show a positive Joy's Law trend, and that this trend does not show a statistically significant dependence on the magnetic flux.

Current sheet oscillations in the magnetic filament approach

Magnetic filament approach is applied for modeling of nonlinear “kink”-like flapping oscillations of thin magnetic flux tubes in the Earth’s magnetotail current sheet. A discrete approximation for the magnetic flux tube was derived on a basis of the Hamiltonian formulation of the problem. The obtained system of ordinary differential equations was integrated by method of Rosenbrock, which is suitable for stiff equations. The two-dimensional exact Kan’s solution of the Vlasov equations was used to set the background equilibrium conditions for magnetic field and plasma. Boundary conditions for the magnetic filament were found to be dependent on the ratio of the ionospheric conductivity and the Alfvén conductivity of the magnetic tube. It was shown that an enhancement of this ratio leads to the corresponding increase of the frequency of the flapping oscillations. For some special case of boundary conditions, when the magnetic perturbations vanish at the boundaries, the calculated frequency of the “kink”-like flapping oscillations is rather close to that predicted by the “double gradient” analytical model. For others cases, the obtained frequency of the flapping oscillations is somewhat larger than that from the “double gradient” theory. The frequency of the nonlinear flapping oscillations was found to be a decreasing function of the amplitude.

LONGITUDINAL OSCILLATIONS IN DENSITY STRATIFIED AND EXPANDING SOLAR WAVEGUIDES

Waves and oscillations can provide vital information about the internal structure of waveguides they propagate in. Here, we analytically investigate the effects of density and magnetic stratification on linear longitudinal magnetohydrodynamic (MHD) waves. The focus of this paper is to study the eigenmodes of these oscillations. It is our specific aim is to understand what happens to these MHD waves generated in flux tubes with non-constant (e.g., expanding or magnetic bottle) cross-sectional area and density variations. The governing equation of the longitudinal mode is derived and solved analytically and numerically. In particular, the limit of the thin flux tube approximation is examined. The general solution describing the slow longitudinal MHD waves in an expanding magnetic flux tube with constant density is found. Longitudinal MHD waves in density stratified loops with constant magnetic field are also analyzed. From analytical solutions, the frequency ratio of the first overtone and fundamental mode is investigated in stratified waveguides. For small expansion, a linear dependence between the frequency ratio and the expansion factor is found. From numerical calculations it was found that the frequency ratio strongly depends on the density profile chosen and, in general, the numerical results are in agreement with the analytical results. The relevance of these results for solar magneto-seismology is discussed.