Model calculations of the rising motion of a prominence loop

Model calculations are presented for the rising motion of the top section of a prominence loop, which is represented by a straight flux rope immersed in a coronal medium permeated with a bipolar magnetic field. Initially the prominence is at rest, in equilibrium with the surrounding coronal medium. When the magnetic monopoles that account for the source current for the bipolar field strengthen, the upward hydromagnetic buoyancy force overcomes the downward gravitational force so that the prominence is initiated into rising motion. The illustrative examples show that prominences can move away from the solar surface by the action of the hydromagnetic buoyancy force, which is preponderant with the diamagnetic force due to the current carried by the prominence interacting with the coronal magnetic field produced by the photospheric currents, if the changes in the photospheric magnetic field are sufficiently large.     

Hydromagnetic buoyancy force in the solar atmosphere

An extraneous magnetized body, either a flux tube or a plasmoid, immersed in the solar atmosphere is subjected to a hydromagnetic buoyancy force. It results from the peripheral inhomogeneity of ambient hydromagnetic pressure, which is caused or enhanced by the presence of the extraneous body. This extra-caused force acts at various mass elements of the immersed body through its distribution as a nearly uniform force density, just like the gravitational force. Since hydromagnetic buoyancy force comprises hydrostatic buoyancy force, hydrodynamic lift force, and magnetostatic diamagnetic force, this constitutes a magnetohydrodynamic generalization of Archimedes' principle which deals with hydrostatic buoyancy force.In the solar atmosphere hydromagnetic buoyancy force has an obliquely upward direction, with a component in the direction opposite to the downward gravity. It provides an upward force to counterbalance or even to exceed the downward gravitational force. Such an upward force is the dynamic cause for the stationary equilibrium of quiescent prominences and outward motion of coronal transients.     

A magnetohydrodynamic theory of coronal loop transients

A magnetohydrodynamic theory is presented for coronal loop transients. It is shown that the heliocentrifugal motion of a transient loop, as exhibited by the translational displacement of the axis of the loop, is driven by the magnetohydrodynamic buoyancy force exerted by the ambient medium. Self-induced hydromagnetic force, which includes the magnetic force produced by the internally driven current and the thermal force produced by the pressure imbalance between the internal and external gas pressures, causes the peripheral expansion of the loop, as exhibited by the lateral broadening and longitudinal stretching. This contention is substantiated by an analysis based on a model structure for a coronal loop.Besides accounting for the acceleration and expansion of a transient loop, this magnetohydrodynamic theory also provides an explanation for the initial ejection of a coronal loop from stationary equilibrium. Magnetic unwinding in consequence of abrupt magnetic activities at the solar surface will cause the periphery of a stationary coronal loop to expand. The increase in volume will enhance the magnetohydrodynamic buyoyancy force to exceed the gravitational force. Once a coronal loop is ejected from the solar surface, it will be continually accelerated and undergo expansion. Eventually a transient loop will blend with the ambient solar wind. This is also indicated by the theory presented in this paper.     

A method to calculate electric currents in quiescent prominences

A 2-dimensional model of the magnetic field associated with quiescent prominences is presented. The coronal field is assumed to be current-free, currents are only allowed in the photosphere and inside the prominence. The prominence is taken to be infinitely thin. For this model a method is given to calculate the field configuration from the observed normal component of the field both in the photosphere and the prominence. The normal field components are inferred from disc observations and H limb observations. The sheet currents inside the prominence are calculated and the resulting Lorentz force is compared with the gravitational force. Within the range of uncertainty in the total hydrogen density of quiescent prominences it is possible to give models where the gravity is balanced by the Lorentz force.     

An acceleration mechanism for loop transients in the outer corona

The heliocentrifugal motion of coronal loop transients is likely driven largely by the buoyant force exerted by the ambient medium. In the outer corona where the solar wind is well formed, the buoyant force results mainly from the rapid outward decrease in the ambient pressure of the solar wind. The contribution from magnetic buoyancy is not so significant as in the vicinity of the solar surface. Therefore, the pertinent features of the loop transients in the outer corona are basically gasdynamical. As a conspicuous part of coronal expansion, the motion of the compressible masses in the transient loops is largely controlled by thermal forces. The translational motion of heliocentrifugal expansion is driven by the hydrodynamic buoyant force, and the lateral motion of peripheral expansion is driven by the pressure difference between the dense plasma of the ejecta and the tenuous plasma of the ambient medium.     

