Full Nonlinear Evolution Research Articles

Resonant capture of stars by black hole binaries: extreme eccentricity excitation

ABSTRACT Massive black hole (MBH) binaries in galactic nuclei are one of the leading sources of ${\sim}$mHz gravitational waves (GWs) for future missions such as Laser Interferometer Space Antenna (LISA). However, the poor sky localization of GW interferometers will make it challenging to identify the host galaxy of MBH mergers absent an electromagnetic counterpart. One such counterpart is the tidal disruption of a star that has been captured into mean motion resonance with the inspiralling binary. Here we investigate the production of tidal disruption events (TDEs) through capture into, and subsequent evolution in, orbital resonance. We examine the full non-linear evolution of planar autoresonance for stars that lock into autoresonance with a shrinking MBH binary. Capture into the 2:1 resonance is guaranteed for any realistic astrophysical parameters (given a relatively small MBH binary mass ratio), and the captured star eventually attains an eccentricity $e\approx 1$, leading to a TDE. Stellar discs can be produced around MBHs following an active galactic nucleus episode, and we estimate the TDE rates from resonant capture produced when a secondary MBH begins inspiralling through such a disc. In some cases, the last resonant TDE can occur within a decade of the eventual LISA signal, helping to localize the GW event.

Gauge preheating with full general relativity

We study gauge preheating following pseudoscalar-driven inflation in full general relativity. We implement the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) scheme to solve the full nonlinear evolution of the metric alongside the dynamics of the pseudoscalar and gauge fields. The dynamics of the background and emission of gravitational waves are broadly consistent with simulations in a Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime. We find large, localized overdensities in the BSSN simulations of order δ = δρ/ρ ∼ 30, and the dimensionless power spectrum of δ peaks above unity. These overdense regions are seeded on length scales only slightly smaller than the horizon, and have a compactness C ∼ 0.1. The scale of peak compactness is shorter than the Jeans length, which implies that pressure of the matter fields plays an important role in the evolution of these objects.

On reactive thin liquid films falling down a vertical cylinder

Chemical reactions in thin liquid films falling down vertical cylinders are widely present in various industrial settings, such as in combustion chambers of internal combustion engines. However, the effect of chemical reactions, especially the non-isothermal reactions, on the dynamics of thin films remains poorly understood. Therefore, in this paper, we systematically study the dynamics of a thin liquid film flowing down vertical cylinders in the presence of an exothermic or endothermic chemical reaction. A reduced model based on the assumption that the film thickness is much smaller than the cylinder radius is firstly derived. We then examine the effect of chemical reactions on the linear stability of the evolution equation as well as its nonlinear behaviors, including the profile and propagation speed of steady traveling waves. We find that the size and propagation speed of sliding beads on the cylinder are suppressed for an exothermic chemical reaction and promoted for an endothermic chemical reaction, respectively. In the end, we perform direct numerical simulations of the full nonlinear evolution equation, which are consistent with the predications from linear stability analysis and nonlinear traveling wave solutions. Our results provide new insight into the influence of chemical reactions on the dynamics of thin films falling down the cylindrical substrate.

Holographic Bjorken flow of a hot and dense fluid in the vicinity of a critical point

In this paper we use the gauge/gravity duality to perform the first systematic study of the onset of hydrodynamic behavior in a hot and dense far-from-equilibrium strongly coupled relativistic fluid with a critical point. By employing a top-down holographic construction that stems from string theory, we numerically obtain the full nonlinear evolution of the far-from-equilibrium system undergoing a Bjorken expansion and address the following question: how does hydrodynamic behavior emerge in the vicinity of a critical point in the phase diagram? For the top-down holographic system analyzed in the present work, we find that the approach to hydrodynamics is strongly affected by the presence of the critical point: the closer the ratio between the chemical potential and the temperature is to its critical value, the longer it takes for the system to be well described by the equations of viscous hydrodynamics.

Spectral and dynamical analysis of a single vortex ring in anisotropic harmonically trapped three-dimensional Bose-Einstein condensates

In the present work, motivated by numerous recent experimental developments we revisit the dynamics of a single vortex ring in anisotropic harmonic traps. At the theoretical level, we start from a general Lagrangian dynamically capturing the evolution of a vortex ring and not only consider its spectrum of linearized excitations, but also explore the full nonlinear dynamical evolution of the ring as a vortical filament. The theory predicts that the ring is stable for $1 \leq \lambda \leq 2$, where $\lambda=\omega_z/\omega_r$ is the ratio of the trapping frequencies along the $z$ and $r$ axes, i.e., for spherical to slightly oblate condensates. We compare this prediction with direct numerical simulations of the full 3D Gross-Pitaevskii equation (GPE) capturing the linearization spectrum of the ring for different values of the chemical potential and as a function of the anisotropy parameter $\lambda$. We identify this result as being only asymptotically valid as the chemical potential $\mu \rightarrow \infty$, revealing how the stability interval narrows and, in particular, its upper bound decreases for finite $\mu$. Finally, we compare at the dynamical level the results of the GPE with the ones effectively capturing the ring dynamics, revealing the unstable evolution for different values of $\lambda$, as well as the good agreement between the two.

