Coronal Magnetic Flux Rope Equilibria and Magnetic Helicity

1 INTRODUCTIONObservations show that the magnetic helicity of solar magnetic structures has a predominantsign in each hemisphere of the Sun, positive in the southern hemisphere and negative in thenorthern, regardless of the solar cycle (Rust, 1994). The magnetic helicity is strictly conservedin the frame of ideal MHD (WOltjer, 1958), and approximately conserved in the presence ofresistive dissipation and magnetic reconnection in a highly conductive plajsma (Taylor, 1974;Berger, 1984; H…     

Initiation of cmes by magnetic flux emergence

The initiation of solar Coronal Mass Ejections (CMEs) is studied in the framework of numerical magnetohydrodynamics (MHD). The initial CME model includes a magnetic flux rope in spherical, axi-symmetric geometry. The initial configuration consists of a magnetic flux rope embedded in a gravitationally stratified solar atmosphere with a background dipole magnetic field. The flux rope is in equilibrium due to an image current below the photosphere. An emerging flux triggering mechanism is used to make this equilibrium system unstable. When the magnetic flux emerges within the filament below the flux rope, this results in a catastrophic behavior similar to previous models. As a result, the flux rope rises and a current sheet forms below it. It is shown that the magnetic reconnection in the current sheet below the flux rope in combination with the outward curvature forces results in a fast ejection of the flux rope as observed for solar CMEs. We have done a parametric study of the emerging flux rate.     

Numerical Experiments Base on the Catastrophe Model of Solar Eruptionstwo

On the basis of the catastrophe model developed by Isenberg et al., we have used the NIRVANA code to perform the magnetohydrodynamics (MHD) numerical experiments to look into the various behaviors of the coronal magnetic configuration that includes a current-carrying flux rope for modelling the prominence levitation in the corona. These behaviors include the evolution of the equilibrium height of magnetic flux rope with the background magnetic field, the corresponding interior equilibrium of magnetic flux rope, the dynamic properties of magnetic flux rope after the system loses equilibrium, as well as the impact of the reference radius on the equilibrium height of magnetic flux rope. In our calculations, an empirical model of the coronal density distribution given by Sittler & Guhathakurta is used, and the physical dissipation is included. Our experiments show that a deviation between the simulated equilibrium height of magnetic flux rope and the theoretical result of Isenberg et al. exists, but it is not apparent, and the evolutionary features of the two results are similar. If the magnetic flux rope is initially located at the stable branch of the theoretical equilibrium curve, the magnetic flux rope will quickly reach the equilibrium position after several rounds of oscillations as a result of the self-adjustment of the system; when the system is located at the critical point it will quickly lose equilibrium and evolve to the eruptive state; the impact of the variation of reference radius on the equilibrium height of magnetic flux rope is consistent with the prediction of the theory; in the eruptive state, the kinetic properties of magnetic flux rope are consistent with the results given by the Lin-Forbes model and observation, and the fast-mode shock in front of the magnetic flux rope is observed in our experiments; furthermore, because that the dissipation is included in our numerical experiments, the energy conversion from the magnetic energy to other forms of energy is very apparent in the eruptive process.     

Catastrophe of Coronal Magnetic Flux Ropes Caused by Photospheric Motions

Using a 2.5-D, time-dependent ideal MHD model in Cartesian coordinates, we carried out numerical simulations to investigate the equilibrium and evolution properties of a magnetic configuration that consists of a coronal magnetic flux rope and a partly open photospheric background field, which is equivalent to that produced by a two-patch magnetic source on the photospheric surface. The axial and annular magnetic fluxes of the flux rope are given and fixed. The global magnetic configuration evolves in response to three types of changes of the background field: decreasing of the distance between the two sources, shrinking of the size of each source, and increasing of the shear in the closed component of the background field. As a result, the geometrical parameters of the flux rope, i.e. the height of the rope axis, the half-width of the rope and the length of the vertical current sheet below the rope, change due to the variation of the background field. It is shown that for a given coronal magnetic flux rope in a partly open background field, the variation of the geometrical parameters of the flux rope displays a catastrophic behavior, namely, there exists a critical point for each case, at which an infinitesimal change of the background field leads to a loss of equilibrium, and thus a jump of the flux rope. The implication of such a catastrophe in solar active phenomena is briefly discussed.     

