The Stability of Toroidal Magnetic Fields in Stellar Interiors

The stability of a toroidal magnetic field, B~, in the radiative zones of stars is investigated In cylindri- cal coordinates ~ a necessary condition for stability, for sufficiently large x/n, is given by -g~(a/az)1n(B~/p) > 0 and [-gm - (2c2/~)](ö/0t~)1n(B~/pz~) > 0, where x is the thermal diffusivity, n the magnetic diffusivity, g the gravitational acceleration, p the density, and c2 the isothermal sound speed. Since g2 is negative or positive according as z is positive or negative, B~/p must be an increasing function of z for positive z and an increasing function of - z for negative z, in order to insure stability. In those regions of the star where g~ > 2c2/w, B~/pzz~ will have to be an increasing function of ~ for the configuration to be stable Where g~ < 2c2/w, stability will require B~j,/pw to decrease with increasing t~. In the absence of a gravitational field, stability requires that (9/ôz)1n(B~/p) = 0 and - (3,/&~) in (B~/p~) > 0 (with no restriction on the magnitude of x/n) Stability criteria are also presented for several interesting limiting situations, e g, the adiabatic one, x = n = 0. Even though a magnetic field configuration may be stable to adiabatic perturbations, the non-zero thermal conductivity of the medium may promote an instability Such a diffusive instability is studied in detail Gravitational potential energy, which under adiabatic conditions normally provides a stabilizing influence (Schwarzschild's stability criterion), can reverse its role and be the source of the magnetic in- stability