Resistively-limited current sheet implosions in planar anti-parallel (1D) and null-point containing (2D) magnetic field geometries

Implosive formation of current sheets is a fundamental plasma process. Previous studies focused on the early time evolution, while here our primary aim is to explore the longer-term evolution, which may be critical for determining the efficiency of energy release. To address this problem, we investigate two closely related problems, namely: (i) 1D, pinched anti-parallel magnetic fields and (ii) 2D, null point containing fields which are locally imbalanced (“null-collapse” or “X-point collapse”). Within the framework of resistive MHD, we simulate the full nonlinear evolution through three distinct phases: the initial implosion, its eventual halting mechanism, and subsequent evolution post-halting. In a parameter study, we find that the scaling with resistivity of current sheet properties at the halting time is in good agreement—in both geometries—with that inferred from a known 1D similarity solution. We find that the halting of the implosions occurs rapidly after reaching the diffusion scale by sudden Ohmic heating of the dense plasma within the current sheet, which provides a pressure gradient sufficient to oppose further collapse and decelerate the converging flow. This back-pressure grows to exceed that required for force balance and so the post-implosion evolution is characterised by the consequences of the current sheet “bouncing” outwards. These are: (i) the launching of propagating fast MHD waves (shocks) outwards and (ii) the width-wise expansion of the current sheet itself. The expansion is only observed to stall in the 2D case, where the pressurisation is relieved by outflow in the reconnection jets. In the 2D case, we quantify the maximum amount of current sheet expansion as it scales with resistivity and analyse the structure of the reconnection region, which forms post-expansion, replete with Petschek-type slow shocks and fast termination shocks.

Models of interacting pairs of thin, quasi-geostrophic vortices: steady-state solutions and nonlinear stability

We study pairwise interactions of elliptical quasi-geostrophic (QG) vortices as the limiting case of vanishingly thin uniform potential vorticity ellipsoids. In this limit, the product of the vertical extent of the ellipsoid and the potential vorticity within it is held fixed to a finite non-zero constant. Such elliptical “lenses” inherit the property that, in isolation, they steadily rotate without changing shape. Here, we use this property to extend both standard moment models and Hamiltonian ellipsoidal models to approximate the dynamical interaction of such elliptical lenses. By neglecting non-elliptical deformations, the simplified models reduce the dynamics to just four degrees of freedom per vortex. For simplicity, we focus on pairwise interactions between identical elliptical vortices initially separated in both the horizontal and vertical directions. The dynamics of the simplified models are compared with the full QG dynamics of the system, and show good agreement as expected for sufficiently distant lenses. The results reveal the existence of families of steadily rotating equilibria in the initial horizontal and vertical separation parameter space. For sufficiently large vertical separations, equilibria with varying shape exist for all horizontal separations. Below a critical vertical separation (stretched by the constant ratio of buoyancy to Coriolis frequencies N / f), comparable to the mean radius of either vortex, a gap opens in horizontal separation where no equilibria are possible. Solutions near the edge of this gap are unstable. In the full QG system, equilibria at the edge of the gap exhibit corners (infinite curvature) along their boundaries. Comparisons of the model results with the full nonlinear QG evolution show that the early stages of the instability are captured by the Hamiltonian elliptical model but not by the moment model that inaccurately estimates shorter-range interactions.

Nonlinear disturbance growth during sedimentation in dilute fibre suspensions

AbstractDisturbances in a dilute fibre suspension are studied with an Eulerian approach. Based on a linear stability analysis, it is shown that inertia and hydrodynamic diffusion damp perturbations at long wavelengths and short wavelengths, respectively, leading to a wavenumber selection. For small but finite Reynolds number of the fluid bulk motion, the most unstable wavenumber is a finite value, which increases with Reynolds number. Furthermore, the diffusion narrows the range of unstable wavenumbers. Numerical simulations of the full nonlinear evolution in time of a normal-mode perturbation show that the induced flow may either die out or saturate on a finite amplitude. The character of this long-time behaviour is dictated by the wavenumber and the presence or absence, as well as nature, of the translational and rotational diffusivities.

