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This paper is the first in a series aimed at understanding the long-termevolution of neutron star magnetic fields. We model the stellar matter as anelectrically neutral and lightly ionized plasma composed of three movingparticle species: neutrons, protons, and electrons, which can be converted intoeach other by weak interactions (beta decays), suffer binary collisions, and beaffected by each other's macroscopic electromagnetic fields. Since theevolution of the magnetic field occurs over thousands of years or more,compared to dynamical time scales (sound and Alfv\'en) of milliseconds toseconds, we use a slow-motion approximation in which we neglect the inertialterms in the equations of motion for the particles. We restrict ourselves to aone-dimensional geometry in which the magnetic field points in one Cartesiandirection but varies only along an orthogonal direction. We study the evolutionof the system in three different ways: (i) estimating time scales directly fromthe equations, guided by physical intuition; (ii) a normal-mode analysis in thelimit of a nearly uniform system; and (iii) a finite-difference numericalintegration of the equations of motion. We find good agreement between ouranalytical normal-mode solutions and the numerical simulations. We show thatthe magnetic field and the particles evolve through successivequasi-equilibrium states, on time scales that can be understood by physicalarguments. Depending of the parameter values the magnetic field can evolve byohmic diffusion or by ambipolar diffusion, the latter being limited either byinterparticle collisions or by relaxation to chemical equilibrium through betadecays. The numerical simulations are further validated by verifying that theysatisfy the known conservation laws also in highly non-linear situations.

Magnetic Field Evolution in Neutron Stars: One-Dimensional Multi-Fluid Model