Time-dependent interaction of granules with magnetic flux tubes

The time-dependent interaction of the granulation velocity field with a magnetic flux tube is investigated here. It is seen that when a magnetic field line is displaced normal to itself so as to simulate thebuffeting action of granules, a flow of gas is initiated along the field. By choosing a lateral velocity field which is consistent with observations of granules, it is found that the resulting gas motion is a downward flow with a velocity compatible with the observed downflow in isolated photospheric flux tubes. It is therefore proposed that the observed photospheric downflow is a manifestation of the interaction of granules with flux tubes.     

On the origin of strong downdrafts associated with the birth of sunspots

During the emerging phase of sunspot pores strong downdrafts are observed in photospheric lines, whereas the upward flow is not detected within the observational accuracy. For the study of the origin of these downdrafts, we present the numerical solutions of non-steady hydrodynamic equations in one dimension along the rising magnetic flux tubes. In these solutions, it is assumed that the solar gas inside the tube is initially in the hydrostatic HSRA and then the tube rises with a given velocity and configuration. The results reproduce well the observed radial velocity distributions, hence it is concluded that the observed strong downdrafts originate from the sliding matter along the rising magnetic flux tubes.     

Nonlinear development of convective instability within slender flux tubes. I. Adiabatic flow

The nonlinear development of convective instability within slender flux tubes is studied using the method of characteristics. It is seen that the initial magnetic field influences the development of the instability. The asymptotic state of the unstable tube depends on the boundary conditions. Flux tubes subjected to ’open’ boundary conditions show a better evidence for field amplification than those subjected to ’closed’ boundary conditions. In either case, convective instability results in the generation of significant gas flow within slender flux tubes.     

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.     

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.     

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.     

3-D non-linear evolution of a magnetic flux tube in a spherical shell: The isentropic case

We present recent 3-D MHD numerical simulations of the non-linear dynamical evolution of magnetic flux tubes in an adiabatically stratified convection zone in spherical geometry, using the anelastic spherical harmonic (ASH) code.We seek to understand the mechanism of emergence of strong toroidal fields from the base of the solar convection zone to the solar surface as active regions. We confirm the results obtained in cartesian geometry that flux tubes that are not twisted split into two counter vortices before reaching the top of the convection zone. Moreover, we find that twisted tubes undergo the poleward-slip instability due to an unbalanced magnetic curvature force which gives the tube a poleward motion both in the non-rotating and in the rotating case. This poleward drift is found to be more pronounced on tubes originally located at high latitudes. Finally, rotation is found to decrease the rise velocity of the flux tubes through the convection zone, especially when the tube is introduced at low latitudes. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Development of structure in pores and sunspots: flows around axisymmetric magnetic flux tubes

Flux elements, pores and sunspots form a family of magnetic features observed at the solar surface. As a first step towards developing a fully non-linear model of the structure of these features and of the dynamics of their interaction with solar convection, we conduct numerical experiments on idealized axisymmetric flux tubes in a compressible convecting atmosphere in cylindrical boxes of radius up to 8 times their depth. We find that the magnetic field strength of the flux tubes is roughly independent of both distance from the centre and the total flux content of the flux tube, but that the angle of inclination from the vertical of the field at the edge of the tube increases with flux content. In all our calculations, fluid motion converges on the flux tube at the surface. The results compare favourably with observations of pores; in contrast, large sunspots lie at the centre of an out-flowing moat cell. We conjecture that there is an inflow hidden beneath the penumbrae of large spots, and that this inflow is responsible for the remarkable longevity of such features.     

Morphology of the magnetic field near Mars and the role of the magnetic crustal anomalies: Dayside region

We have studied the morphology of magnetic flux tubes near Mars and have found that the magnetic field lines near Mars forms a wing-like flux tube structure downstream of the bow shock. These magnetic flux tubes are concentrated close to the plane, which contains the center of Mars, the interplanetary magnetic field, and the Mars-Sun line. Regions near Mars on dayside were found to be magnetically connected to the region downstream of the bow shock in the sunlight. The study suggests that the photoelectrons that were observed on the nightside far from Mars are associated with magnetic field lines which are, or which were, magnetically connected to the Martian dayside region.     

