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Astron. Astrophys. 321, 685-690 (1997)

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1. Introduction

Observational data on the surface magnetic field of many white dwarfs and neutron stars provide evidence for the complex character of their magnetic configurations. In the atmospheres of white dwarfs, the field strength can be deduced from Zeeman broadened absorption lines while the field direction can be inferred from polarization measurements (see, e.g., Chanmugam 1992 for a general review). It turns out that the surface magnetic configurations of many white dwarfs are likely to consist of a mixture of different modes rather than a simple dipole structure. Thus, Meggitt & Wickramasinghe (1989) and Ferraro & Wickramasinghe (1989) have found the presence of the quadrupole mode in atmospheres of some white dwarfs, Achilleos et al. (1992) have concluded that the field of the magnetic white dwarf Feige 7 can be represented as the sum of the dipole and quadrupole components with the strengths approximately 2/3 and 1/3 of the total field strength, respectively. There is also evidence for a complex magnetic field structure of white dwarfs in AM Her like systems (see Chanmugam 1992 for details). Probably, the magnetic configurations of at least some neutron stars differ from the pure centered dipole as well. For instance, it has been pointed out by Krolik (1991) that the fields of millisecond pulsars may consist of few multipole components, and an admixture of higher order modes can account for the unusual pulse morphology of these objects. Thus, it seems that departures from the simplest dipole configuration are widespread both in white dwarfs and neutron stars.

The problem of the origin and evolution of the magnetic field in white dwarfs and neutron stars is still far from being solved. Therefore, one cannot exclude that a complex distribution of the magnetic field originates from the initial conditions which are subject to many uncertainties. The decay of magnetic eigenmodes in white dwarfs has been first considered by Chanmugam & Gabriel (1972) who argued that the decay time scale of the lowest multipoles is of the order of [FORMULA] yr but high order modes can evolve on a shorter time scale. Wendel et al.(1987) have analysed in detail the influence of the cooling of white dwarfs on the time evolution of individual modes, and confirmed that the decay of the fundamental eigenmode proceeds on a time scale [FORMULA] yr. These calculations showed that it is more likely to observe complex magnetic configurations in relatively young white dwarfs if high order modes are generated during the initial evolutionary stages.

However, a complex magnetic configuration can be created also from a simple one due to non-linear magnetohydrodynamic effects in the course of the evolution of magnetized stars. One possibility is associated with a dependence of transport processes on the magnetic field. Thus, the conductive properties of magnetized plasma are anisotropic and the electric resistivities along and across the field may differ. Besides, the presence of the magnetic field induces the so called Hall current which is perpendicular both to the electric, [FORMULA], and magnetic, [FORMULA], fields. This current is non-dissipative since it does not contribute directly to an increase in the density of entropy, [FORMULA],


where [FORMULA] is the current density, [FORMULA] and [FORMULA] are the components of the resistivity tensor; the subscripts [FORMULA] and [FORMULA] mean the component of the corresponding quantities along and across the magnetic field. However, the Hall current couples different modes and alters a magnetic configuration by redistributing the energy among modes. In this way, the Hall current can affect indirectly the rate of the magnetic field dissipation and, in principle, can increase this rate substantially for strongly magnetized stars. Numerical simulations first done by Urpin & Shalybkov (1991) for the simplest toroidal magnetic configuration confirmed this conclusion. More recently several attempts have been made to consider more complex magnetic configurations including a poloidal field (see, e.g., Naito & Kojima 1994; Muslimov 1994; Muslimov et al.1995). Naito & Kojima (1994) studied the effect of the Hall current expanding the vector potential of the magnetic field by a set of Legendre polynomials up to [FORMULA], but they restricted their calculations to the case when the Hall effect is small in comparison with the ohmic dissipation. Muslimov (1994) and Muslimov et al.(1995) analysed a transfer of the energy among the modes assuming that the magnetic configuration consists originally of dipole, quadrupole and lowest order toroidal components. They argued that this configuration is the simplest one influenced by the Hall effect. In both these papers, the authors neglected the influence of the poloidal components on the toroidal one in order to simplify the numerical calculations. This simplification causes some doubts, however, since the obtained results do not even reproduce well known phenomena (like helicoidal oscillations of the magnetic field) associated with the Hall effect.

In the present paper, we focus on the most important phenomena caused by the Hall effect keeping in mind applications to particular astrophysical objects (white dwarfs and neutron stars) for the forthcoming paper. The non-linear magnetic evolution is complicated, and to understand its physical content it is helpful to consider initially a very idealized example which, nevertheless, describes qualitatively the main features of more realistic models. Our adopted model is therefore maximally simplified: we treat the decay of a strong magnetic field in a uniform conducting sphere. The original magnetic configuration is also chosen to be the simplest one: it is assumed that only the lowest order poloidal mode, corresponding to the dipole field outside the sphere, is presented at [FORMULA]. Our numerical calculations illustrate the process of the generation of higher order poloidal and toroidal modes from the simple original configuration and the evolution of these modes.

The paper is organized as follows. Sect. 2 presents the main equations governing the evolution of the magnetic field in the presence of the Hall effect. Sect. 3 describes the numerical method and results of our calculations. Principal conclusions are summarized in Sect. 4.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998