Line-tied Magnetic Fields Research Articles

A complete topological invariant for braided magnetic fields

A topological flux function is introduced to quantify the topology of magnetic braids: non-zero line-tied magnetic fields whose field lines all connect between two boundaries. This scalar function is an ideal invariant defined on a cross-section of the magnetic field, whose integral over the cross-section yields the relative magnetic helicity. Recognising that the topological flux function is an action in the Hamiltonian formulation of the field line equations, a simple formula for its differential is obtained. We use this to prove that the topological flux function uniquely characterises the field line mapping and hence the magnetic topology. A simple example is presented.

CURRENT SINGULARITIES IN LINE-TIED THREE-DIMENSIONAL MAGNETIC FIELDS

This paper considers the current distributions that derive from finite amplitude perturbations of line-tied magnetic fields comprising hyperbolic field structures. The initial equilibrium on which we principally focus is a planar magnetic X-point threaded by a uniform axial field. This field is line-tied on all surfaces but subject to three-dimensional (3D) disturbances that alter the initial topology. Results of ideal relaxation simulations are presented which illustrate how intense current structures form that can be related, through the influence of line-tying, to the quasi-separatrix layers (QSLs) of the initial configuration. It is demonstrated that the location within the QSL that attracts the current, and its scaling properties, are strongly dependent on the relative dimensions of the QSL with respect to the line-tied boundaries. These results are contrasted with the behavior of a line-tied 3D field containing an isolated null point. In this case, it is found that the dominant current always forms at the null, but that the collapse is inhibited when the null is closer to a line-tied boundary.

Unique topological characterization of braided magnetic fields

We introduce a topological flux function to quantify the topology of magnetic braids: non-zero, line-tied magnetic fields whose field lines all connect between two boundaries. This scalar function is an ideal invariant defined on a cross-section of the magnetic field, and measures the average poloidal magnetic flux around any given field line, or the average pairwise crossing number between a given field line and all others. Moreover, its integral over the cross-section yields the relative magnetic helicity. Using the fact that the flux function is also an action in the Hamiltonian formulation of the field line equations, we prove that it uniquely characterizes the field line mapping and hence the magnetic topology.

EXPLOSIVE INSTABILITY AND CORONAL HEATING

The observed energy-loss rate from the solar corona implies that the coronal magnetic field has a critical angle at which energy is released. It has been hypothesized that at this critical angle an explosive would occur, leading to an enhanced conversion of magnetic energy into heat. In earlier investigations, we have shown that a shear-dependent magnetohydrodynamic process called has many of the distinctive features of the hypothetical explosive instability. In this paper, we give the first demonstration that this occurs in a system with line-tied magnetic fields and boundary shearing—basically the situation described by Parker. We also show that, as the disturbance due to secondary instability attains finite amplitude, there is a transition to turbulence which leads to enhanced dissipation of magnetic and kinetic energy. These results are obtained from numerical simulations performed with a new parallelized, viscoresistive, three-dimensional code that solves the cold plasma equations. The code employs a Fourier collocation—finite difference spatial discretization, and uses a third-order Runge-Kutta temporal discretization.

Nonequilibrium and current sheet formation in line-tied magnetic fields

Parker’s model of coronal heating [E. N. Parker, Astrophys. J. 174, 499 (1972)] is considered within the framework of ideal reduced magnetohydrodynamics. It is shown that there can be at most one smooth magnetostatic equilibrium for a given smooth footpoint mapping between two end plates to which field lines are line-tied. If such a smooth equilibrium is deformed continuously by further footpoint motion so that it becomes unstable, there is no other smooth equilibrium for the plasma to relax to, and the system tends to a nonequilibrium state containing singular currents (“current sheets”). It is shown that this process can occur as the system relaxes asymptotically to a state of minimum energy (possibly in infinite time). Numerical simulations that begin from smooth initial conditions containing current layers are presented. As the current layers become increasingly intense due to footpoint motion and eventually cross a threshold for instability, the magnetic relaxation observed in the simulation shows a tendency to form nonequilibrium states with current sheets. A necessary geometrical criterion that determines the sites of current sheet formation in models without nulls or closed field lines is given. According to this criterion, the rate of velocity amplification, analogous to the Lyapunov exponent in nonlinear dynamics, becomes unbounded at singularities.

The form of ideal current layers in line-tied magnetic fields

view Abstract Citations (80) References (30) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Form of Ideal Current Layers in Line-tied Magnetic Fields Longcope, D. W. ; Strauss, H. R. Abstract It has been argued that a magnetic field which is initially continuous and is line-tied to rigid boundaries in a continuous manner cannot develop tangential discontinuities or current sheets. This would appear to have many consequences in those theories of reconnection and coronal heating which are based on the existence of such current sheets. It is shown here that while the nonexistence of current sheet may hold in a strict sense, it is possible for simple magnetic geometries to spontaneously develop current layers of nonzero thickness which are indistinguishable, in a practical sense, from genuine current sheets. The thickness of these layers can easily be more than six orders of magnitude smaller than the apparent length scale of the initial equilibrium. We suggest that numerical magnetohydrodynamics simulations have encountered such features, but lacked sufficient resolution to distinguish them from current sheets. Turbulent motion of photospheric footpoints will generate this type of current layer in about one eddy turnover. Publication: The Astrophysical Journal Pub Date: December 1994 DOI: 10.1086/175045 Bibcode: 1994ApJ...437..851L Keywords: Current Sheets; Jacobi Matrix Method; Magnetohydrodynamics; Solar Corona; Solar Magnetic Field; Astronomical Models; Boundary Layer Stability; Current Density; Mapping; Numerical Analysis; Photosphere; Solar Physics; MAGNETOHYDRODYNAMICS: MHD; SUN: CORONA; SUN: MAGNETIC FIELDS full text sources ADS |

The stability of line-tied coronal loops and an extended Suydam criterion

The initiation of prominence eruptions and solar flares can be described in terms of an MHD instability. When the footpoint of a coronal magnetic structure is anchored in the dense photosphere, magnetic energy is slowly built up until critical conditions are exceeded and an instability is triggered. The significance of an extended Suydam criterion for testing line-tied magnetic fields is illustrated by comparing with numerical instability thresholds for a variety of different modes. It is shown that when the extended Suydam criterion predicts an instability, for a nearly force free field, then there exists a global instability with a lower threshold. When the gas pressure is significant localised modes are destabilised slightly before a global mode.

Resistive tearing in line-tied magnetic fields: Slab geometry

The resistive tearing-mode instability of a current carrying plasma sheet is investigated including the stabilising photospheric line-tying boundary conditions. This end condition prohibits a single Fourier mode and so requires a series expansion in harmonics of the fundamental sheet excitation. Equilibria in which there exist field lines that do not connect to the photosphere are unstable provided the ratio of the sheet length to characteristic transverse scale is smaller than a critical value that depends on the equilibrium profile. Line-tying has a strong stabilising effect on the fundamental periodic mode. That tearing mode harmonic which develops close to the instability threshold, leads to a configuration with one X point and one 0 point. Its linear growth rate follows the usual constant-ψ scaling with resistivity γ ∼ S-3/5, where S is the magnetic Reynolds number.