Philip E. Hardee

Three-dimensional Relativistic Magnetohydrodynamic Simulations of Magnetized Spine-Sheath Relativistic Jets

Numerical simulations of weakly and strongly magnetized relativistic jets embedded in a weakly magnetized and strongly magnetized stationary or weakly relativistic (v = c/2) sheath have been performed. A magnetic field parallel to the flow is used in these simulations performed by the new general relativistic magnetohydrodynamic numerical code RAISHIN used in its relativistic magnetohydrodynamic (RMHD) configuration. In the numerical simulations, the Lorentz factor ? = 2.5 jet is precessed to break the initial equilibrium configuration. In the simulations, sound speeds are c/ in the weakly magnetized simulations and 0.3c in the strongly magnetized simulations. The Alfv?n wave speed is 0.07c in the weakly magnetized simulations and 0.56c in the strongly magnetized simulations. The results of the numerical simulations are compared to theoretical predictions from a normal mode analysis of the linearized RMHD equations capable of describing a uniform axially magnetized cylindrical relativistic jet embedded in a uniform axially magnetized relativistically moving sheath. The theoretical dispersion relation allows investigation of effects associated with maximum possible sound speeds, Alfv?n wave speeds near light speed, and relativistic sheath speeds. The prediction of increased stability of the weakly magnetized system resulting from c/2 sheath speeds and the stabilization of the strongly magnetized system resulting from c/2 sheath speeds is verified by the numerical simulation results.

Jet Stability and the Generation of Superluminal and Stationary Components

We present a numerical simulation of the response of an expanding relativistic jet to the ejection of a superluminal component. The simulation has been performed with a relativistic time-dependent hydrodynamical code from which simulated radio maps are computed by integrating the transfer equations for synchrotron radiation. The interaction of the superluminal component with the underlying jet results in the formation of multiple conical shocks behind the main perturbation. These trailing components can be easily distinguished because they appear to be released from the primary superluminal component instead of being ejected from the core. Their oblique nature should also result in distinct polarization properties. Those appearing closer to the core show small apparent motions and a very slow secular decrease in brightness and could be identified as stationary components. Those appearing farther downstream are weaker and can reach superluminal apparent motions. The existence of these trailing components indicates that not all observed components necessarily represent major perturbations at the jet inlet; rather, multiple emission components can be generated by a single disturbance in the jet. While the superluminal component associated with the primary perturbation exhibits a rather stable pattern speed, trailing components have velocities that increase with distance from the core but move at less than the jet speed. The trailing components exhibit motion and structure consistent with the triggering of pinch modes by the superluminal component. The increase in velocity of the trailing components is an indirect consequence of the acceleration of the expanding fluid, which is assumed to be relativistically hot; if observed, such accelerations would therefore favor an electron-positron (as opposed to proton rest mass) dominated jet.

On Three-dimensional Structures in Relativistic Hydrodynamic Jets

The appearance of wavelike helical structures on steady relativistic jets is studied using a normal mode analysis of the linearized —uid equations. Helical structures produced by the normal modes scale relative to the resonant (most unstable) wavelength and not with the absolute wavelength. The resonant wave- length of the normal modes can be less than the jet radius even on highly relativistic jets. High-pressure regions helically twisted around the jet beam may be con—ned close to the jet surface, penetrate deeply into the jet interior, or be con—ned to the jet interior. The high-pressure regions range from thin and ribbon-like to thick and tubelike depending on the mode and wavelength. The wave speeds can be sig- ni—cantly diUerent at diUerent wavelengths but are less than the —ow speed. The highest wave speed for the jets studied has a Lorentz factor somewhat more than half that of the underlying —ow speed. A maximum pressure —uctuation criterion found through comparison between theory and a set of relativistic axisymmetric jet simulations is applied to estimate the maximum amplitudes of the helical, elliptical, and triangular normal modes. Transverse velocity —uctuations for these asymmetric modes are up to twice the amplitude of those associated with the axisymmetric pinch mode. The maximum ampli- tude of jet distortions and the accompanying velocity —uctuations at, for example, the resonant wave- length decreases as the Lorentz factor increases. Long-wavelength helical surface mode and shorter wavelength helical —rst body mode generated structures should be the most signi—cant. Emission from high-pressure regions as they twist around the jet beam can vary signi—cantly as a result of angular variation in the —ow direction associated with normal mode structures if they are viewed at about the beaming angle h \ 1/c. Variation in the Doppler boost factor can lead to brightness asymmetries by factors up to 6 as long-wavelength helical structure produced by the helical surface mode winds around the jet. Higher order surface modes and —rst body modes produce less variation. Angular variation in the —ow direction associated with the helical mode appears consistent with precess- ing jet models that have been proposed to explain the variability in 3C 273 and BL Lac object AO 0235)164. In particular, cyclic angular variation in the —ow direction produced by the normal modes could produce the activity seen in BL Lac object OJ 287. Jet precession provides a mechanism for trig- gering the helical modes on multiple length scales, e.g., the galactic superluminal GRO J1655(40. Subject headings: BL Lacertae objects: individual (OJ 287) ¨ galaxies: activegalaxies: jets ¨ hydrodynamicsinstabilitiesrelativity

