arXiv subjects

Rainer Hollerbach

Publications and source records attributed to Rainer Hollerbach.

At least 37 records · Page 2Linked to original sources

arXiv2017

Magnetic Axis Drift and Magnetic Spot Formation in Neutron Stars with Toroidal Fields

We explore magnetic field configurations that lead to the formation of magnetic spots on the surface of neutron stars, and to the displacement of the magnetic dipole axis. We find that a toroidally dominated magnetic field is essential for the generation of a single spot with a strong magnetic field. Once a spot forms, it survives for several million years, even after the total magnetic field has decayed significantly. We find that the dipole axis is not stationary with respect to the neutron star's surface and does not in general coincide with the location of the magnetic spot. This is due to non-axisymmetric instabilities of the toroidal field that displace the poloidal dipole axis at rates that may reach $0.4^{\circ}$ per century. A misaligned poloidal dipole axis with the toroidal field leads to more significant displacement of the dipole axis than the fully aligned case. Finally we discuss the evolution of neutron stars with such magnetic fields on the $P-\dot{P}$ diagram and the observational implications. We find that neutron stars spend a very short time before they cross the Death-Line of the $P-\dot{P}$ diagram, compared to their characteristic ages. Moreover, the maximum intensity of their surface magnetic field is substantially higher than the dipole component of the field. We argue that SGR 0418+5729 could be an example of this type of behaviour, having a weak dipole field, yet hosting a magnetic spot responsible for its magnetar behaviour. The evolution on the pulse profile and braking index of the Crab pulsar, which are attributed to an increase of its obliquity, are compatible with the anticipated drift of the magnetic axis.

astro-ph.HE↗

arXiv2017

Far-from-Equilibrium Time Evolution between two Gamma Distributions

Many systems in nature and laboratories are far from equilibrium and exhibit significant fluctuations, invalidating the key assumptions of small fluctuations and short memory time in or near equilibrium. A full knowledge of Probability Distribution Functions (PDFs), especially time-dependent PDFs, becomes essential in understanding far-from-equilibrium processes. We consider a stochastic logistic model with multiplicative noise, which has gamma distributions as stationary PDFs. We numerically solve the transient relaxation problem, and show that as the strength of the stochastic noise increases the time-dependent PDFs increasingly deviate from gamma distributions. For sufficiently strong noise a transition occurs whereby the PDF never reaches a stationary state, but instead forms a peak that becomes ever more narrowly concentrated at the origin. The addition of an arbitrarily small amount of additive noise regularizes these solutions, and re-establishes the existence of stationary solutions. In addition to diagnostic quantities such as mean value, standard deviation, skewness and kurtosis, the transitions between different solutions are analyzed in terms of entropy and information length, the total number of statistically distinguishable states that a system passes through in time.

cond-mat.stat-mech↗

arXiv2017

Dynamo Action in a Quasi-Keplerian Taylor-Couette Flow

We numerically compute the flow of an electrically conducting fluid in a Taylor-Couette geometry where the rotation rates of the inner and outer cylinders satisfy $Ω_o/Ω_i=(r_o/r_i)^{-3/2}$. In this quasi-Keplerian regime a non-magnetic system would be Rayleigh-stable for all Reynolds numbers $Re$, and the resulting purely azimuthal flow incapable of kinematic dynamo action for all magnetic Reynolds numbers $Rm$. For $Re=10^4$ and $Rm=10^5$ we demonstrate the existence of a finite-amplitude dynamo, whereby a suitable initial condition yields mutually sustaining turbulence and magnetic fields, even though neither could exist without the other. This dynamo solution results in significantly increased outward angular momentum transport, with the bulk of the transport being by Maxwell rather than Reynolds stresses.