Magnetic flux tubes — nonlinear strings in relativistic magnetohydrodynamics

A special Lagrangian coordinate system is introduced, in which the magnetic flux tube is the main object of investigation. The flux tube is shown to behave as a nonlinear relativistic string in relation to tangential force. General and particular variational principles are also discussed. With the introduction of a Lagrangian coordinate system the relativistic magnetohydrodynamic equation of motion reduces to a set of nonlinear string equations. From these results it follows that a highly conductive plasma with a frozen-in magnetic field can be considered as a gas of nonlinear strings interacting through pressure forces. A method is developed which allows the multi-dimensional nonlinear problems of RMHD (relativistic MHD) with boundary layers to be reduced to a set of two-dimensional problems for flux tubes, i.e., nonlinear strings.     

Statistics of depleted flux tubes in the jovian magnetosphere

On many of its passes through the Io torus the Galileo spacecraft has detected the presence of what appear to be thin magnetic flux tubes with fields somewhat higher than their surroundings. On these flux tubes the magnetic pressure is sufficiently above the pressure of neighboring tubes that it is possible the plasma contributions to the pressure within these tubes are depleted. Due to their short duration, they are only detectable in high time-resolution magnetometer data. Herein we survey all high time-resolution data that are available over the full Galileo mission and present a final statistical study. These tubes occupy 0.32% of the torus outside the orbit of Io. None are found inside. Their strength indicates that the ratio of the thermal pressure to magnetic pressure in the outer torus is about 2%. Comparison of the observed electron density in the neighborhood of these tubes indicates that the ion temperature is in the range 30-100 eV, consistent with other estimates. The amount of magnetic flux transported by these thin tubes could supply the amount of magnetic flux mass-loaded and transported to the magnetotail if the inward velocity is about 300 times that of the outward transport. Finally, the thin flux tubes are found in clusters, as they would occur if they resulted from the breakup of larger flux tubes.     

Twist transport in strongly torsioned astrophysical flux tubes

Riemannian geometrical effects on the expansion of the electron magnetohydrodynamical (EMH) superconductivity modeled twisted nonplanar thin magnetic flux tubes are considered. A solution is found which represents almost incompressible plasma flows, where the twist of flux tube is computed in terms of the continuous variation of its cross-section. It is shown that the twist increases in regions where twisted flux tube expands as in Parker’s conjecture. From computation of compression along the tube we show that when the torsion is weak a centrifugal or vorticity effect on the longitudinal direction of the tube enhances the screening effect on the “superconductor”. Throughout the paper we consider helical flux tubes where torsion and curvature of the tube are constants. Thus we show that the Parker’s conjecture is valid in a continuos manner for these type II superconducting twisted flux tubes. Throughout the paper we adopt the approximation that the radial component of the magnetic field varies so slowly along the tube axis that it can be approximated to zero along the tube. It is suggested that the models discussed here may also be applied to DNA and nanotubes.     

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.     

The rise of twisted magnetic flux tubes

Sunspots are caused by the eruption of magnetic flux tubes through the solar photosphere: current theories of the internal magnetic field of the Sun suggest that such tubes must rise relatively unscathed from the base of the convection zone. In order to understand how the structure of the magnetic field within a buoyant flux tube affects its stability as it rises, we have considered the quasi-two-dimensional rise of isolated magnetic flux tubes through an adiabatically stratified atmosphere. The magnetic field is initially helical; we have investigated a range of initial field configurations, varying the distribution and strength of the twist of the field.     

The stability of a magnetic flux element in a horizontally stratified compressible plasma

The configuration of a magnetic flux element in a static, compressible, gravitationally stratified plasma is considered. Under isothermal conditions an exact force-free solution is given for a two-dimensional cartesian flux sheath but for an axi-symmetric element, i.e. a flux tube, approximate solutions, applicable only to thin flux tubes, are obtained.When the isothermal condition is relaxed within the flux tube, non-force-free solutions are obtained exhibiting either a temperature excess or deficit on the axis of the flux tube. Using these solutions, the total potential energy of such a tube and its surroundings may be calculated. By comparing this with the total energy of an equivalent system ofn flux tubes the temperature excess or deficit in these tubes corresponding to the adiabatic subdivision of the single tube is determined.It is shown that under non-adiabatic conditions the subdivision process must be endothermic (i.e., external energy is required) if the temperature within the original tube is significantly less than its surroundings but exothermic if the temperatures are comparable. Thus it is conjectured that magnetic structures are less susceptible to subdivision if they are significantly cooler than their surroundings.     

