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At the high temperatures present inside recently formed neutron stars($T\gtrsim 5\times 10^{8}\, \text{K}$), the particles in their cores are in the"strong-coupling" regime, in which collisional forces make them behave as asingle, stably stratified, and thus non-barotropic fluid. In this regime,axially symmetric hydromagnetic quasi-equilibrium states are possible, whichare only constrained to have a vanishing azimuthal Lorentz force. In suchequilibria, the particle species are not in chemical ($\beta$) equilibrium, so$\beta$ decays (Urca reactions) tend to restore the chemical equilibrium,inducing fluid motions that change the magnetic field configuration. If thestars remained hot for a sufficiently long time, this evolution wouldeventually lead to a chemical equilibrium state, in which the fluid isbarotropic and the magnetic field, if axially-symmetric, satisfies thenon-linear Grad-Shafranov equation. In this work, we present a numerical schemethat decouples the magnetic and thermal evolution, enabling to efficientlyperform, for the first time, long-term magneto-thermal simulations in thisregime for different magnetic field strengths and geometries. Our resultsdemonstrate that, even for magnetar-strength fields $\gtrsim 10^{16} \,\mathrm{G}$, the feedback from the magnetic evolution on the thermal evolutionis negligible. Thus, as the core passively cools, the Urca reactions quicklybecome inefficient at restoring chemical equilibrium, so the magnetic fieldevolves very little, and the Grad-Shafranov equilibrium is not attained in thisregime. Therefore, any substantial evolution of the core magnetic field mustoccur later, in the cooler "weak-coupling" regime ($T\lesssim 5\times 10^8 \,\mathrm{K}$), in which Urca reactions are effectively frozen and ambipolardiffusion becomes relevant.

Magneto-thermal evolution in the cores of adolescent neutron stars: The Grad-Shafranov equilibrium is never reached in the 'strong-coupling' regime