We use covariant methods to analyse the nonlinear evolution of self-gravitating, non-relativistic media. The formalism is first applied to imperfect fluids, aiming at the kinematic effects of viscosity, before extended to inhomogeneous magnetised environments. The nonlinear electrodynamic formulae are derived and successively applied to electrically resistive and to highly conductive fluids. By nature, the covariant equations isolate the magnetic effects on the kinematics and the dynamics of the medium, combining mathematical transparency and physical clarity. Employing the Newtonian analogue of the relativistic 1+3 covariant treatment, also facilitates the direct comparison with the earlier relativistic studies and helps to identify the differences in an unambiguous way. The purpose of this work is to set the framework and take a first step towards the detailed analytical study of complex nonlinear systems, like non-relativistic astrophysical plasmas and collapsing protogalactic clouds.
We study and develop constraint preserving boundary conditions for the Newtonian magnetohydrodynamic equations and analyze the behavior of the numerical solution upon considering different possible options.
We present the generalization of a recently introduced modified gravitational potential for self-gravitating fluids. The use of this potential allows for an accurate approximation of general relativistic effects in an otherwise Newtonian hydrodynamics code also in cases of rapid rotation. We test this approach in numerical simulations of astrophysical scenarios related to compact stars, like supernova core collapse with both a simplified and detailed microphysical description of matter, and rotating neutron stars in equilibrium. We assess the quality of the new potential, and demonstrate that …
We investigate the possibility to approximate relativistic effects in hydrodynamical simulations of stellar core collapse and post-bounce evolution by using a modified gravitational potential in an otherwise standard Newtonian hydrodynamic code. Different modifications of the effective relativistic potential introduced by Rampp & Janka (2002) are discussed. Corresponding hydrostatic solutions are compared with solutions of the TOV equations, and hydrodynamic simulations with two different codes are compared with fully relativistic results. One code is applied for one- and two-dimensional calcu…
Using the (3+1) formalism in general relativity, we perform the post-Newtonian(PN) approximation to clarify what sort of gauge condition is suitable for numerical analysis of coalescing compact binary neutron stars and gravitational waves from them. We adopt a kind of transverse gauge condition to determine the shift vector. On the other hand, for determination of the time slice, we adopt three slice conditions(conformal slice, maximal slice and harmonic slice) and discuss their properties. Using these conditions, the PN hydrodynamic equations are obtained up through the 2.5PN order including …
We present a detailed study of the hydrodynamical interactions in a Newtonian black hole-neutron star binary during the last stages of inspiral. We consider close binaries which are tidally locked, use a stiff equation of state (with an adiabatic index Gamma=3) throughout, and explore the effect of different initial mass ratios on the evolution of the system. We calculate the gravitational radiation signal in the quadrupole approximation. Our calculations are carried out using a Smooth Particle Hydrodynamics (SPH) code.
We examine the local conditions for radiative damping and driving of short wavelength, propagating hydrodynamic and magnetohydrodynamic (MHD) waves in static, optically thick, stratified equilibria. We show that so-called strange modes in stellar oscillation theory and magnetic photon bubbles are intimately related and are both fundamentally driven by the background radiation flux acting on compressible waves. We identify the necessary criteria for unstable driving of these waves, and show that this driving can exist in both gas and radiation pressure dominated media, as well as pure Thomson s…
Relativistic cosmological perturbation analyses can be made based on several different fundamental gauge conditions. In the pressureless limit the variables in certain gauge conditions show the correct Newtonian behaviors. Considering the general curvature ($K$) and the cosmological constant ($Λ$) in the background medium, the perturbed density in the comoving gauge, and the perturbed velocity and the perturbed potential in the zero-shear gauge show the same behavior as the Newtonian ones in general scales. In the first part, we elaborate these Newtonian correspondences. In the second part, us…
The post-Newtonian (PN) approximation, based on the assumptions of weak gravitational fields and slow motions, provides a way to estimate general relativistic effects in the fully nonlinear evolution stage of the large-scale cosmic structures. We extend Chandrasekhar's first order PN (1PN) hydrodynamics based on the Minkowski background to the Robertson-Walker background. We assume the presence of Friedmann's cosmological spacetime as a background. In the background we include the three-space curvature, the cosmological constant and general pressure. In the Newtonian order and 1PN order we inc…
We demonstrate, both at the Newtonian and (general) relativistic levels, the existence of a generalization of Kelvin's circulation theorem (for pure fluids) which is applicable to perfect magnetohydrodynamics. The argument is based on the least action principle for magnetohydrodynamic flow. Examples of the new conservation law are furnished. The new theorem should be helpful in identifying new kinds of vortex phenomena distinct from magnetic ropes or fluid vortices.