Nonlinear evolution of instabilities behind spheres and disks

We have performed precise and systematic experiments with PIV in order to measure the velocity field in the wake of a solid sphere and of a disk in a water channel, in the range of intermediate Reynolds number in which stationary and oscillatory instabilities appear, including the hairpin shedding regime. From these experimental data, we study the modal decomposition of the streamwise vorticity in an unsteady case and we describe the full nonlinear evolution of the bifurcating branches. We compare these results with recent theoretical and numerical studies on instability in the vortex shedding process at these intermediate Reynolds numbers.

A systematic numerical study of the tidal instability in a rotating triaxial ellipsoid

The full non-linear evolution of the tidal instability is studied numerically in an ellipsoidal fluid domain relevant for planetary cores applications. Our numerical model, based on a finite element method, is first validated by reproducing some known analytical results. This model is then used to address open questions that were up to now inaccessible using theoretical and experimental approaches. Growth rates and mode selection of the instability are systematically studied as a function of the aspect ratio of the ellipsoid and as a function of the inclination of the rotation axis compared to the deformation plane. We also quantify the saturation amplitude of the flow driven by the instability and calculate the viscous dissipation that it causes. This tidal dissipation can be of major importance for some geophysical situations and we thus derive general scaling laws which are applied to typical planetary cores.

A moving boundary model motivated by electric breakdown: II. Initial value problem

An interfacial approximation of the streamer stage in the evolution of sparks and lightning can be formulated as a Laplacian growth model regularized by a ‘kinetic undercooling’ boundary condition. Using this model we study both the linearized and the full nonlinear evolution of small perturbations of a uniformly translating circle. Within the linear approximation analytical and numerical results show that perturbations are advected to the back of the circle, where they decay. An initially analytic interface stays analytic for all finite times, but singularities from outside the physical region approach the interface for t → ∞ , which results in some anomalous relaxation at the back of the circle. For the nonlinear evolution numerical results indicate that the circle is the asymptotic attractor for small perturbations, but larger perturbations may lead to branching. We also present results for more general initial shapes, which demonstrate that regularization by kinetic undercooling cannot guarantee smooth interfaces globally in time.

A mechanism of Marangoni instability in evaporating thin liquid films due to soluble surfactant

In film coating and other applications involving thin liquid films, surfactants are typically employed to suppress the usually undesirable instabilities driven by surface phenomena. Yet, in the present study a mechanism of Marangoni instability in evaporating thin films is presented and analyzed, which has its origin on the effects of a soluble surfactant. As the film thins due to evaporation, thickness perturbations lead to surfactant concentration perturbations, which in turn drive film motion and tend to enhance uneven drying. A thin-film analysis is applied and evolution equations for the film thickness and the surfactant concentration are derived and analyzed by the techniques of linear stability and numerical simulation. In the linear analysis a nonautonomous system is obtained for the film thickness and surfactant concentration perturbations, which shows that the instability will manifest itself provided that an appropriate Marangoni number is relatively large and the surfactant solubility in the bulk is large as well. On the other hand, low solubility in the bulk, diffusion, and the effect of surfactant on interfacial mobility through the surface viscosity are found to suppress disturbance growth. Direct numerical simulations of the full nonlinear evolution equations confirm those results and add to the picture obtained for the physical system behavior. Estimates of the relevant dimensionless parameters suggest that the conditions for instability may be met in relatively thick films, on the order of tens of microns, for which the effects of molecular forces and disjoining pressure are not dominant. Moreover, the stabilizing effects of diffusion and interfacial mobility are not likely to become significant unless the films are much thinner, i.e., on the order of 1 μm or below.

Grid based linear neutrino perturbations in cosmological N-body simulations

We present a novel, fast and precise method for including the effect of light neutrinos in cosmological N-body simulations. The effect of the neutrino component is included by using the linear theory neutrino perturbations in the calculation of the gravitational potential in the N-body simulation. By comparing this new method with the full non-linear evolution first presented in [1], where the neutrino component was treated as particles, we find that the new method calculates the matter power spectrum with an accuracy better than 1% for ∑mν ≲ 0.5 eV at z = 0. This error scales approximately as (∑mν)2, making the new linear neutrino method extremely accurate for a total neutrino mass in the range 0.05−0.3 eV. At z = 1 the error is below 0.3% for ∑mν ≲ 0.5 eV and becomes negligible at higher redshifts. This new method is computationally much more efficient than representing the neutrino component by N-body particles.

