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The braking torque that dictates the timing properties of magnetars is closely tied to the large-scale dipolar magnetic field on their surface. The formation of this field has been a topic of ongoing debate. One proposed mechanism, based on macroscopic principles, involves an inverse cascade within the neutron star’s crust. However, this phenomenon has not been observed in realistic simulations. In this study, we provide compelling evidence supporting the feasibility of the inverse cascading process in the presence of an initial helical magnetic field within realistic neutron star crusts and discuss its contribution to the amplification of the large-scale magnetic field. Our findings, derived from a systematic investigation that considers various coordinate systems, peak wavenumber positions, crustal thicknesses, magnetic boundary conditions, and magnetic Lundquist numbers, reveal that the specific geometry of the crustal domain–with its extreme aspect ratio–requires an initial peak wavenumber from small-scale structures for the inverse cascade to occur. However, this same aspect ratio confines the cascade to structures on the scale of the crust, making the formation of a large-scale dipolar surface field unlikely. Despite these limitations, the inverse cascade remains a significant factor in the magnetic field evolution within the crust and may help explain highly magnetized objects with weak surface dipolar fields, such as low-field magnetars and central compact objects. 

ABSTRACT The thermal evolution of isolated neutron stars is a key element in unravelling their internal structure and composition and establishing evolutionary connections among different observational subclasses. Previous studies have predominantly focused on one-dimensional or axisymmetric two-dimensional models. In this study, we present the thermal evolution component of the novel three-dimensional magnetothermal code MATINS (MAgneto-Thermal evolution of Isolated Neutron Star). MATINS employs a finite volume scheme and integrates a realistic background structure, along with state-of-the-art microphysical calculations for the conductivities, neutrino emissivities, heat capacity, and superfluid gap models. This paper outlines the methodology employed to solve the thermal evolution equations in MATINS, along with the microphysical implementation that is essential for the thermal component. We test the accuracy of the code and present simulations with non-evolving magnetic fields of different configurations (all with electrical currents confined to the crust and a magnetic field that does not thread the core), to produce temperature maps of the neutron star surface. Additionally, for a specific magnetic field configuration, we show one fully coupled evolution of magnetic field and temperature. Subsequently, we use a ray-tracing code to link the neutron star surface temperature maps obtained by MATINS with the phase-resolved spectra and pulsed profiles that would be detected by distant observers. This study, together with our previous article focused on the magnetic formalism, presents in detail the most advanced evolutionary code for isolated neutron stars, with the aim of comparison with their timing properties, thermal luminosities and the associated X-ray light curves.