physics.flu-dyn↗

arXiv2017

Transport properties of the Azimuthal Magnetorotational Instability

The magnetorotational instability (MRI) is thought to be a powerful source of turbulence in Keplerian accretion disks. Motivated by recent laboratory experiments, we study the MRI driven by an azimuthal magnetic field in an electrically conducting fluid sheared between two concentric rotating cylinders. By adjusting the rotation rates of the cylinders, we approximate angular velocity profiles $ω\propto r^{q}$. We perform direct numerical simulations of a steep profile close to the Rayleigh line $q \gtrsim -2 $ and a quasi-Keplerian profile $q \approx -3/2$ and cover wide ranges of Reynolds ($Re\le 4\cdot10^4$) and magnetic Prandtl numbers ($0\le Pm \le 1$). In the quasi-Keplerian case, the onset of instability depends on the magnetic Reynolds number, with $Rm_c \approx 50$, and angular momentum transport scales as $\sqrt{Pm} Re^2$ in the turbulent regime. The ratio of Maxwell to Reynolds stresses is set by $Rm$. At the onset of instability both stresses have similar magnitude, whereas the Reynolds stress vanishes or becomes even negative as $Rm$ increases. For the profile close to the Rayleigh line, the instability shares these properties as long as $Pm\gtrsim0.1$, but exhibits a markedly different character if $Pm\rightarrow 0$, where the onset of instability is governed by the Reynolds number, with $Re_c \approx 1250$, transport is via Reynolds stresses and scales as $Re^2$. At intermediate $Pm=0.01$ we observe a continuous transition from one regime to the other, with a crossover at $Rm=\mathcal{O}(100)$. Our results give a comprehensive picture of angular momentum transport of the MRI with an imposed azimuthal field.

astro-ph.SR↗

arXiv2017

Time-dependent probability density functions and information geometry in stochastic logistic and Gompertz models

A probabilistic description is essential for understanding growth processes far from equilibrium. In this paper, we compute time-dependent Probability Density Functions (PDFs) in order to investigate stochastic logistic and Gompertz models, which are two of the most popular growth models. We consider different types of short-correlated internal (multiplicative) and external (additive) stochastic noises and compare the time-dependent PDFs in the two models, elucidating the effects of the additive and multiplicative noises on the form of PDFs. We demonstrate an interesting transition from a unimodal to a bimodal PDF as the multiplicative noise increases for a fixed value of the additive noise. A much weaker (leaky) attractor in the Gompertz model leads to a significant (singular) growth of the population of a very small size. We point out the limitation of using stationary PDFs, mean value and variance in understanding statistical properties of the growth far from equilibrium, highlighting the importance of time-dependent PDFs. We further compare these two models from the perspective of information change that occurs during the growth process. Specifically, we define an infinitesimal distance at any time by comparing two PDFs at times infinitesimally apart and sum these distances in time. The total distance along the trajectory quantifies the total number of different states that the system undergoes in time, and is called the information length. We show that the time-evolution of the two models become more similar when measured in units of the information length and point out the merit of using the information length in unifying and understanding the dynamic evolution of different growth processes.

nlin.AO↗

arXiv2017

Phase Slip Solutions in Magnetically Modulated Taylor-Couette Flow

We numerically investigate Taylor-Couette flow in a wide-gap configuration, with $r_i/r_o=1/2$, the inner cylinder rotating, and the outer cylinder stationary. The fluid is taken to be electrically conducting, and a magnetic field of the form $B_z\approx(1 + \cos(2πz/z_0))/2$ is externally imposed, where the wavelength $z_0=50(r_o-r_i)$. Taylor vortices form where the field is weak, but not where it is strong. As the Reynolds number measuring the rotation rate is increased, the initial onset of vortices involves phase slip events, whereby pairs of Taylor vortices are periodically formed and then drift outward, away from the midplane where $B_z=0$. Subsequent bifurcations lead to a variety of other solutions, including ones both symmetric and asymmetric about the midplane. For even larger Reynolds numbers a different type of phase slip arises, in which vortices form at the outer edges of the pattern and drift inward, disappearing abruptly at a certain point. These solutions can also be symmetric or asymmetric about the midplane, and co-exist at the same Reynolds number. Many of the dynamics of these phase slip solutions are qualitatively similar to previous results in geometrically ramped Taylor-Couette flows.