Stability of the AdS Soliton Spacetime

Stability of the AdS Soliton Spacetime Gungwon Kang Korea Institute of Science and Technology Information, Daejeon 305-806 Changheon Ohy and Chul H. Leez Department of Physics and BK21 Division of Advanced Research and Education in Physics, Hanyang University, Seoul 133-791 (Received 21 August 2007) The AdS soliton spacetime is a vacuum static solution of ve-dimensional Einstein equations with a cosmological constant. It is asymptotically AdS and regular everywhere without having any curvature singularity or an event horizon inside. Recently, Horowitz and Myers conjectured that this spacetime is the ground state among all spacetime solutions that are asymptotically AdS. If such a conjecture is valid, the spacetime should be stable under small perturbations. We have investigated this issue by numerically studying the full nonlinear evolution of a certain class of perturbations. Our results show that some metric components decay out whereas one metric component grows, indicating a sort of instability. Motivated by this result, we also performed a linearized stability analysis for small perturbations in the same class. Interestingly, our result in such a linearized analysis does not show any indication of instability. Finally, we discuss the implications of our results on the stability of the AdS soliton background. PACS numbers: 04.25.Dm, 04.25.Nx, 04.65.+e

Solving the high energy evolution equation including running coupling corrections

We study the solution of the nonlinear Balitsky-Kovchegov evolution equation with the recently calculated running coupling corrections [I. I. Balitsky, Phys. Rev. D 75, 014001 (2007). and Y. Kovchegov and H. Weigert, Nucl. Phys. A784, 188 (2007).]. Performing a numerical solution we confirm the earlier result of Albacete et al. [Phys. Rev. D 71, 014003 (2005).] (obtained by exploring several possible scales for the running coupling) that the high energy evolution with the running coupling leads to a universal scaling behavior for the dipole-nucleus scattering amplitude, which is independent of the initial conditions. It is important to stress that the running coupling corrections calculated recently significantly change the shape of the scaling function as compared to the fixed coupling case, in particular, leading to a considerable increase in the anomalous dimension and to a slow-down of the evolution with rapidity. We then concentrate on elucidating the differences between the two recent calculations of the running coupling corrections. We explain that the difference is due to an extra contribution to the evolution kernel, referred to as the subtraction term, which arises when running coupling corrections are included. These subtraction terms were neglected in both recent calculations. We evaluate numerically the subtractionmore » terms for both calculations, and demonstrate that when the subtraction terms are added back to the evolution kernels obtained in the two works the resulting dipole amplitudes agree with each other. We then use the complete running coupling kernel including the subtraction term to find the numerical solution of the resulting full nonlinear evolution equation with the running coupling corrections. Again the scaling regime is recovered at very large rapidity with the scaling function unaltered by the subtraction term.« less

Exact soliton solutions and nonlinear modulation instability in spinor Bose-Einstein condensates

We find one-, two-, and three-component solitons of the polar and ferromagnetic (FM) types in the general (nonintegrable) model of a spinor (three-component) model of the Bose-Einstein condensate, based on a system of three nonlinearly coupled Gross-Pitaevskii equations. The stability of the solitons is studied by means of direct simulations and, in a part, analytically, using linearized equations for small perturbations. Global stability of the solitons is considered by means of an energy comparison. As a result, ground-state and metastable soliton states of the FM and polar types are identified. For the special integrable version of the model, we develop the Darboux transformation (DT). As an application of the DT, analytical solutions are obtained that display full nonlinear evolution of the modulational instability of a continuous-wave state seeded by a small spatially periodic perturbation. Additionally, by dint of direct simulations, we demonstrate that solitons of both the polar and FM types, found in the integrable system, are structurally stable; i.e., they are robust under random changes of the relevant nonlinear coefficient in time.

