Relaxed equilibrium configurations to model fossil fields

Abstract

Context. The understanding of fossil fields' origin, topology, and stability is one of the corner stones of the stellar magnetism theory. On one hand, since they survive on secular time scales, they may modify the structure and the evolution of their host stars. On the other hand, they must have a complex stable structure since it has been demonstrated that the simplest purely poloidal or toroidal fields are unstable on dynamical time scales. In this context, the only stable configuration found today is the one resulting from a numerical simulation by Braithwaite and collaborators who studied the evolution of an initial stochastic magnetic field, which is found to relax on a mixed stable configuration (poloidal and toroidal) that seems to be in equilibrium and then diffuses. Aims. We investigate an equilibrium field in a semi-analytical way. In this first article, we study the barotropic magnetohydrostatic equilibrium states. Methods. The problem reduces to a Grad-Shafranov-like equation with arbitrary functions. These functions are constrained by deriving the lowest-energy equilibrium states for given invariants of the considered axisymmetric problem, in particular for a given helicity known to be one of the causes of such problems. These theoretical results were applied to realistic stellar cases, the solar radiative core and the envelope of an Ap star, and discussed. In both cases we assumed that the field is initially confined in the stellar radiation zone.Results. The generalization of the force-free Taylor's relaxation states studied in laboratory experiments (in spheromaks) that become non force-free in the self-gravitating stellar case are obtained. The case of general baroclinic equilibrium states will be studied in Paper II.

Seismic modelling of the late Be stars HD 181231 and HD 175869 observed with CoRoT: a laboratory for mixing processes

“…However, for massive (non-Bp) stars, detected fields usually are dipolar (see e.g. Neiner et al 2003), a predicted behaviour for fossil fields (Braithwaite & Spruit 2004;Duez & Mathis 2010). A few nondipolar examples are known, such as τ Sco (Donati et al 2006), but they remain very rare.…”

Dynamics of fossil magnetic fields in massive star interiors

“…Second, non force-free equilibria have been identified in plasma physics as the result of MHD relaxation (Montgomery & Phillips 1988). Third, as shown by Duez & Mathis (2010), this family of equilibria is a generalization of Taylor states (force-free relaxed equilibria in plasma laboratory experiments; see Taylor 1974) in a stellar context, where the stable stratification of the medium plays a crucial role.…”

“…This is due to the stable stratification and to the different orders of spatial derivatives involved in the variation of E mag and H. The reached equilibrium is thus the one of minimum energy for given magnetic helicity and mass encompassed in magnetic flux tubes. This can be determined applying a variational method where we minimize E with respect to H and M Ψ as described by Woltjer (1959) and Duez & Mathis (2010). This leads to the purely dipolar MHS barotropic state (in the hydrodynamic meaning of the term, i.e.…”

γ Pegasi: testing Vega-like magnetic fields in B stars

“…The primordial field evolves to an equilibrium state: a twisted torus inside the star, that appears most of the time as a simple oblique dipole field at the surface (Braithwaite & Nordlund 2006;Duez & Mathis 2010). After the discovery of very weak magnetic fields in Vega and Sirius, two scenarios have been proposed to explain the existence of such weak fields.…”