Magnetic Energy of Force-Free Fields with Detached Field Lines

Using an axisymmetrical ideal MHD model in spherical coordinates, we present a numerical study of magnetic configurations characterized by a levitating flux rope embedded in a bipolar background field whose normal field at the solar surface is the same or very close to that of a central dipole. The characteristic plasma β (the ratio between gas pressure and magnetic pressure) is taken to be sosmall (β= 10^-4) that the magnetic field is close to being force-free. The system as a whole is then let evolve quasi-statically with a slow increase of either the annular magnetic flux or the axial magnetic flux of the rope, and the total magneticenergy of the system grows accordingly. It is found that there exists an energy threshold: the flux rope sticks to the solar surface in equilibrium if the magneticenergy of the system is below the threshold, whereas it loses equilibrium if the threshold is exceeded. The energy threshold is found to be larger than that of thecorresponding fully-open magnetic field by a factor of nearly 1.08 irrespective as towhether the background field is completely closed or partly open, or whether the magnetic energy is enhanced by an increase of annular or axial flux of the rope.This gives an example showing that a force-free magnetic field may have an energy larger than the corresponding open field energy if part of the field lines is allowed tobe detached from the solar surface. The implication of such a conclusion in coronal mass ejections is briefly discussed and some comments are made on the maximum energy of force-free magnetic fields.     

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.     

Catastrophe of coronal magnetic flux ropes in partially open magnetic fields

Using a 2.5-D, time-dependent ideal MHD model in Cartesian coordinates, a numerical study is carried out to find equilibrium solutions associated with a magnetic flux rope in the corona. The ambient magnetic field is partially open, consisting of a closed arcade in the center and an open field at the flank. The coronal magnetic flux rope is characterized by its magnetic properties, including the axial and annular magnetic fluxes and the magnetic helicity, and its geometrical features, including the height of the rope axis, the halfwidth of the rope and the length of the vertical current sheet below the rope. It is shown that for a given partially open ambient magnetic field, the dependence of the geometrical features on the magnetic properties displays a catastrophic behavior, namely, there exists a certain critical point, across which an infinitesimal enhancement of the magnetic parameters causes a finite jump of the geometrical parameters for the rope. The amplitude of the jump depends on the extent to which the ambient magnetic field in open, and approaches to zero when the ambient magnetic field becomes completely closed. The implication of such a catastrophe in solar active phenomena is briefly discussed.     

Catastrophe of coronal magnetic rope in partly open multipolar magnetic field

Catastrophe of coronal magnetic rope embedded in a partly open multipolar background magnetic field is studied by using a 2-dimensional, 3-component ideal MHD model in spherical coordinates. The background field is composed of three closed bipolar fields of a coronal streamer and an open field with an equatorial current sheet. The magnetic rope lies below the central bipolar field, and it is characterized by its annular and axial magnetic fluxes. For a given annual flux, there is a critical value of the axial flux, and for a given axial flux, there is a critical value of annual flux such that, below the critical value, the magnetic rope is attached to the solar surface and the system stays in equilibrium, but when the critical value is exceeded, the magnetic rope breaks free and erupts upward. This implies that catastrophe can occur in a coronal magnetic rope embedded in a partly open multipolar background magnetic field. Our computation gives a threshold value of magnetic energy that is about 15% greater than the energy of the partly open magnetic field (the central bipolar field open and the fields on either side closed). The excess energy may serve as source for solar explosions such as coronal mass ejections.     

Damped Oscillations of Coronal Loops

A mechanism of damped oscillations of a coronal loop is investigated. The loop is treated as a thin toroidal flux rope with two stationary photospheric footpoints, carrying both toroidal and poloidal currents. The forces and the flux-rope dynamics are described within the framework of ideal magnetohydrodynamics (MHD). The main features of the theory are the following: i) Oscillatory motions are determined by the Lorentz force that acts on curved current-carrying plasma structures and ii) damping is caused by drag that provides the momentum coupling between the flux rope and the ambient coronal plasma. The oscillation is restricted to the vertical plane of the flux rope. The initial equilibrium flux rope is set into oscillation by a pulse of upflow of the ambient plasma. The theory is applied to two events of oscillating loops observed by the Transition Region and Coronal Explorer (TRACE). It is shown that the Lorentz force and drag with a reasonable value of the coupling coefficient (c _d ) and without anomalous dissipation are able to accurately account for the observed damped oscillations. The analysis shows that the variations in the observed intensity can be explained by the minor radial expansion and contraction. For the two events, the values of the drag coefficient consistent with the observed damping times are in the range c _d ≈2 – 5, with specific values being dependent on parameters such as the loop density, ambient magnetic field, and the loop geometry. This range is consistent with a previous MHD simulation study and with values used to reproduce the observed trajectories of coronal mass ejections (CMEs).     

