The high quiescent x-ray luminosity observed in some magnetars is widely attributed to the decay and evolution of their ultrastrong magnetic fields. Several dissipation mechanisms have been proposed, each operating with different efficiencies depending on the region of the star. In this context, ambipolar diffusion, i.e., the relative motion of charged particles with respect to neutrons in the neutron star core, has been proposed as a promising candidate due to its strong dependence on magnetic field strength and its capacity to convert magnetic energy into heat. We perform axisymmetric magnetohydrodynamic simulations to study the long-term magnetic evolution of a neutron star core composed of normal (non-Cooper-paired) matter under the influence of ambipolar diffusion. The core is modeled as a two-fluid system consisting of neutrons and a charged-particle fluid (protons and electrons), coupled to the magnetic field. Simulations are performed both at constant and variable temperatures. In the latter case, a strategy that decouples the magnetic and thermal evolution is employed, enabling efficient thermal modeling across a range of initial magnetic field strengths. At constant temperature, we obtain the expected result where neutrons reach diffusive equilibrium, the Lorentz force is balanced by chemical potential gradients of charged particles, and the magnetic field satisfies a nonlinear Grad-Shafranov equation. When thermal evolution is included, fields $B\gtrsim 5\times {10}^{15}\mathrm{G}$ can balance ambipolar heating and neutrino cooling, delaying the evolution over $\sim {10}^3[B/(5\times {10}^{15}\mathrm{G}){]}^{-6/5}\mathrm{yr}$. Although the surface luminosity is enhanced compared with passive cooling, the heating from ambipolar diffusion alone is insufficient to fully explain the persistent x-ray emission observed in magnetars.

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• Magnetohydrodynamics
• Neutron star cooling
• Neutron stars & pulsars