physics.flu-dyn↗

arXiv2017

Shercliff layers in strongly magnetic cylindrical Taylor-Couette flow

We numerically compute axisymmetric Taylor-Couette flow in the presence of axially periodic magnetic fields, with Hartmann numbers up to $Ha^2=10^7$. The geometry of the field singles out special field lines on which Shercliff layers form. These are simple shear layers for insulating boundaries, versus super-rotating or counter-rotating layers for conducting boundaries. Some field configurations have previously studied spherical analogs, but fundamentally new configurations also exist, having no spherical analogs. Finally, we explore the influence of azimuthal fields $B_ϕ\sim r^{-1}{\bf\hat e}_ϕ$ on these layers, and show that the flow is suppressed for conducting boundaries but enhanced for insulating boundaries.

physics.flu-dyn↗

arXiv2017

Absolute versus convective helical magnetorotational instabilities in Taylor-Couette flows

We study magnetic Taylor-Couette flow in a system having nondimensional radii $r_i=1$ and $r_o=2$, and periodic in the axial direction with wavelengths $h\ge100$. The rotation ratio of the inner and outer cylinders is adjusted to be slightly in the Rayleigh-stable regime, where magnetic fields are required to destabilize the flow, in this case triggering the axisymmetric helical magnetorotational instability (HMRI). Two choices of imposed magnetic field are considered, both having the same azimuthal component $B_ϕ=r^{-1}$, but differing axial components. The first choice has $B_z=0.1$, and yields the familiar HMRI, consisting of unidirectionally traveling waves. The second choice has $B_z\approx0.1\sin(2πz/h)$, and yields HMRI waves that travel in opposite directions depending on the sign of $B_z$. The first configuration corresponds to a convective instability, the second to an absolute instability. The two variants behave very similarly regarding both linear onset as well as nonlinear equilibration.

physics.flu-dyn↗

arXiv2016

Axisymmetric pulse train solutions in narrow-gap spherical Couette flow

We numerically compute the flow induced in a spherical shell by fixing the outer sphere and rotating the inner one. The aspect ratio $ε=(r_o-r_i)/r_i$ is set at 0.04 and 0.02, and in each case the Reynolds number measuring the inner sphere's rotation rate is increased to $\sim10\%$ beyond the first bifurcation from the basic state flow. For $ε=0.04$ the initial bifurcations are the same as in previous numerical work at $ε=0.154$, and result in steady one- and two-vortex states. Further bifurcations yield travelling wave solutions similar to previous analytic results valid in the $ε\to0$ limit. For $ε=0.02$ the steady one-vortex state no longer exists, and the first bifurcation is directly to these travelling wave solutions, consisting of pulse trains of Taylor vortices travelling toward the equator from both hemispheres, and annihilating there in distinct phase-slip events. We explore these time-dependent solutions in detail, and find that they can be both equatorially symmetric and asymmetric, as well as periodic or quasi-periodic in time.

physics.flu-dyn↗

arXiv2016

Suppression of a kinematic dynamo by large shear

We numerically solve the magnetic induction equation in a spherical shell geometry, with a kinematically prescribed axisymmetric flow that consists of a superposition of a small-scale helical flow and a large-scale shear flow. The small-scale flow is chosen to be a local analog of the classical Roberts cells, consisting of strongly helical vortex rolls. The large-scale flow is a shearing motion in either the radial or the latitudinal directions. In the absence of large-scale shear, the small-scale flow is an efficient dynamo, in agreement with previous results. Adding increasingly large shear flows strongly suppresses the dynamo efficiency, indicating that shear is not always a favourable ingredient in dynamo action.

