Magnetic field dynamics in isolated neutron stars with an external dipole field

Neutron stars can harbor extremely strong magnetic fields, yet the structure and stability of their magnetic field configuration remain poorly understood. Observations of pulsars indicate that the large-scale external field is predominantly dipolar far from the star, while the internal magnetic configurations are largely unconstrained. We investigate the dynamical stability of magnetized neutron stars through long-term numerical-relativity simulations. We explore a range of models with an initial external dipole field and mixed poloidal-toroidal internal field where the energy of the toroidal component varies up to $80\%$ of the magnetic energy. We find that the internal magnetic field relaxes toward a dynamically stable mixed poloidal-toroidal geometry, in which the toroidal component contributes to $\lesssim10\%$ of the total magnetic energy both in the exterior and in the interior. This configuration emerges within one Alfv\'en time following the saturation of the Tayler instabilities and also aided by gravitational-wave emission. These results suggest that long-lived neutron star magnetic fields are strongly constrained toward stable mixed configurations, with important implications for pulsar emission models, magnetar evolution, and the interpretation of gravitational-wave signals from magnetized remnants.