We describe a new method to include magnetic fields into smooth particle hydrodynamics. The derivation of the self-gravitating hydrodynamics equations from a variational principle is discussed in some detail. The non-dissipative magnetic field evolution is instantiated by advecting so-called Euler potentials. This approach enforces the crucial $\nabla\cdot\vec{B}=0$-constraint by construction. These recent developments are implemented in our three-dimensional, self-gravitating magnetohydrodynamics code MAGMA. A suite of tests is presented that demonstrates the superiority of this new approach …
Turbulence in an incompressible fluid with and without a magnetic field as well as moderately compressible MHD turbulence are compared. For the magnetohydrodynamic (MHD) models the probability distribution functions of the velocity components perpendicular to the external magnetic field are like the incompressible hydrodynamic (HD) model while those parallel to the field have a smaller range of velocities. The probability distribution functions of the transverse velocity increments for the MHD models decline slower than the incompressible HD model. The similarity of incompressible HD and both …
This paper presents an initial survey of the properties of accretion flows in the Kerr metric from three-dimensional, general relativistic magnetohydrodynamic simulations of accretion tori. We consider three fiducial models of tori around rotating, both prograde and retrograde, and nonrotating black holes; these three fiducial models are also contrasted with axisymmetric simulations and a pseudo-Newtonian simulation with equivalent initial conditions to delineate the limitations of these approximations.
We performed relativistic magnetohydrodynamic simulations of the hydrodynamic boosting mechanism for relativistic jets explored by Aloy & Rezzolla (2006) using the RAISHIN code. Simulation results show that the presence of a magnetic field changes the properties of the shock interface between the tenuous, overpressured jet ($V^z_j$) flowing tangentially to a dense external medium. Magnetic fields can lead to more efficient acceleration of the jet, in comparison to the pure-hydrodynamic case. A ``poloidal'' magnetic field ($B^z$), tangent to the interface and parallel to the jet flow, produces …
In this paper we examine various issues closely related to the ongoing discussion on the nature of the Blandford-Znajek mechanism of extraction of rotational energy of black holes. In particular, we show that switch-on and switch-off shocks are allowed by the shock equations of relativistic MHD and have similar properties to their Newtonian counterparts. Just like in Newtonian MHD they are limits of fast and slow shock solutions and as such they may be classified as weakly evolutionary shocks. The analysis of Punsly's MHD waveguide problem shows that its solution cannot have the form of a trav…
We apply our recently developed code for spherically symmetric, fully general relativistic (GR) Lagrangian hydrodynamics and multigroup flux-limited diffusion neutrino transport to examine the effects of GR on the hydrodynamics and transport during collapse, bounce, and the critical shock reheating phase of core collapse supernovae. Comparisons of models computed with GR versus Newtonian hydrodynamics show that collapse to bounce takes slightly less time in the GR limit, and that the shock propagates slightly farther out in radius before receding. After a secondary quasistatic rise in the shoc…
We performed a series of high-resolution (up to 1024^3) direct numerical simulations of hydro and MHD strong turbulence. We found that for simulations with normal viscosity the slopes for spectra of MHD are similar, although slightly more shallower than for hydro simulations. However, for simulations with hyperviscosity the slopes were very different, for instance, the slopes for hydro simulations showed pronounced and well-defined bottleneck effect, while the MHD slopes were relatively much less affected. We believe that this is indicative of MHD strong turbulence being less local than Kolmog…
We present the results of three-dimensional hydrodynamical simulations of the final stages of inspiral in a black hole-neutron star binary, when the separation is comparable to the stellar radius. We use a Newtonian Smooth Particle Hydrodynamics (SPH) code to model the evolution of the system, and take the neutron star to be a polytrope with a soft (adiabatic index G=2 and G=5/3) equation of state and the black hole to be a Newtonian point mass. The only non-Newtonian effect we include is a gravitational radiation back reaction force, computed in the quadrupole approximation for point masses. …
We present a numerical study of the hydrodynamics in the final stages of inspiral of a black hole-neutron star binary, when the binary separation becomes comparable to the stellar radius. We use a Newtonian three-dimensional Smooth Particle Hydrodynamics (SPH) code, and model the neutron star with a soft (adiabatic index Gamma=5/3) polytropic equation of state and the black hole as a Newtonian point mass which accretes matter via an absorbing boundary at the Schwarzschild radius. Our initial conditions correspond to tidally locked binaries in equilibrium, and we have explored configurations wi…
Two pseudo-Newtonian potentials, which approximate the angular and epicyclic frequencies of the relativistic accretion disk around rotating (and counter rotating) compact objects, are presented. One of them, the Logarithmically Modified Potential, is a better approximation for the frequencies while the other, the Second-order Expanded potential, also reproduces the specific energy for circular orbits in close agreement with the General Relativistic values. These potentials may be included in time dependent hydrodynamical simulations to study the temporal behavior of such accretion disks.
