## 4. DiscussionWe have presented calculations of the non-linear field decay influenced by the Hall current. Our study has been limited to an illustrative case of the field decay in a uniform conducting sphere. The adopted model is maximally simplified but it describes well the main characteristic features of a field dissipation process in the case when the magnetic field is strong enough to magnetize plasma. In contrast to previous studies (see, e.g., Muslimov 1994, Muslimov et al. 1995) which used unjustified simplifications to solve the problem, our analysis is based on direct 2D-simulations. Calculations show that even the evolution of the simplest initial magnetic configuration corresponding to the dipole field outside the sphere is rather complicated. The Hall current changes drastically the behaviour of the magnetic field due to coupling and energy transfer among modes. As a result, a complex magnetic configuration can be created from the simplest one in the course of evolution. Starting from a pure dipole field (), the generated magnetic configuration consists of the poloidal harmonics with and (other multipoles are substantially weaker) and a toroidal component which is anti-symmetric to the equatorial plane. The most remarkable feature of the decay is an oscillating behaviour of individual modes. Both the newly generated harmonics and the dipole mode undergo oscillations associated with a non-dissipative energy exchange among modes. The nature of these oscillations can be easily understood if one consider as the example of linear waves in a magnetized plasma. For illustration, consider the behaviour of small perturbations of the toroidal and poloidal magnetic fields, and , respectively, in a conducting sphere in the presence of an unperturbed poloidal magnetic field, . For the sake of simplicity, we assume that the wavelength of perturbations is short in comparison with both the radius of the sphere and the length scale of , thus perturbations can be described in the local approximation. A behaviour of small perturbations is governed by the linearized Eqs. (8) and (9) which read Like in the case of a large scale field, the Hall current associated with the terms proportional to couples the poloidal and toroidal components of the magnetic field. In the local approximation, one can assume that both and are proportional to where and are the frequency and wave vector of the wave, respectively. For the plane wave, Eqs. (11) and (12) yield a set of algebraic equations, where is the inverse ohmic decay time of the wave. Eqs. (13) and (14) illustrate most clearly the coupled behaviour of the field components. The Hall current associated with changes the strength and, hence, the energy of the poloidal field. In its turn, the toroidal field is influenced by the Hall current produced by the poloidal field. Since the energy exchange among modes is non-dissipative, the Hall currents give a contribution only to Re whereas Im is completely determined by the ohmic dissipation. The dispersion relation corresponding to the set of Eqs. (13) and (14) is In a strong magnetic field with , this equation describes waves with the period and with an amplitude slowly decreasing due to the ohmic dissipation on a time scale . These short wavelength modes (sometimes called helicoidal) are determined by the Hall effect and can exist only in a strongly magnetized plasma. They are completely analogous to the oscillations considered above in a large scale magnetic field. Like small scale modes, oscillations of the total magnetic field have a period determined by the Hall time scale, . Both the newly generated modes and the original dipole undergo oscillations, thus the dipole moment can even increase during some evolutionary phases. The amplitude of the oscillations may reach a relatively high value depending on the initial Hall parameter. For example, a varying fraction of the dipole component may be as strong as of the total field strength for the considered range of . Higher order modes may also contain an appreciable fraction of the total field. Thus, the poloidal mode with can provide about of the field strength at the surface, the toroidal field which is concentrated in the interior reaches of the total field strength. Other field components are substantially weaker and do not in practice influence on the magnetic evolution. The ohmic decay of the magnetic field diminishes the influence of the Hall current which is linearly dependent on the field. Due to this, the period of magnetic variations becomes longer whereas their amplitude becomes smaller. At the late evolutionary stage when the magnetic field is reduced by dissipation to such an extent that , the Hall timescale is comparable with the ohmic one (or longer) and oscillations are smoothed. During the further evolution, the behaviour of the field is weakly influenced by the Hall effect and does practically not differ from the linear decay case. As was mentioned, the Hall current does not contribute directly to the rate of the ohmic dissipation. However, the Hall drift changes the distribution of the magnetic field and makes its configuration more complex. The dissipation rate is strongly sensitive to a curvature of the magnetic field lines and, therefore, a generation of higher order modes associated with the Hall current may accelerate the dissipation of the field. Calculations first done by Urpin & Shalybkov (1991) confirm this qualitative conclusion. It was argued that the behaviour of the magnetic energy integrated over the volume departs notably from the standard ohmic dissipation if the initial magnetic field is sufficiently strong (). For weaker initial fields, departures from the linear decay are not significant. © European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |