A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: THE GENERAL THEORY OF ORBITS

Second‐Order Parabolic Differential Equations

Hydrodynamic and hydromagnetic stability

In this paper we study a class of three-dimensional magnetohydrodynamic model problems that may be useful to understand the role of twisted flux ropes in coronal mass ejections. We construct in a half-space a series of force-free bipolar configurations with different helicity contents and bring them into an evolution by imposing to their footpoints on the boundary slow motions converging toward the inversion line. For all the cases that have been computed, this process leads, after a phase of quasi-static evolution, to the formation of a twisted flux rope by a reconnection process and to the global disruption of the configuration. In contrast with the results of some previous studies, however, the rope is never in equilibrium. It thus appears that the presence of a rope in the preeruptive phase is not a necessary condition for the disruption but may be the product of the disruption itself. Moreover, the helicity keeps an almost constant value during the evolution, and the problem of the origin of the helicity content of an eruptive configuration appears to be that of the initial force-free state. In addition to these numerical simulations, we report some new relations for the time variations of the energy and the magnetic helicity and develop a simple analytical model in which the magnetic field evolution exhibits essential features quite similar to those observed during the quasi-static phase in the numerics.

Coronal Mass Ejection: Initiation, Magnetic Helicity, and Flux Ropes. I. Boundary Motion-driven Evolution

Quantum fields and extended objects in space-times with constant curvature spatial section

The existence of force-free magnetic fields in the exterior domain of some compact simply connected surface S is proved via an iteration scheme. The iteration starts with an arbitrary exterior vacuum field, which contains flux tubes originating and ending on S. At one cross-section of such a flux tube with S an arbitrary function α is prescribed. For small values of α (in the Holder-norm 1, λ; 0 < λ < 1) the iteration is shown to converge to a force-free field with the prescribed values of α in a flux tube which is close to the vacuum flux tube and α≡ 0 outside. The force-free field is close (in the Holder- norm 1,λ) to the starting vacuum field, in particular, it has the same field line topology, the same boundary values on S and satisfies the same decay conditions in spatial infinity. It is in general three-dimensional and requires no continuous symmetries.

On the existence of force-free magnetic fields with small nonconstant α in exterior domains

Nonlinear conditional stability of the Benard problem with rotation and free boundaries is studied in this paper in the case of Prandtl number Pr = 1 by means of a generalized energy functional. Previously used functionals to study this problem fail in the case Pr = 1. A new functional is proposed and coincidence of linear and nonlinear stability boundary is proved for moderate Taylor numbers T. The coincidence ends at the same value \( T = 80\pi^4\) as in the case Pr > 1. The marginal Rayleigh number of nonlinear conditional stability grows asymptotically with the square root of the Taylor number as opposed to the critical Rayleigh number of linearized stability which grows with the 2/3 power of T.

Nonlinear stability of the rotating Bénard problem, the case Pr = 1

Solutions of the kinematic dynamo problem with a purely toroidal magnetic field in a spherical volume of conducting fluid are called invisible, since the magnetic field is completely confined to the fluid volume and vanishes at the conductor-insulator interface, thus being invisible to an external observer who is not able to penetrate the interface. This paper presents evidence that such solutions do not exist. More precisely, it is shown that the toroidal scalar T, if it satisfies a certain regularity condition, decays monotonically with respect to the norm in the case of a spherical fluid volume with radius R and with respect to the norm in the case of a plane layer. The result is valid for nonsteady radially varying conductivity, possibly moving boundaries and a nonsteady, compressible flow field constrained in such a way that no poloidal magnetic field is generated. If the mixed norm is replaced by the uniform (but weaker) L 1-norm, the decay result implies boundedness of T—apart from a const...

On the invisible dynamo

A note on nonlinear stability of plane parallel shear flows

Taylor’s theory of relaxed toroidal plasmas (states of lowest energy with fixed total magnetic helicity) is extended to include a vacuum between the plasma and the wall. In the extended variational problem, one prescribes, in addition to the helicity and the magnetic fluxes whose conservation follows from the perfect conductivity of the wall, the fluxes whose conservation follows from the assumption that the plasma-vacuum interface is also perfectly conducting (if the wall is a magnetic surface, then one has the toroidal and the poloidal flux in the vacuum). Vanishing of the first energy variation implies a pressureless free-boundary magnetohydrostatic equilibrium with a Beltrami magnetic field in the plasma, and in general with a surface current in the interface. Positivity of the second variation implies that the equilibrium is stable according to ideal magnetohydrodynamics, that it is a relaxed state according to Taylor’s theory if the interface is replaced by a wall, and that the surface current is no...

Ralf Kaiser

Papers

On the existence of force-free magnetic fields with small nonconstant α in exterior domains

Nonlinear stability of the rotating Bénard problem, the case Pr = 1

On the invisible dynamo

A note on nonlinear stability of plane parallel shear flows

Relaxed plasma-vacuum systems