Alfvén wave interaction with inhomogeneous plasmas: acceleration and energy cascade towards small-scales

Abstract. Investigating the process of electron acceleration in auroral regions, we present a study of the temporal evolution of the interaction of Alfvén waves (AW) with a plasma inhomogeneous in a direction transverse to the static magnetic field. This type of inhomogeneity is typical of the density cavities extended along the magnetic field in auroral acceleration regions. We use self-consistent Particle In Cell (PIC) simulations which are able to reproduce the full nonlinear evolution of the electromagnetic waves, as well as the trajectories of ions and electrons in phase space. Physical processes are studied down to the ion Larmor radius and electron skin depth scales. We show that the AW propagation on sharp density gradients leads to the formation of a significant parallel (to the magnetic field) electric field (E-field). It results from an electric charge separation generated on the density gradients by the polarization drift associated with the time varying AW E-field. Its amplitude may reach a few percents of the AW E-field. This parallel component accelerates electrons up to keV energies over a distance of a few hundred Debye lengths, and induces the formation of electron beams. These beams trigger electrostatic plasma instabilities which evolve toward the formation of nonlinear electrostatic structures (identified as electron holes and double layers). When the electrostatic turbulence is fully developed we show that it reduces the further wave/particle exchange. This sequence of mechanisms is analyzed with the program WHAMP, to identify the instabilities at work and wavelet analysis techniques are used to characterize the regime of energy conversions (from electromagnetic to electrostatic structures, from large to small length scales). This study elucidates a possible scenario to account for the particle acceleration and the wave dissipation in inhomogeneous plasmas. It would consist of successive phases of acceleration along the magnetic field, the development of an electrostatic turbulence, the thermalization and the heating of the plasma. Space plasma physics (charged particle motion and acceleration; numerical studies).

Data assimilation with an extended Kalman filter for impact-produced shock-wave dynamics

Model assimilation of data strives to determine optimally the state of an evolving physical system from a limited number of observations. The present study represents the first attempt of applying the extended Kalman filter (EKF) method of data assimilation to shock-wave dynamics induced by a high-speed impact. EKF solves the full nonlinear state evolution and estimates its associated error-covariance matrix in time. The state variables obtained by the blending of past model evolution with currently available data, along with their associated minimized errors (or uncertainties), are then used as initial conditions for further prediction until the next time at which data becomes available. In this study, a one-dimensional (1D) finite-difference code is used along with data measured from a 1D flyer plate experiment. An ensemble simulation suggests that the nonlinearity of the modeled system can be reasonably tracked by EKF. The results demonstrate that the EKF assimilation of a limited amount of pressure data, measured at the middle of the target plate alone, helps track the evolution of all the state variables. The fidelity of EKF is further investigated with numerically generated synthetic data from so-called “identical-twin experiments”, in which the true state is known and various measurement techniques and strategies can be made easily simulated. We find that the EKF method can effectively assimilate the density fields, which are distributed sparsely in time to mimic radiographic data, into the modeled system.

Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization

We present a novel linearization method to calculate accurate eye diagrams and bit error rates (BERs) for arbitrary optical transmission systems and apply it to a dispersion-managed soliton (DMS) system. In this approach, we calculate the full nonlinear evolution using Monte Carlo methods. However, we analyze the data at the receiver assuming that the nonlinear interaction of the noise with itself in an appropriate basis set is negligible during transmission. Noise-noise beating due to the quadratic nonlinearity in the receiver is kept. We apply this approach to a highly nonlinear DMS system, which is a stringent test of our approach. In this case, we cannot simply use a Fourier basis to linearize, but we must first separate the phase and timing jitters. Once that is done, the remaining Fourier amplitudes of the noise obey a multivariate Gaussian distribution, the timing jitter is Gaussian distributed, and the phase jitter obeys a Jacobi-/spl Theta/ distribution, which is the periodic analogue of a Gaussian distribution. We have carefully validated the linearization assumption through extensive Monte Carlo simulations. Once the effect of timing jitter is restored at the receiver, we calculate complete eye diagrams and the probability density functions for the marks and spaces. This new method is far more accurate than the currently accepted approach of simply fitting Gaussian curves to the distributions of the marks and spaces. In addition, we present a deterministic solution alternative to the Monte Carlo method.

Exact and truncated dynamics in nonequilibrium field theory

Nonperturbative dynamics of quantum fields out of equilibrium is often described by the time evolution of a hierarchy of correlation functions, using approximation methods such as Hartree, large ${N}_{f},$ and $n\mathrm{PI}$-effective action techniques. These truncation schemes can be implemented equally well in a classical statistical system, where results can be tested by comparison with the complete nonlinear evolution obtained by numerical methods. For a $(1+1)$-dimensional scalar field we find that the early-time behavior is reproduced qualitatively by the Hartree dynamics. The inclusion of direct scattering improves this to the quantitative level. We show that the emergence of nonthermal temperature profiles at intermediate times can be understood in terms of the fixed points of the evolution equations in the Hartree approximation. The form of the profile depends explicitly on the initial ensemble. While the truncated evolution equations do not seem to be able to get away from the fixed point, the full nonlinear evolution shows thermalization with a (surprisingly) slow relaxation.