A Comparison of the Morphology and Stability of Relativistic and Nonrelativistic Jets

We compare results from a relativistic and a nonrelativistic set of two-dimensional axisymmetric jet simulations. For a set of five relativistic simulations that either increase the Lorentz factor or decrease the adiabatic index, we compute nonrelativistic simulations with equally useful power or thrust. We examine these simulations for morphological and dynamical differences, focusing on the velocity field, the width of the cocoon, the age of the jets, and the internal structure of the jet itself. The primary result of these comparisons is that the velocity field of nonrelativistic jet simulations cannot be scaled up to give the spatial distribution of Lorentz factors seen in relativistic simulations. Since the local Lorentz factor plays a major role in determining the total intensity for parsec-scale extragalactic jets, this suggests that a nonrelativistic simulation cannot yield the proper intensity distribution for a relativistic jet. Another general result is that each relativistic jet and its nonrelativistic equivalents have similar ages (in dynamical time units, ≡R/aa, where R is the initial radius of a cylindrical jet and aa is the sound speed in the ambient medium). Also, jets with a larger Lorentz factor have a smaller cocoon size. In addition to these comparisons, we have completed four new relativistic simulations to investigate the effect of varying thermal pressure on relativistic jets. The simulations confirm that faster (larger Lorentz factor) and colder jets are more stable, with smaller amplitude and longer wavelength internal variations. However, an exception to this occurs for the hottest jets, which appear the most stable. The apparent stability of these jets does not follow from linear normal mode analysis, which suggests that there are available growing Kelvin-Helmholtz modes. However, these modes are not excited because of a lack of perturbations able to couple to them. As an example of how these simulations can be applied to the interpretation of observations, we use our results to estimate some parameters of Cygnus A. Although none of these estimates alone can determine if the jets in Cyg A are relativistic or nonrelativistic, estimates for the age and the jet to ambient density ratio confirm values for these parameters estimated by other means.

THREE-DIMENSIONAL RELATIVISTIC MAGNETOHYDRODYNAMIC SIMULATIONS OF CURRENT-DRIVEN INSTABILITY. I. INSTABILITY OF A STATIC COLUMN

We have investigated the development of current-driven (CD) kink instability through three-dimensional relativistic magnetohydrodynamic simulations. A static force-free equilibrium helical magnetic configuration is considered in order to study the influence of the initial configuration on the linear and nonlinear evolution of the instability. We found that the initial configuration is strongly distorted but not disrupted by the kink instability. The instability develops as predicted by linear theory. In the nonlinear regime, the kink amplitude continues to increase up to the terminal simulation time, albeit at different rates, for all but one simulation. The growth rate and nonlinear evolution of the CD kink instability depend moderately on the density profile and strongly on the magnetic pitch profile. The growth rate of the kink mode is reduced in the linear regime by an increase in the magnetic pitch with radius and reaches the nonlinear regime at a later time than the case with constant helical pitch. On the other hand, the growth rate of the kink mode is increased in the linear regime by a decrease in the magnetic pitch with radius and reaches the nonlinear regime sooner than the case with constant magnetic pitch. Kink amplitude growth in the nonlinear regime for decreasing magnetic pitch leads to a slender helically twisted column wrapped by magnetic field. On the other hand, kink amplitude growth in the nonlinear regime nearly ceases for increasing magnetic pitch.