Eruptive prominences and coronal transients

The observed interrelationship between coronal transients and eruptive prominences is used as the basis for a theoretical MHD model of these events. The model begins with an equilibrium configuration consisting of a coronal loop or arcade with a filament lying underneath with its axis oriented perpendicular to the overlying field. The lifting of the filament from the solar surface produces an increase in magnetic pressure under the helmet which drives it outward. This increased pressure is associated with the internal field of the filament as well as the field beneath it. The underlying field could be that which produced the filament eruption or, alternatively, reconnected field lines formed by the inward collapse of the legs of the transient towards the neutral line beneath the rising prominence. We do not attempt to explain the filament eruption which may be due to internal forces in the prominence or, alternatively, from forces imposed from beneath as would be produced by emerging flux. In the latter case, the filament is passive and merely acts as a tracer for the more fundamental underlying process.It is shown that the outward force per unit mass produced by the driving magnetic field and the inward restoring forces in the overlying field due to magnetic tension and gravity all decrease with distance at the same rate - namely, as the inverse square of the distance from the solar center. Hence, the ratio of net outward to inward force is independent of radial distance from the Sun. A stability analysis shows that this situation is one of neutral stability.A mathematical model of this physical process is described in which the MHD equations in simplified form, neglecting gas pressure forces, are solved in time for the velocity, width, density, and magnetic field strength of the transient. The solutions show that the velocity increases sharply close to the Sun but quickly approaches a constant value. The width increases linearly with radial distance. Both of these results are in agreement with observations. An examination of the forces exerted on the legs of the transient shows that their motion should be horizontally inward.On leave from the High Altitude Observatory, National Center for Atmospheric Research, Boulder, Colo., U.S.A.     

The shape of buoyant coronal loops in a magnetic field and the eruption of coronal transients and prominences

The equilibrium and non-equilibrium properties of a coronal loop embedded in a stratified isothermal atmosphere are investigated. The shape of the loop is determined by a balance between magnetic tension, buoyancy, and external pressure gradients. The footpoints of the loop are anchored in the photosphere; if they are moved too far apart, no equilibrium is possible and the loop erupts upwards. This critical separation is independent of the pressure differential between the loop and the external medium if the loop has enhanced magnetic field, but varies if instead the loop pressure is increased. The maximum width is proportional to the larger of the gravitational scale-height and the length-scale of the ambient field. In some circumstances, it is shown that multiple solutions exist for the tube path. These results may be relevant to the eruption of prominences during the preflare phase of two-ribbon flares and to the onset of coronal loop transients. Such eruptions may occur if the footpoint separation, internal pressure or internal magnetic field are too great.     

A simulation study of the formation of a bipolar magnetic structure

Some recent solar observations show that a bipolar magnetic flux in an active region tends to disappear in situ, in less than one solar rotation, without evidence of spreading. This feature is difficult to explain if it is assumed that magnetic buoyancy is the dominant force in controlling dynamics of a magnetic flux tube, since the assumption implies no other force to submerge the tube. These observations may be explained by assuming that a convective motion is the major cause for the formation of a $\text{Ω}$-shaped geometry of the magnetic flux tube, but that the flux tube thus arisen is submerged by the counteracting Lorentz force as the convective motion decays. A two-dimensional MHD simulation method is used to demonstrate this possibility.     

Large‐scale magnetic flux concentrations from turbulent stresses

In this study we provide the first numerical demonstration of the effects of turbulence on the mean Lorentz force and the resulting formation of large‐scale magnetic structures. Using three‐dimensional direct numerical simulations (DNS) of forced turbulence we show that an imposed mean magnetic field leads to a decrease of the turbulent hydromagnetic pressure and tension. This phenomenon is quantified by determining the relevant functions that relate the sum of the turbulent Reynolds and Maxwell stresses with the Maxwell stress of the mean magnetic field. Using such a parameterization, we show by means of two‐dimensional and three‐dimensional mean‐field numerical modelling that an isentropic density stratified layer becomes unstable in the presence of a uniform imposed magnetic field. This large‐scale instability results in the formation of loop‐like magnetic structures which are concentrated at the top of the stratified layer. In three dimensions these structures resemble the appearance of bipolar magnetic regions in the Sun. The results of DNS and mean‐field numerical modelling are in good agreement with theoretical predictions. We discuss our model in the context of a distributed solar dynamo where active regions and sunspots might be rather shallow phenomena (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The formation of prominences by thermal instability: A numerical study

A two-dimensional model of prominence formation in a region containing a magnetic neutral sheet is constructed for a variety of initial conditions, assuming the coronal plasma to be described by the usual hydromagnetic approximation, with infinite electric conductivity. In each case the magnetic field is initially vertical, varying antisymmetrically with respect to the neutral sheet, to a maximum value at a distance of 70 000 km from the neutral sheet. In the first case, the plasma is initially in hydrostatic equilibrium, whereas in successive cases, the pressure is assumed to be of such a value that the plasma is in lateral equilibrium of total pressure (gas plus magnetic). In a variation of this case, the value of the solar gravitational field was artificially reduced, and the effects considered. Large lateral motions are produced in each case, thus apparently inhibiting the condensation of prominences, with the exception of the unrealistic case of artificially reduced gravity. The results suggest that consideration either of a third component of the magnetic field (horizontal and parallel to the neutral sheet), or a finite conductivity, allowing magnetic recombination across the neutral sheet, or both, would more realistically represent the problem and might thus show the development of prominences.     

