Tilting instability of magnetically confined spheromaks

We consider the tilting instability of a magnetically confined spheromak using three-dimensional magnetohydrodynamic and relativistic particle-in-cell calculations with an application to astrophysical plasmas, specifically those occurring in magnetar magnetospheres. The instability is driven by the counter-alignment of the spheromak's intrinsic magnetic dipole with the external magnetic field. Initially, the spheromak rotates – tilts – trying to lower its magnetic potential energy. As a result, a current sheet forms between the internal magnetic field of a spheromak and the confining field. Magnetic reconnection sets in; this leads to the annihilation of the newly counter-aligned magnetic flux of the spheromak. This occurs on a few Alfvén time scales. In the case of a higher-order (second-order) spheromak, the internal core is first pushed out of the envelope, resulting in formation of two nearly independent tilting spheromaks. Thus, the magnetically twisted outer shell cannot stabilize the inner core. During dissipation, helicity of the initial spheromak is carried away by torsional Alfvén waves, violating the assumptions of the Taylor relaxation theorem. In applications to magnetar giant flares, fast development of tilting instabilities and no stabilization of the higher-order spheromaks make it unlikely that trapped spheromaks are responsible for the tail emission lasting hundreds of seconds.

A relaxation theorem for a differential inclusion with “maxima”

We consider a Cauchy problem associated to a nonconvex differential inclusion with “maxima” and we prove a Filippov type existence result. This result allows to obtain a relaxation theorem for the problem considered.

A Derivation of the Gibbs Equation and the Determination of Change in Gibbs Entropy from Calorimetry

In this paper, we give a succinct derivation of the fundamental equation of classical equilibrium thermodynamics, namely the Gibbs equation. This derivation builds on our equilibrium relaxation theorem for systems in contact with a heat reservoir. We reinforce the comments made over a century ago, pointing out that Clausius’ strict inequality for a system of interest is within Clausius’ set of definitions, logically undefined. Using a specific definition of temperature that we have recently introduced and which is valid for both reversible and irreversible processes, we can define a property that we call the change in calorimetric entropy for these processes. We then demonstrate the instantaneous equivalence of the change in calorimetric entropy, which is defined using heat transfer and our definition of temperature, and the change in Gibbs entropy, which is defined in terms of the full N-particle phase space distribution function. The result shows that the change in Gibbs entropy can be expressed in terms of physical quantities.

A RELAXATION RESULT FOR AN INHOMOGENEOUS FUNCTIONAL PRESERVING POINT-LIKE AND CURVE-LIKE SINGULARITIES IN IMAGE PROCESSING

In this paper we address a relaxation theorem for a new integral functional of the calculus of variations defined on the space of functions in [Formula: see text] whose gradient is an Lp-vector field with distributional divergence given by a Radon measure. The result holds for integrand of type f(x, Δu) without any coerciveness condition, with respect to the second variable, and C1-continuity assumptions with respect to the spatial variable x.