Magnetohydrodynamic Models of Axisymmetric Protostellar Jets

We present the results of a series of axisymmetric time-dependent magnetohydrodynamic (MHD) simulations of the propagation of cooling, overdense jets, motivated by the properties of outflows associated with young stellar objects. A variety of initial field strengths and configurations are explored for both steady and time-variable (pulsed) jets. Even apparently weak magnetic fields with strengths B < 60 micro-G in the pre-shocked jet beam can have a significant effect on the dynamics, for example by altering the density, width, and fragmentation of thin shells formed by cooling gas. A linear analysis predicts that axisymmetric pinch modes of the MHD Kelvin-Helmholtz instability should grow only slowly for the highly supermagnetosonic jets studied here; we find no evidence for them in our simulations. Some of our models appear unstable to current-driven pinch modes, however the resulting pressure and density variations induced in the jet beam are not large, making this mechanism an unlikely source of emission knots in the jet beam. In the case of pulsed jets, radial hoop stresses confine shocked jet material in the pulses to the axis, resulting in a higher density in the pulses in comparison to purely hydrodynamic models.

The Stability of Radiatively Cooling Jets. II. Nonlinear Evolution

We use two-dimensional time-dependent hydrodynamical simulations to follow the growth of the Kelvin-Helmholtz (K-H) instability in cooling jets into the nonlinear regime. We focus primarily on asymmetric modes that give rise to transverse displacements of the jet beam. A variety of Mach numbers and two different cooling curves are studied. The growth rates of waves in the linear regime measured from the numerical simulations are in excellent agreement with the predictions of the linear stability analysis presented in the first paper in this series. In the nonlinear regime, the simulations show that asymmetric modes of the K-H instability can affect the structure and evolution of cooling jets in a number of ways. We find that jets in which the growth rate of the sinusoidal surface wave has a maximum at a so-called resonant frequency can be dominated by large-amplitude sinusoidal oscillations near this frequency. Eventually, growth of this wave can disrupt the jet. On the other hand, nonlinear body waves tend to produce low-amplitude wiggles in the shape of the jet but can result in strong shocks in the jet beam. In cooling jets, these shocks can produce dense knots and filaments of cooling gas within the jet. Ripples in the surface of the jet beam caused by both surface and body waves generate oblique shock spurs driven into the ambient gas. Our simulations show these shock spurs can accelerate ambient gas at large distances from the jet beam to low velocities, which represents a new mechanism by which low-velocity bipolar outflows may be driven by high-velocity jets. Rapid entrainment and acceleration of ambient gas may also occur if the jet is disrupted. For parameters typical of protostellar jets, the frequency at which K-H growth is a maximum (or highest frequency to which the entire jet can respond dynamically) will be associated with perturbations with a period of ~200 yr. Higher frequency (shorter period) perturbations excite waves associated with body modes that produce internal shocks and only small-amplitude wiggles within the jet. The fact that most observed systems show no evidence for large-amplitude sinusoidal oscillation leading to disruption is indicative that the perturbation frequencies are generally large, consistent with the suggestion that protostellar jets arise from the inner regions (r < 1 AU) of accretion disks.

Dynamics and Structure of Three-dimensional Poloidally Magnetized Supermagnetosonic Jets

