We investigate the density-shear instability in Hall-MHD via numerical simulation of the full non-linear problem, in the context of magnetar activity. We confirm the development of the instability of a plane-parallel magnetic field with an appropriate intensity and electron density profile, in accordance with analytic theory. We find that the instability also appears for a monotonically decreasing electron number density and magnetic field, a plane-parallel analogue of an azimuthal or meridional magnetic field in the crust of a magnetar. The growth rate of the instability depends on the Hall properties of the field (magnetic field intensity, electron number density and the corresponding scale-heights), while being insensitive to weak resistivity. Since the Hall effect is the driving process for the evolution of the crustal magnetic field of magnetars, we argue that this instability is critical for systems containing strong meridional or azimuthal fields. We find that this process mediates the formation of localised structures with much stronger magnetic field than the average, which can lead to magnetar activity and accelerate the dissipation of the field and consequently the production of Ohmic heating. Assuming a 5 × 10 14 G magnetic field at the base of crust, we anticipate that magnetic field as strong as 10 15 G will easily develop in regions of typical size of a few 10 2 meters, containing magnetic energy of 10 43 erg, sufficient to power magnetar bursts. These active regions are more likely to appear in the magnetic equator where the tangential magnetic field is stronger.

Aims. In the first few minutes of a newborn neutron star's life the Hall effect and differential rotation may both be important. We demonstrate that these two ingredients are sufficient for generating a "shear-Hall instability" and for studying its excitation conditions, growth rates, and characteristic magnetic field patterns. Methods. We numerically solve the induction equation in a spherical shell, with a kinematically prescribed differential rotation profile Ω(s), where s is the cylindrical radius. The Hall term is linearized about an imposed uniform axial field. The linear stability of individual azimuthal modes, both axisymmetric and non-axisymmetric, is then investigated. Results. For the shear-Hall instability to occur, the axial field must be parallel to the rotation axis if Ω(s) decreases outward, whereas if Ω(s) increases outward it must be anti-parallel. The instability draws its energy from the differential rotation, and occurs on the short rotational timescale rather than on the much longer Hall timescale. It operates most efficiently if the Hall time is comparable to the diffusion time. Depending on the precise field strengths B 0 , either axisymmetric or non-axisymmetric modes may be the most unstable. Conclusions. Even if the differential rotation in newborn neutron stars is quenched within minutes, the shear-Hall instability may nevertheless amplify any seed magnetic fields by many orders of magnitude.

We investigate the decay of a large-scale magnetic field in the context of incompressible, two-dimensional magnetohydrodynamic turbulence. It is well established that a very weak mean field, of strength significantly below equipartition value, induces a small-scale field strong enough to inhibit the process of turbulent magnetic diffusion. In light of ever-increasing computer power, we revisit this problem to investigate fluids and magnetic Reynolds numbers that were previously inaccessible. Furthermore, by exploiting the relation between the turbulent diffusion of the magnetic potential and that of the magnetic field, we are able to calculate the turbulent magnetic diffusivity extremely accurately through the imposition of a uniform mean magnetic field. We confirm the strong dependence of the turbulent diffusivity on the product of the magnetic Reynolds number and the energy of the large-scale magnetic field. We compare our findings with various theoretical descriptions of this process.

We investigate the stability of the Hall-MHD system and determine its importance for neutron stars at their birth, when they still consist of differentially rotating plasma permeated by extremely strong magnetic fields. We solve the linearised Hall-MHD equations in a spherical shell threaded by a homogeneous magnetic field. With the fluid/flow coupling and the Hall effect included, the magnetorotational instability and the Hall effect are both acting together. Results differ for magnetic fields aligned with the rotation axis and anti-parallel magnetic fields. For a positive alignment of the magnetic field the instability grows on a rotational time-scale for any sufficiently large magnetic Reynolds number. Even the magnetic fields which are stable against the MRI due to the magnetic diffusion are now susceptible to the shear-Hall instability. In contrast, the negative alignment places strong restrictions on the growth and the magnitude of the fields, hindering the effectiveness of the Hall-MRI. While non-axisymmetric modes of the MRI can be suppressed by strong enough rotation, there is no such restriction when the Hall effect is present. The implications for the magnitude and the topology of the magnetic field of a young neutron star may be significant.