The magnetic non-equilibrium of buoyant flux tubes in the solar corona

The equilibrium shape of a slender flux tube in the stratified solar atmosphere is studied. The path is determined by a balance between the downwards magnetic tension, which depends on the curvature of the loop, and the upwards buoyancy force. Previous results for untwisted slender tubes are extended to include twisted tubes embedded in an external magnetic field.The path of an untwisted tube in an atmosphere with an ambient magnetic field is calculated. For a given footpoint separation, the height of the tube is lowered by increasing the strength of the external magnetic field. If the footpoints are slowly moved apart, the tube rises, until a threshold separation is reached beyond which there is no possible equilibrium height. This threshold width does not depend on the strength of the external field.The effects of twisting up a curved loop are studied, using an extension of results obtained for slender curved tubes with a straight axis. It is shown that for a twisted tube of given width, there can be two possible values of the equilibrium height. If, however, the tube is twisted more than a certain amount or if the footpoints are too widely separated there is no equilibrium. The critical footpoint separation for non-equilibrium is smaller for a twisted tube that an untwisted one.Twisting a tube or moving its feet apart is thus likely to result in non-equilibrium, causing the tube to rise indefinitely under the influence of the unbalanced buoyant force. It is suggested that this phenomenon could be important in the preflare stage of a large two-ribbon solar flare, by causing the initial slow rise of an active region filament. As well as being involved in the onset of an erupting prominence, this non-equilibrium may also be relevant to the formation of coronal loop transients.     

Magnetic field and differential rotation in the radiative zone of the Sun

The problem of the interaction between magnetic fields and differential rotation in the radiative zone of the Sun is investigated. It is demonstrated that effects of magnetic buoyancy can be neglected in the analysis of this interaction. It is shown that hydromagnetic torsional waves propagating from the solar core cannot be responsible for the 22-year solar cycle. A possible geometry of the magnetic field that conforms with stationary differential rotation is considered. A verifying method for hypotheses on the structure of the magnetic field and torsional oscillations in the radiative zone of the Sun is proposed based on helioseismic data.     

Differences between solar wind plasmoids and ideal magnetohydrodynamic filaments

Plasma irregularities present in the solar wind are plasmoids, i.e. plasma-magnetic field entities. These actual plasmoids differ from ideal magnetohydrodynamic (MHD) filaments. Indeed, (1) their “skin” is not infinitely thin but has a physical thickness which is determined by the gyromotion of the thermal ions and electrons, (2) they are of finite extent and their magnetic flux is interconnected with the interplanetary magnetic flux, (3) when they penetrate into the magnetosphere their magnetic field lines become rooted in the ionosphere (i.e. in a medium with finite transverse conductivity), (4) the external Lorentz force acting on their boundary surface depends on the orientation of their magnetic moment with respect to the external magnetic field, (5) when their mechanical equilibrium is disturbed, hydromagnetic oscillations can be generated. It is also suggested that the front side of all solar wind plasmoids which have penetrated into the magnetosphere is the inner edge of the magnetospheric boundary layer while the magnetopause is considered to be the surface where the magnetospheric plasma ceases to have a trapped pitch angle distribution.     

Some recent developments in the theoretical dynamics of magnetic fields

This article describes recent developments in the theoretical investigation of magnetostatic equilibrium in the presence of gravity, nonequilibrium in hydromagnetics, and classical problems in hydromagnetic stability. The construction of magnetostatic equilibria has progressed beyond geometrically idealized systems, such as the axisymmetric system, to fully three-dimensional systems capable of modelling realistic solar structures. Nonequilibrium in a magnetic field with an arbitrary interweaving of lines of force due to random footpoint motion is a novel and subtle property with important implications for the solar atmosphere. Work begun by Parker and subsequent developments are described. To the extent quasi-static solar structures are approximated by stable equilibrium, ideal hydromagnetic stability theory provides a first insight into how stability is achieved in the solar environment. A qualitative physical picture based on recent stability analyses is given. The article places emphasis on understanding basic principles and issues rather than detailed results which can be found in the published literature.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

The magnetic field transfer in the solar convective zone

The large-scale azimuth magnetic field is pumping to the bottom of the solar convective zone due to the diamagnetic action of turbulent conductive fluids. When the field at the bottom is of about 10³ G, an equilibrium is established between diamagnetic pumping and buoyancy.If, in addition to the density gradient, an additional anisotropy exists (for instance, due to rotation), another mechanism of the magnetic field transfer appears, the efficiency of which greatly depends on the magnitude of the anistropy parameter.     