A set of three-dimensional magnetohydrodynamical simulations of supermagnetosonic magnetized jets has been performed. The jets contain an equipartition primarily poloidal magnetic field, and the effect of jet density on jet dynamics and structure is evaluated. The jet is precessed at the origin to break the symmetry and to excite Kelvin-Helmholtz-unstable helical modes. In the linear limit, observed structures are similar in all simulations and can be produced by structures predicted to arise as a result of instability. The amplitude of various unstable modes is evaluated. Most unstable modes do not reach the maximum amplitudes estimated from the linear theory by computing displacement surfaces associated with the modes. Surprisingly, even these large-amplitude distortions are fitted reasonably well by displacement surfaces computed from the linear theory. Large-amplitude helical and elliptical distortions lead to significant differences in the nonlinear development of the jets as a function of the jet density. Jets less dense than the surrounding medium entrain material, lose energy through shock heating, and slow down relatively rapidly once large-amplitude distortions develop as a result of instability. Jets more dense than the surrounding medium lose much less energy as they entrain and accelerate the surrounding medium. The dense jet maintains a high-speed spine that exhibits large-amplitude helical twisting and elliptical distortion over considerable distance without disruption of internal jet structures as happens for the less dense jets. This dense high-speed spine is surrounded by a less dense sheath consisting of slower moving jet fluid and magnetic field mixed with the external medium. Simulated synchrotron intensity and fractional polarization images from these calculations provide a considerably improved connection between simulation results and jet observations than do images made using the fluid variables alone. Intensity structure in the dense jet simulation appears remarkably similar to structure observed in the Cygnus A jet. These simulations suggest that the extended jets in high-power radio sources propagate to such large distances without disruption by entrainment because they are surrounded by a lobe or cocoon whose density is less than the jet density.

Stability Properties of Strongly Magnetized Spine-Sheath Relativistic Jets

We derive linearized relativistic magnetohydrodynamic (RMHD) equations describing a uniform axially magnetized cylindrical relativistic jet spine embedded in a uniform axially magnetized relativistically moving sheath. The displacement current is retained in the equations, so that effects associated with Alfven wave propagation near light speed can be studied. A dispersion relation for the normal modes is obtained. Analytical solutions for the normal modes in the low- and high-frequency limits are found, and a general stability condition is determined. A trans-Alfvenic and even a super-Alfvenic relativistic jet spine can be stable to velocity shear-driven Kelvin-Helmholtz modes. The resonance condition for maximum growth of the normal modes is obtained in the kinetically and magnetically dominated regimes. Numerical solution of the dispersion relation verifies the analytical solutions and is used to study the regime of high sound and Alfven speeds.

Effects of Magnetic Fields on Mass Entrainment of Supermagnetosonic Jets

We have performed three-dimensional MHD equilibrium jet simulations that have been designed to excite the Kelvin-Helmholtz (K-H) instability and allow us to investigate the spatial growth of the mixing layer between a magnetized jet and an initially unmagnetized external medium. These simulations differ in the magnetic field strength and orientation, in the jet-to-ambient density ratio, and in the amplitude of the initial (velocity) perturbation. We calculate as a direct measure of mass entrainment the mass of magnetized or high axial velocity material and compare the growth of entrained mass with dynamical and potentially observable properties of a jet. An equipartition toroidal field (whose magnetic pressure is approximately equal to the jets thermal pressure) can inhibit the growth of the K-H instability and reduce mass entrainment significantly. An equipartition axial field has a slight stabilizing effect and reduces mass entrainment relative to a weak field for comparable magnetosonic Mach number. As the jet and ambient medium are mixed, the width of simulated total synchrotron intensity images increases and the fractional polarization of the jet decreases. In these simulations the jet evolves to a fast-moving magnetized spine surrounded by a slower moving, less magnetized sheath. These centralized spines are more easily disrupted in jets less dense than the surrounding medium. As a consequence of their greater instability, simulations with axial magnetic fields are more likely than simulations with toroidal magnetic fields to filament into axial (matter) streams. In both the width of the simulated intensity images and in the mass of magnetized material, we see evidence for a linear growth stage, a nonlinear growth stage, and a final saturation stage. Results from a normal-mode analysis suggest that the initial linear stage of mixing coincides with growth of the K-H-unstable normal surface modes and a spatial progression from higher to lower order modes. The nonlinear stage coincides with large-amplitude elliptical distortion to the jet cross section and filamentation of the jet. A subset of simulations was run at lower (9 zones/jet radius) and higher (25 zones/jet radius) spatial resolutions than the typical moderate spatial resolution of 15 zones/jet radius. We find that the instability saturates at approximately the same amount of entrained mass (relative to the jet mass) in the moderate- and high-resolution cases, although the transition between the linear and nonlinear stages is closer to the inlet in the higher resolution simulations.