Convective intensification of magnetic fields in the quiet Sun

Kilogauss-strength magnetic fields are often observed in intergranular lanes at the photosphere in the quiet Sun. Such fields are stronger than the equipartition field B _e, corresponding to a magnetic energy density that matches the kinetic energy density of photospheric convection, and comparable with the field B _p that exerts a magnetic pressure equal to the ambient gas pressure. We present an idealized numerical model of three-dimensional compressible magnetoconvection at the photosphere, for a range of values of the magnetic Reynolds number. In the absence of a magnetic field, the convection is highly supercritical and characterized by a pattern of vigorous, time-dependent, 'granular' motions. When a weak magnetic field is imposed upon the convection, magnetic flux is swept into the convective downflows where it forms localized concentrations. Unless this process is significantly inhibited by magnetic diffusion, the resulting fields are often much greater than B _e and the high magnetic pressure in these flux elements leads to their being partially evacuated. Some of these flux elements contains ultraintense magnetic fields that are significantly greater than B _p. Such fields are contained by a combination of the thermal pressure of the gas and the dynamic pressure of the convective motion, and they are constantly evolving. These ultraintense fields develop owing to non-linear interactions between magnetic fields and convection; they cannot be explained in terms of 'convective collapse' within a thin flux tube that remains in overall pressure equilibrium with its surroundings.     

On the appearance of magnetic flux in the solar photosphere

Ideas and models for the appearance of photospheric magnetic structure are confronted with observational data. Some findings are: The magnetic flux emerging in an active region consists of a bundle of flux tubes which were already concentrated before penetrating into the photosphere. A model of a rising bunch of flux tubes joining into a few strands at larger depths describes the coalescence of spots near the leading and following edges of the active region while more flux may surface near the center of the region. There is no observational evidence for appreciable helical twists in the flux bundles.Throughout the region's lifetime the magnetic elements move coherently, the whole magnetic structure rotates faster than the quiet photosphere. In active regions the convective flow at scales larger than the granulation is arrested by the magnetic structure. The long-lived supergranular cells around spots and in the enhanced network in turn determine the decay properties of spots and facular clusters. The modulation of the convective flow by the magnetic structure explains the slow dispersal of faculae.The hierarchy of magnetic elements (sunspots-pores-knots-facular clusters-facular points) may be explained by a set of magnetostatic flux tube models in the top of the convection zone. The underlying assumptions are that the heat flow along the magnetic field is reduced and that there is no heat exchange across the field except by radiation.A tentative model is proposed to account for the amplification, ascent and emergence of intense flux bundles. The assumptions are: (i) the field is concentrated in toroidal bundles by differential rotation, (ii) in the deep convection zone flux bundles are contained by the external turbulent pressure, and (iii) for field strengths up to the equipartition value efficient lateral heat exchange is possible. After a loop has surfaced radiative cooling and subsequent convective downflow reduce the temperature in the top of the flux tubes which then contract to field strengths well above the local equipartition value. There the heat flow is channelled along the field, which creates the conditions for the magnetostatic flux tube models without requiring a blocking of the heat flow somewhere within the tubes.The paper contains a brief review on the evolution of the magnetic field from the emergence in active regions up to the enigmatic disappearance, and a list of topics for further observational investigation.     

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.     

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^° to 40^° 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 10²⁰ Mx and 10²¹ Mx, and more simulations of the previously investigated case of 10²² 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 ≥?40 kG are good candidates for the progenitors of large (10²¹ Mx to 10²² Mx) solar active regions, which agrees with the results from Article 1 for flux tubes of 10²² 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.