physics.plasm-ph↗

arXiv2016

Dynamical model for spindown of solar-type stars

Since their formation, stars slow down their rotation rates by the removal of angular momentum from their surfaces, e.g. via stellar winds. Despite the complexity of the processes involved, a traditional model, where the removal of angular momentum loss by magnetic fields is prescribed, has provided a useful framework to understand observational relations between stellar rotation and age and magnetic field strength. Here, a spindown model is proposed where loss of angular momentum by magnetic fields is evolved dynamically, instead of being kinematically prescribed. To this end, we evolve the stellar rotation and magnetic field simultaneously over stellar evolution time by extending our previous work on a dynamo model which incorporates the nonlinear feedback mechanisms on rotation and magnetic fields. Our extended model reproduces key observations and explains the presence of the two branches of (fast and slow rotating) stars which have different relations between rotation rate $Ω$ vs. time (age), magnetic field strength $|B|$ vs. rotation rate, and frequency of magnetic field $ω_{cyc}$ vs. rotation rate. For fast rotating stars we find: (i) an exponential spindown $Ω\propto e^{-1.35t}$, with $t$ measured in Gyrs, (ii) $|B|$ saturates for higher rotation rate, (iii) $ω_{cyc} \propto Ω^{0.85}$. For slow rotating stars we obtain: (i) a power law spindown $Ω\propto t^{-0.52}$, (ii) $|B|$ scales almost linearly with rotation rate, (iii) $ω_{cyc} \propto Ω^{1.16}$. The results obtained are in good agreement with observations. The Vaughan-Preston gap is consistently explained in our model by the shortest spindown timescale in this transition from fast to slow rotators. Our results highlight the importance of self-regulation of magnetic fields and rotation by direct and indirect interactions involving nonlinear feedback in stellar evolution.

astro-ph.SR↗

arXiv2016

Resistive Tearing Instability in Electron-MHD: Application to Neutron Star Crusts

We study a resistive tearing instability developing in a system evolving through the combined effect of Hall drift in the Electron-MHD limit and Ohmic dissipation. We explore first the exponential growth of the instability in the linear case and we find the fastest growing mode, the corresponding eigenvalues and dispersion relation. The instability growth rate scales as $γ\propto B^{2/3} σ^{-1/3}$ where $B$ is the magnetic field and $σ$ the electrical conductivity. We confirm the development of the tearing resistive instability in the fully non-linear case, in a plane parallel configuration where the magnetic field polarity reverses, through simulations of systems initiating in Hall equilibrium with some superimposed perturbation. Following a transient phase, during which there is some minor rearrangement of the magnetic field, the perturbation grows exponentially. Once the instability is fully developed the magnetic field forms the characteristic islands and X-type reconnection points, where Ohmic decay is enhanced. We discuss the implications of this instability for the local magnetic field evolution in neutron stars' crusts, proposing that it can contribute to heating near the surface of the star, as suggested by models of magnetar post-burst cooling. In particular, we find that a current sheet a few meters thick, covering as little as $1\%$ of the total surface can provide $10^{42}~$erg in thermal energy within a few days. We briefly discuss applications of this instability in other systems where the Hall effect operates such as protoplanetary discs and space plasmas.

astro-ph.HE↗

arXiv2016

Magnetic field evolution in magnetar crusts through three dimensional simulations

Current models of magnetars require extremely strong magnetic fields to explain their observed quiescent and bursting emission, implying that the field strength within the star's outer crust is orders of magnitude larger than the dipole component inferred from spin-down measurements. This presents a serious challenge to theories of magnetic field generation in a proto-neutron star. Here, we present detailed modelling of the evolution of the magnetic field in the crust of a neutron star through 3-D simulations. We find that, in the plausible scenario of equipartition of energy between global-scale poloidal and toroidal magnetic components, magnetic instabilities transfer energy to non-axisymmetric, kilometre-sized magnetic features, in which the local field strength can greatly exceed that of the global-scale field. These intense small-scale magnetic features can induce high energy bursts through local crust yielding, and the localised enhancement of Ohmic heating can power the star's persistent emission. Thus, the observed diversity in magnetar behaviour can be explained with mixed poloidal-toroidal fields of comparable energies.

