2. Basic equations
Faraday's induction equation and Ampere's equation governing the evolution of the magnetic, , and electric, , fields read in the magnetohydrodynamic approximation
where is the current density. Neglecting fluid motions, the generalized Ohm's law may be written in the form (see, e.g., Lifshitz & Pitaevskii 1981)
where is the so called Hall component of the resistivity tensor, ; .
The conductive properties of the internal region of white dwarfs and the neutron star crust are determined by degenerate electrons. With a sufficient accuracy, the electron scattering can be described in the relaxation time approximation. Then, if the magnetic field is non-quantizing, the parallel and perpendicular to the magnetic field components of the resistivity tensor are related by where is the electric resistivity at (see Urpin & Yakovlev 1980). Ohm's law (3) can then be rewritten as
where is the Hall parameter, and are the gyrofrequency and relaxation time of electrons, respectively. Note that the Hall parameter depends linearly on the magnetic field. At , the evolution of the magnetic field is weakly influenced by the Hall effect whereas at this influence is especially pronounced. The magnetization condition, , can be fulfiled for the relatively wide range of the magnetic fields detected in white dwarfs and neutron stars (see for more details Urpin & Yakovlev 1980).
For the sake of simplicity, we assume the conducting sphere to be uniform thus both and are independent of coordinates. Then, the induction equation (1) transforms to
One can represent the magnetic field as the sum of poloidal, , and toroidal, , field components. The present paper is addressed to the evolution of the axisymmetric magnetic configuration thus one has in this case (see, e.g., Chandrasekhar & Prendergast 1956)
where is the unit vector in the azimuthal direction; r, , are spherical coordinates. At , the toroidal and poloidal components evolve independently whereas at they are strongly coupled. If , the eigenmodes decay exponentially with a slower decay for a lower order mode. The decay time scale of the fundamental poloidal mode is (see, e.g., Landau & Lifschitz 1960)
where a is the radius of the sphere.
The induction equation (5) yields two equations governing the evolution of the poloidal and toroidal magnetic fields,
where . Note that is the toroidal vector thus the cross-product is zero. It is seen from Eq. (9) that even if the original magnetic configuration is purely poloidal, a toroidal field will be generated due to the Hall current associated with the term proportional to . In its turn, the toroidal field will generate high-order poloidal modes due to the non-linear term on the right hand side of Eq. (8). However, the Hall current can not provide a generation of poloidal modes if the original magnetic configuration is purely toroidal. Thus, in order to analyse the influence of the Hall effect on the evolution of the magnetic field there is no need to consider a complicated original configuration consisting at least of three modes as was pointed out by Muslimov (1994) and Muslimov et al.(1995). Even the simplest dipole magnetic configuration is influenced by the Hall current, and the evolution of this configuration may illustrate the main qualitative features of the non-linear field decay. That is why we suppose the initial field to be poloidal with the -component of the vector potential, , given by
In the uniform conducting sphere, this distribution corresponds to the fundamental mode of the dipole field with the surface strength at the magnetic pole.
The boundary condition requires the magnetic field to be continuous at the surface, . Since we assume that the sphere is isolated in a vacuum, it implies that the toroidal component vanishes at the surface. The poloidal field, , should be matched to the magnetic multipole moments at . Besides, both the toroidal field, , and the potential, , go to zero at from a symmetry.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998