A hydromagnetic theory of geomagnetic storms and auroras

A theory of geomagnetic storms, auroras and associated effects is further developed. It depends on motions in the Earth's exosphere or magnetosphere initiated by a combination of pressure and frictional drag of the solar wind and modified and extended by electric fields and currents in the ionosphere. Motion may be non-divergent, streamline flow opposed only by Lorentz forces in the ionosphere and not propagating to Earth, or divergent, non-streamline motion opposed by Lorentz forces in the Earth. The two types of motion are coupled in the E region where the former is identified with free flow of Hall current and the generation of non-streamline motion. The latter is identified with blockage of Hall current, the creation of a polarization field and hence the generation of streamline motion.

A theory of all components of a geomagnetic storm is given in terms of combinations of these motions, and their distant, ionospheric and earth currents. This includes a new theory of the preliminary reverse part of the DS field and the transition from the sudden commencement to the main phase of the DS field. It is extended to introduce briefly a theory of auroras based mainly on ionospheric drifts caused by the magnetospheric motions.     

Cosmic electric currents and the generalized Bennett relation

A generalized form of the Bennett pinch is studied in both cylindrical geometry and plane-parallel geometry. In this kind of pinch electromagnetic forces, kinetic pressure gradient forces, centrifugal forces, and gravitational forces may act. For each of the two geometries considered a generalized Bennett relation is derived. By means of these relations it is possible to describe among other things the pure Bennett pinch, Jean's criterion in one and two dimensions, force-free magnetic fields, gravitationally balanced magnetic pressures, and continuous transitions between these states. The theory is applied to electric currents in the magnetosphere, in the solar atmosphere, and in the interstellar medium. It is pointed out that the currents in the solar atmosphere and in the interstellar medium may lead to pinches that are of vital importance to the phenomena of solar flares and star formation, respectively.Paper dedicated to Professor Hannes Alfvén on the occasion of his 80th birthday, 30 May 1988.     

Formation of solar prominences by photospheric shearing motions

A numerical simulation is performed to investigate the prominence formation in a magnetic arcade by photospheric shearing motions. A two-and-a-half-dimensional magnetohydrodynamic (MHD) code is used, in which the gravitational force, radiative cooling, thermal conduction and a simplified form of coronal heating are included. It is found that a footpoint shear induces an expansion of the magnetic arcade and cooling of the plasma in it. Simultaneously the denser material from the lower part of the arcade is pulled up by the expanding field lines. A local enhancement of radiative cooling is thus effected, which leads to the onset of thermal instability and the condensation of coronal plasma. The condensed material grows vertically to form a sheet-like structure making dips on field lines, leading to the formation of the Kippenhahn- Schlüter type prominence. The mass of the prominence is found to be supplied not only by the condensation of the material in the vicinity but also by the siphon-type upflows. The upward growth of the vertical sheet-structure of the prominence is saturated at a certain stage and the newly condensed material is found to slide down from above the prominence along magnetic field lines. This drainage of material leads to the formation of an arc-shaped cavity of low density and low pressure around the prominence. The problem of force and heat balance is addressed and the prominence is found to be not in a static equilibrium but in a dynamic interaction with its environment.     

Magnetic fields,loop prominences and the great flares of August, 1972

Lines of magnetic force, computed under the assumption that the solar corona is free of electric currents, have been compared with loop prominence systems associated with three flares in August, 1972. The computed fields closely match the observations of loops at a height of 40000 km at times 3–4 h after onset of the associated flares. Inferred magnetic field intensities in the loops range from 1300 G where the loops converge into a sunspot to 50–80 G at 40 000 km above the photosphere. The first-seen and lowest-lying loops are sheared with respect to the calculated fields. Higher loops conform more closely to the current-free fieldlines. A model of Barnes and Sturrock is used to relate the degree of shear to the excess magnetic energy available during the flare of August 7. On various lines of evidence, it is suggested that magnetic energy was available to accelerate particles not only during the impulsive phase of the flare, but also during the following 2–3 h. The particle acceleration region seems to be in the magnetic fields just above the visible loops. The bright outer edges of the flare ribbons are identified as particle impact regions. The dense knots of loop prominence material fall to the ribbons' inner edges.On leave from Tel Aviv University, Tel Aviv, Israel.