astro-ph.SR↗

arXiv2016

Generalised quasi-linear approximation of the HMRI

Motivated by recent advances in Direct Statistical Simulation (DSS) of astrophysical phenomena such as out of equilibrium jets, we perform a Direct Numerical Simulation (DNS) of the helical magnetorotational instability (HMRI) under the generalised quasilinear approximation (GQL). This approximation generalises the quasilinear approximation (QL) to include the self-consistent interaction of large-scale modes, interpolating between fully nonlinear DNS and QL DNS whilst still remaining formally linear in the small scales. In this paper we address whether GQL can more accurately describe low-order statistics of axisymmetric HMRI when compared with QL by performing DNS under various degrees of GQL approximation. We utilise various diagnostics, such as energy spectra in addition to first and second cumulants, for calculations performed for a range of Reynolds and Hartmann numbers (describing rotation and imposed magnetic field strength respectively). We find that GQL performs significantly better than QL in describing the statistics of the HMRI even when relatively few large-scale modes are kept in the formalism. We conclude that DSS based on GQL (GCE2) will be significantly more accurate than that based on QL (CE2).

physics.flu-dyn↗

arXiv2015

Nonaxisymmetric linear instability of cylindrical magnetohydrodynamic Taylor-Couette flow

We consider the nonaxisymmetric modes of instability present in Taylor-Couette flow under the application of helical magnetic fields, mainly for magnetic Prandtl numbers close to the inductionless limit, and conduct a full examination of marginal stability in the resulting parameter space. We allow for the azimuthal magnetic field to be generated by a combination of currents in the inner cylinder and fluid itself, and introduce a parameter governing the relation between the strength of these currents. A set of governing eigenvalue equations for the nonaxisymmetric modes of instability are derived and solved by spectral collocation with Chebyshev polynomials over the relevant parameter space, with the resulting instabilities examined in detail. We find that by altering the azimuthal magnetic field profiles the azimuthal magnetorotational instability, nonaxisymmetric helical magnetorotational instability, and Tayler instability yield interesting dynamics, such as different preferred mode types, and modes with azimuthal wave number $m>1$. Finally, a comparison is given to the recent WKB analysis performed by Kirillov $et$ $al$. [Kirillov, Stefani, and Fukumoto, J. Fluid Mech. 760, 591 (2014)] and its validity in the linear regime.

physics.flu-dyn↗

arXiv2015

Magnetar Activity via the Density-Shear Instability in Hall-MHD

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 $5\times10^{14}$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.

astro-ph.HE↗

arXiv2015

Three-dimensional simulations of the magnetic stress in a neutron star crust

We present the first fully self-consistent three-dimensional model of a neutron star's magnetic field, generated by electric currents in the star's crust via the Hall effect. We find that the global-scale field converges to a Hall-attractor state, as seen in recent axisymmetric models, but that small-scale features in the magnetic field survive even on much longer timescales. These small-scale features propagate toward the dipole equator, where the crustal electric currents organize themselves into a strong equatorial jet. By calculating the distribution of magnetic stresses in the crust, we predict that neutron stars with fields stronger than $10^{14}$G can still be subject to starquakes more than $10^5$yr after their formation.

astro-ph.SR↗

arXiv2014

Density-shear instability in electron MHD

We discuss a novel instability in inertia-less electron magneto-hydrodynamics (EMHD), which arises from a combination of electron velocity shear and electron density gradients. The unstable modes have a lengthscale longer than the transverse density scale, and a growth-rate of the order of the inverse Hall timescale. We suggest that this density-shear instability may be of importance in magnetic reconnection regions on scales smaller than the ion skin depth, and in neutron star crusts. We demonstrate that the so-called Hall drift instability, previously argued to be relevant in neutron star crusts, is a resistive tearing instability rather than an instability of the Hall term itself. We argue that the density-shear instability is of greater significance in neutron stars than the tearing instability, because it generally has a faster growth-rate and is less sensitive to geometry and boundary conditions. We prove that, for uniform electron density, EMHD is "at least as stable" as regular, incompressible MHD, in the sense that any field configuration that is stable in MHD is also stable in EMHD. We present a connection between the density-shear instability in EMHD and the magneto-buoyancy instability in anelastic MHD.

physics.plasm-ph↗