AMPLIFICATION OF NEUTRON-STAR MAGNETIC-FIELDS BY THERMOELECTRIC EFFECTS .4. AVERAGED SMALL-SCALE MODES AND SELECTION-RULES FOR LARGE-SCALE MODES

The investigation of neutron star magnetic field evolution driven by t hermoelectric effects is continued. As in previous papers the magnetic held is assumed to be located in the liquid layer of the neutron star 's envelope. The main result obtained until now was a rapid growth of small-scale toroidal fields with multipolarities in the order of n sim ilar or equal to 1000. Large-scale fields (like dipole or quadrupole m odes) which decay in linear approximation, can, in general, be amplifi ed by nonlinear interaction with small-scale modes. Two difficulties a rise in this problem: (i) the handling of a large number of small-scal e modes to be taken into account, and (ii) the as a matter of principl e unknown structure of the seed held. The problem is solved by introdu cing averaged small-scale modes, which get the structure of an individ ual axisymmetric mode. As initial conditions we assume chaotically dis tributed relative weights for the individual small-scale modes. The no nlinear interaction between different modes is analysed analytically. When restricting the problem to the diagonal (and strongest) part of t he interaction coefficients, it is found that poloidal magnetic fields cannot be amplified, not even in the most general, non-axisymmetric f ormalism. Toroidal fields can be induced by the nonlinear interaction, if any axis is physically distinguished by the initial conditions. Th e axisymmetric coupling rules hold approximately in the non-axisymmetr ic theory. The result is a preferred induction of axisymmetric, toroid al large-scale modes with even multipolarity (for instance, the quadru pole mode), while the induction of all other large-scale modes (odd-n modes, like dipole mode; even-n, nonaxisymmetric modes; poloidal compo nents) is strongly suppressed within the present model.