E. N. Parker

Equilibrium of magnetic fields with arbitrary interweaving of the lines of force i. discontinuities in the torsion

Abstract It is shown that surfaces ∑α of discontinuity in the torsion α in a force-free field (whose existence was established in the previous paper) have special properties. There is a discontinuity in the magnetic field (i.e. a current sheet) associated with ∑α if the extension of ∑α along the field is terminated anywhere within the fluid, or if two or more ∑α intersect. If the ∑α do not terminate, they extend arbitrarily far along the field from the location of the arbitrary change in winding pattern that produces them. In that case there would be as many ∑α as changes in winding pattern, each ∑α oriented arbitrarily with respect to the others. Intersections would be unavoidable, and discontinuities (current sheets) inevitable companions of the ∑α. Hence sequences of arbitrary patterns of winding and wrapping of the lines of force generally produce current sheets. The current sheets are the principal sources of dissipation of the field in a highly conducting fluid, because the interior of each current ...

The rapid dissipation of magnetic fields in highly conducting fluids

Abstract This paper treats the dynamical conditions that obtain when long straight parallel twisted flux tubes in a highly conducting fluid are packed together in a broad array. It is shown that there is generally no hydrostatic equilibrium. In place of equilibrium there is a dynamical nonequilibrium, leading to neutral point reconnection and progressive coalescence of neighboring tubes (with the same sense of twisting), forming tubes of larger diameter and reduced twist. The magnetic energy in the twisting of each tube declines toward zero, dissipated into small-scale motions of the fluid and thence into heat. The physical implications are numerous. For instance, it has been suggested that the subsurface magnetic field of the sun is composed of close-packed twisted flux tubes. Any such structures are short lived, at best. The footpoints of the filamentary magnetic fields above bipolar magnetic regions on the sun are continually shuffled and rotated by the convection, so that the fields are composed of tw...

Alfén waves in a thermally stratified fluid

Abstract The propagation of Alfven waves along a uniform horizontal field in a highly conducting incompressible fluid, subject to the convective forces produced by a uniform vertical temperature gradient, is treated in a Boussinesq approximation. It is shown that there are exact solutions with large amplitude but restricted form. Their restricted form means that an arbitrary disturbing force produces other motions as well as Alfven waves. An arbitrary initial disturbance of small amplitude produces waves whose state of polarization varies along the direction of propagation. For large amplitudes, however, any mixtures of polarization states causes scattering into new modes.

Spontaneous tangential discontinuities and the optical analogy for static magnetic fields. I. Force-free fields, potential fields, and discontinuities

Abstract The lines of force of a magnetic field extending through an infinitely conducting fluid between footpoints on the planes z=0 and z=L can be wrapped and interwoven by bounded continuous motions of the footpoints into a random sequence of arbitrary topological patterns along the field. The sign of the topological helicity of the winding pattern of the field vanes at random along the field. On the other hand, in force-free equilibrium the relative helicity (α=B. ▿ B/B 2) is rigorously constant along each line of force. Indeed in the limit of an endless random sequence of independent winding patterns along the field, the helicity falls asymptotically to zero and the field becomes curl free. A field with constant helicity along each line of force is obliged to create internal tangential discontinuities (in which the sign of the curl and the helicity is arbitrary) if it is to follow the varying topological helicity imposed by the twists and turns of the succession of independent winding patterns.

Tangential discontinuities and the optical analogy for stationary fields II. The optical analogy

Abstract It is shown that any stationary three-dimensional velocity field or magnetic field is a potential field in the two dimensional subspace of the Bernoulli surfaces S(Q) or isobaric surfaces S(p), respectively. From this it is shown that the streamlines and the lines of force follow the optical ray paths in S(Q) and S(p) for indices of refraction v and B, respectively. This formal analogy shows how the lines are refracted by variations of the pressure applied by the fluid and field on either side. In particular, it is shown how continuous variations of the pressure produce discontinuities (bifurcations) in the field, forming tangential discontinuities.

Tangential discontinuities and the optical analogy for stationary fields III. Zones of exclusion

Abstract This paper presents a number of formal examples of the bifurcation of individual flux surfaces by the pressure maxima imposed by the fields on either side. An approximate necessary and sufficient criterion for the convexity of the pressure maximum is provided, with application to fields with and without gaps in their flux surfaces. Gaps automatically produce tangential discontinuities in almost all cases, by permitting fields otherwise separated by finite distance to come in contact. Both Euclidean and non Euclidean flux surfaces are examined, showing that positive curvature fosters the formation of gaps while negative curvature opposes it. The special conditions for producing single or double gaps are pointed out. The general conditions for producing gaps are so mild as to indicate the special character of the familiar continuous solutions to the force-free equilibrium equations, in which the maxima and minima of the field pressure are so arranged as to provide flux surfaces without gaps.

Tangential discontinuities and the optical analogy for stationary fields. v. formal integration of the force-free field equations

Abstract This paper demonstrates the appearance of tangential discontinuities in deformed force-free fields by direct integration of the field equation ▿ x B = αB. To keep the mathematics tractable the initial field is chosen to be a layer of linear force-free field Bx = + B 0cosqz, By = — B 0sinqz, Bz = 0, anchored at the distant cylindrical surface ϖ = (x 2 + y 2)1/2 = R and deformed by application of a local pressure maximum of scale l centered on the origin x = y = 0. In the limit of large R/l the deformed field remains linear, with α = q[1 + O(l 2/R 2)]. The field equations can be integrated over ϖ = R showing a discontinuity extending along the lines of force crossing the pessure maximum. On the other hand, examination of the continuous solutions to the field equations shows that specification of the normal component on the enclosing boundary ϖ = R completely determines the connectivity throughout the region, in a form unlike the straight across connections of the initial field. The field can escape...

Tangential discontinuities and the optical analogy for stationary fields IV. High speed fluid sheets

Abstract It was shown in the previous paper that a sufficiently strong pressure maximum applied to an equilibrium flux surface, by the fields on either side of the surface, produces a gap in the flux surface. The fields on either side make contact through the gap to produce a surface of tangential discontinuity (current sheet). It is shown in the present paper that there is a high speed sheet of fluid and field sliding over the surface of discontinuity when the applied pressure moves slowly across the flux surface. Conditions in the active X-ray corona of the sun suggest that such sheets are generally present, with velocities of the order of 102 km/sec, but with thicknesses too small to be observed. More substantial high speed sheets of fluid may occur in solar flares.

The hydrodynamics of magnetic nonequilibrium

Abstract The strongest fluid motions produced by the general dynamical nonequilibrium of close-packed twisted flux tubes are associated with the neutral point reconnection of the magnetic field, expelling jets of fluid along the neutral sheet away from each reconnection point at velocities comparable to the Alfven speed. Most of the energy of the reconnecting field goes into the kinetric energy of the jets. It is pointed out that the dynamical nonequilibrium of close packed twisted flux tubes is essentially the problem of two dimensional MHD turbulence. Hence the extensive literature on two dimensional turbulence represents the hydrodynamics of magnetic nonequilibrium. Conversely, many features of the turbulence can be understood as a consequence of the dynamical nonequilibrium. Numerical simulations of two dimensional turbulence show the strong jets of fluid from the points of reconnection. The present paper is concerned primarily with the dynamics of those jets. The simulations show the jets often to be...

Spontaneous tangential discontinuities and the optical analogy for static magnetic fields. VI: Topology of current sheets

Abstract The electric surface current in a tangential discontinuity in a force-free magnetic field is conserved. The direction of the current is halfway between the direction of the continuous fields on either side of the surface of discontinuity. Hence the current sheets, i.e. the surface of tangential discontinuity, have a topology that is distinct from the lines of force of the field. The precise nature of the topology of the current sheet depends upon the form of the winding patterns in the field. Hence, invariant winding patterns and random winding patterns are treated separately. Current sheets may have edges, at the junction of two or more topological separatrices. The current lines may, in special cases, be closed on themselves. The lines of force that lie on either side of a current sheet somewhere pass off the sheet across a junction onto another sheet. In most cases the current sheets extending along a field make an irregular honeycomb. The honeycomb pattern varies along the field if the windin...