Numerical simulations of fast and slow coronal mass ejections

Solar coronal mass ejections (CMEs) show a large variety in their kinematic properties. CMEs originating in active regions and accompanied by strong flares are usually faster and accelerated more impulsively than CMEs associated with filament eruptions outside active regions and weak flares. It has been proposed more than two decades ago that there are two separate types of CMEs, fast (impulsive) CMEs and slow (gradual) CMEs. However, this concept may not be valid, since the large data sets acquired in recent years do not show two distinct peaks in the CME velocity distribution and reveal that both fast and slow CMEs can be accompanied by both weak and strong flares. We present numerical simulations which confirm our earlier analytical result that a flux‐rope CME model permits describing fast and slow CMEs in a unified manner. We consider a force‐free coronal magnetic flux rope embedded in the potential field of model bipolar and quadrupolar active regions. The eruption is driven by the torus instability which occurs if the field overlying the flux rope decreases sufficiently rapidly with height. The acceleration profile depends on the steepness of this field decrease, corresponding to fast CMEs for rapid decrease, as is typical of active regions, and to slow CMEs for gentle decrease, as is typical of the quiet Sun. Complex (quadrupolar) active regions lead to the fastest CMEs. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical experiments on the evolution in coronal magnetic configurations including a filament in response to the change in the photosphere

We investigate equilibrium height of a flux rope, and its internal equilibrium in a realistic plasma environment by carrying out numerical simulations of the evolution of systems including a current-carrying flux rope. We find that the equilibrium height of a flux rope is approximately described by a power-law function of the relative strength of the background field. Our simulations indicate that the flux rope can escape more easily from a weaker background field. This further confirms that a catastrophe in the magnetic configuration of interest can be triggered by a decrease in strength of the background field. Our results show that it takes some time to reach internal equilibrium depending on the initial state of the flux rope. The plasma flow inside the flux rope due to the adjustment for the internal equilibrium of the flux rope remains small and does not last very long when the initial state of the flux rope commences from the stable branch of the theoretical equilibrium curve. This work also confirms the influence of the initial radius of the flux rope in its evolution; the results indicate that a flux rope with a larger initial radius erupts more easily. In addition, by using a realistic plasma environment and a much higher resolution in our simulations,we notice some different characteristics compared to previous studies in Forbes.     

The role of emerging bipoles in the formation of a sunspot penumbra

The generation of magnetic flux in the solar interior and its transport from the convection zone into the photosphere, the chromosphere, and the corona will be in the focus of solar physics research for the next decades. With 4 m class telescopes, one plans to measure essential processes of radiative magneto‐hydrodynamics that are needed to understand the nature of solar magnetic fields. One key‐ingredient to understand the behavior of solar magnetic field is the process of flux emergence into the solar photosphere, and how the magnetic flux reorganizes to form the magnetic phenomena of active regions like sunspots and pores. Here, we present a spectropolarimetric and imaging data set from a region of emerging magnetic flux, in which a proto‐spot without penumbra forms a penumbra. During the formation of the penumbra the area and the magnetic flux of the spot increases. First results of our data analysis demonstrate that the additional magnetic flux, which contributes to the increasing area of the penumbra, is supplied by the region of emerging magnetic flux. We observe emerging bipoles that are aligned such that the spot polarity is closer to the spot. As an emerging bipole separates, the pole of the spot polarity migrates towards the spot, and finally merges with it. We speculate that this is a fundamental process, which makes the sunspot accumulate magnetic flux. As more and more flux is accumulated a penumbra forms and transforms the proto‐spot into a full‐fledged sunspot (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Background magnetic fields during last three cycles of solar activity

This paper describes our studies of evolution of the solar magnetic field with different sign and field strength in the range from –100 G to 100 G. The structure and evolution of large‐scale magnetic fields on the Sun during the last 3 cycles of solar activity is investigated using magnetograph data from the Kitt Peak Solar Observatory. This analysis reveals two groups of the large‐scale magnetic fields evolving differently during the cycles. The first group is represented by relatively weak background fields, and is best observed in the range of 3–10 Gauss. The second group is represented by stronger fields of 75–100 Gauss. The spatial and temporal properties of these groups are described and compared with the total magnetic flux. It is shown that the anomalous behaviour of the total flux during the last cycle can be found only in the second group. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Observational Features of Large-Scale Structures as Revealed by the Catastrophe Model of Solar Eruptions

Large-scale magnetic structures are the main carrier of major eruptions in the solar atmosphere. These structures are rooted in the photosphere and are driven by the unceas-ing motion of the photospheric material through a series of equilibrium configurations. The motion brings energy into the coronal magnetic field until the system ceases to be in equilib-rium. The catastrophe theory for solar eruptions indicates that loss of mechanical equilibrium constitutes the main trigger mechanism of major eruptions, usually shown up as solar flares, eruptive prominences, and coronal mass ejections (CMEs). Magnetic reconnection which takes place at the very beginning of the eruption as a result of plasma instabilities/turbulence inside the current sheet, converts magnetic energy into heating and kinetic energy that are responsible for solar flares, and for accelerating both plasma ejecta (flows and CMEs) and energetic particles. Various manifestations are thus related to one another, and the physics behind these relationships is catastrophe and magnetic reconnection. This work reports on re- cent progress in both theoretical research and observations on eruptive phenomena showing the above manifestations. We start by displaying the properties of large-scale structures in the corona and the related magnetic fields prior to an eruption, and show various morphological features of the disrupting magnetic fields. Then, in the framework of the catastrophe theory, we look into the physics behind those features investigated in a succession of previous works, and discuss the approaches they used.     