In Newtonian and relativistic hydrodynamics the Riemann problem consists of calculating the evolution of a fluid which is initially characterized by two states having different values of uniform rest-mass density, pressure and velocity. When the fluid is allowed to relax, one of three possible wave-patterns is produced, corresponding to the propagation in opposite directions of two nonlinear hydrodynamical waves. New effects emerge in a special relativistic Riemann problem when velocities tangential to the initial discontinuity surface are present. We show that a smooth transition from one wav…
We introduce an adaptation of the well known Tree+SPH numerical scheme to Post Newtonian (PN) hydrodynamics and gravity. Our code solves the (0+1+2.5)PN equations. These equations include Newtonian hydrodynamics and gravity (0PN), the first order relativistic corrections to those (1PN) and the lowest order gravitational radiation terms (2.5PN). We test various aspects of our code using analytically solvable test problems. We then proceed to study the 1PN effects on binary neutron star coalescence by comparing calculations with and without the 1PN terms. We find that the effect of the 1PN terms…
We present a new, completely Lagrangian magnetohydrodynamics code that is based on the SPH method. The equations of self-gravitating hydrodynamics are derived self-consistently from a Lagrangian and account for variable smoothing length (``grad-h''-) terms in both the hydrodynamic and the gravitational acceleration equations. The evolution of the magnetic field is formulated in terms of so-called Euler potentials which are advected with the fluid and thus guarantee the MHD flux-freezing condition. This formulation is equivalent to a vector potential approach and therefore fulfills the $\vec{\n…
We present a hydrodynamical study of the final stages of inspiral in a black hole-neutron star (NS) binary. We use a Newtonian 3D SPH code, and model the NS with a stiff (index G=3 and G=2.5) polytropic equation of state and the black hole as a Newtonian point mass. Our initial conditions correspond to irrotational binaries in equilibrium (approximating the NS as a compressible ellipsoid), and we have explored configurations with different initial mass ratios, 0.2< q=M_ns/M_bh<0.5. The dynamical evolution is followed for ~23ms. We include gravitational radiation losses in the quadrupole approx…
We have developed a new massively-parallel radiation-hydrodynamics code (Cosmos) for Newtonian and relativistic astrophysical problems that also includes radiative cooling, self-gravity, and non-equilibrium, multi-species chemistry. Several numerical methods are implemented for the hydrodynamics, including options for both internal and total energy conserving schemes. Radiation is treated using flux-limited diffusion. The chemistry incorporates 27 reactions, including both collisional and radiative processes for atomic hydrogen and helium gases, and molecular hydrogen chains. In this paper we …
The properties of linear Alfvén, slow, and fast magnetoacoustic waves for uniform plasmas in relativistic magnetohydrodynamics (MHD) are discussed, augmenting the well-known expressions for their phase speeds with knowledge on the group speed. A 3+1 formalism is purposely adopted to make direct comparison with the Newtonian MHD limits easier and to stress the graphical representation of their anisotropic linear wave properties using the phase and group speed diagrams. By drawing these for both the fluid rest frame and for a laboratory Lorentzian frame which sees the plasma move with a three-ve…
We investigate through high resolution 3D simulations the nonlinear evolution of compressible magnetohydrodynamic flows subject to the Kelvin-Helmholtz instability. We confirm in 3D flows the conclusion from our 2D work that even apparently weak magnetic fields embedded in Kelvin-Helmholtz unstable plasma flows can be fundamentally important to nonlinear evolution of the instability. In fact, that statement is strengthened in 3D by this work, because it shows how field line bundles can be stretched and twisted in 3D as the quasi-2D Cat's Eye vortex forms out of the hydrodynamical motions. In o…