Evolution of morphological features of CMEs deduced from catastrophe model of solar eruptions

We describe the evolution of morphological features of the magnetic configuration of CME according to the catastrophe model developed previously. For the parameters chosen for the present work, roughly half of the total mass is nominally contained in the initial flux rope, while the remaining plasma is brought by magnetic reconnection from the corona into the current sheet and from there into the CME bubble. The physical attributes of the difference in the observable features between CME bubble and flare loop system were studied. We tentatively identified distinguishable evolutionary features like the outer shell, the expanding bubble and the flux rope with the leading edge, void and core of the 3-component CME structure. The role of magnetic reconnection is discussed as a possible mechanism for the heating of the prominence material during eruptions. Several aspects of this explanation that need improvement are outlined.     

Coronal mass ejections: Some possibilities for prediction

The onset stage of coronal mass ejections (CMEs) is difficult to observe and is poorly studied. In spite of their practical importance, methods for CME predictions with sufficient lead times are only in the nascent stages of development. The most probable CME mechanism is a catastrophic loss of equilibrium of a large-scale current system in the corona (a flux rope). A twisted magnetic rope is maintained by the tension of field lines of photospheric sources until parameters of the system reach critical values and the equilibrium is lost. Unfortunately, there is low-density plasma (coronal cavity) in most of the rope volume; thus, it is difficult to observe a rope. However, the lower parts of the helical field lines of a rope are fine traps for the dense cold plasma of prominences. Thus, prominences are the best tracers of flux ropes in the corona. The maximal height up to which the rope is in stable equilibrium can be found by analyzing the distribution of the magnetic field generated by photospheric sources in the corona. Comparing this critical height with the actually observed prominence height, one can estimate the probability of the loss of equilibrium by a magnetic rope with a following eruption of prominences and coronal mass ejections.     

Dynamical processes in flux tabes and their role in chromospheric heating

We model the dynamical interaction between magnetic flux tubes and granules in the solar photosphere which leads to the excitation of transverse (kink) and longitudinal (sausage) tube waves. The investigation is motivated by the interpretation of network oscillations in terms of flux tube waves. The calculations show that for magnetic field strengths typical of the network, the energy flux in transverse waves is higher than in longitudinal waves by an order of magnitude. But for weaker fields, such as those that might be found in internetwork regions, the energy fluxes in the two modes are comparable. Using observations of footpoint motions, the energy flux in transverse waves is calculated and the implications for chromospheric heating are pointed out.     

Force-free magnetic fields in the solar atmosphere

Reliable measurements of the solar magnetic field are restricted to the level of the photosphere. For about half a century attempts have been made to calculate the field in the layers above the photosphere, i.e. in the chromosphere and in the corona, from the measured photospheric field. The procedure is known as magnetic field extrapolation. In the superphotospheric parts of active regions the magnetic field is approximately force-free, i.e. electric currents are aligned with the magnetic field. The practical application to solar active regions has been largely confined to constant-α or linear force-free fields, with a spatially constant ratio, α, between the electric current and the magnetic field. We review results obtained from extrapolations with constant-α force-free fields, in particular on magnetic topologies favourable for flares and on magnetic and current helicities. Presently, different methods are being developed to calculate non-constant-α or nonlinear force-free fields from photospheric vector magnetograms. We also briefly discuss these methods and present a comparison of a linear and a nonlinear force-free magnetic field extrapolation applied to the same photospheric boundary data. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A magnetohydrodynamical model for the formation of episodic jets

Episodic ejection of plasma blobs has been observed in many black hole systems. While steady, continuous jets are believed to be associated with large-scale open magnetic fields, what causes the episodic ejection of blobs remains unclear. Here by analogy with the coronal mass ejection on the Sun, we propose a magnetohydrodynamical model for episodic ejections from black holes associated with the closed magnetic fields in an accretion flow. Shear and turbulence of the accretion flow deform the field and result in the formation of a flux rope in the disc corona. Energy and helicity are accumulated and stored until a threshold is reached. The system then loses its equilibrium and the flux rope is thrust outward by the magnetic compression force in a catastrophic way. Our calculations show that for parameters appropriate for the black hole in our Galactic centre, the plasmoid can attain relativistic speeds in about 35 min.