Adam Frank

Numerical magnetohydrodynamics in astrophysics: Algorithm and tests for multidimensional flow

We describe a numerical code to solve the equations for ideal magnetohydrodynamics (MHD). It is based on an explicit finite difference scheme on an Eulerian grid, called the Total Variation Diminishing (TVD) scheme, which is a second-order-accurate extension of the Roe-type upwind scheme. We also describe a nonlinear Riemann solver for ideal MHD, which includes rarefactions as well as shocks and produces exact solutions for two-dimensional magnetic field structures as well as for the three-dimensional ones. The numerical code and the Riemann solver have been used to test each other. Extensive tests encompassing all the possible ideal MHD structures with planar symmetries (ie ~one-dimensional flows) are presented. These include those for which the field structure is two-dimensional ({it i.e.}, those flows often called ``

The magnetohydrodynamic kelvin-helmholtz instability: A two-dimensional numerical study

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The MHD Kelvin-Helmholtz Instability. II. The Roles of Weak and Oblique Fields in Planar Flows

dimensional) as well as those for which the magnetic field plane rotates ({it i.e.,}, those flows often called ``

A Mechanism for the Production of Jets and Ansae in Planetary Nebulae

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Precessing Jets and Point-Symmetric Nebulae

dimensional). Results indicate that the code can resolve strong fast, slow, and magnetosonic shocks within 2-4 cells while more cells are required if shocks become weak. With proper stiffening, rotational discontinuities are resolved within 3-5 cells. Contact discontinuities are also resolved within 3-5 cells with stiffening and 6-8 cells without stiffening, while the stiffening on contact discontinuities in some cases generates numerical oscillations. Tangential discontinuities spread over more than 10 cells. Our tests confirm that slow compound structures with two-dimensional magnetic field are composed of intermediate shocks (so called ``2-4

Hydrodynamical Models of Outflow Collimation in Young Stellar Objects

Using a new numerical code we have carried out two-dimensional simulations of the nonlinear evolution of unstable sheared magnetohydrodynamic flows. We considered two cases: a strong magnetic field (Alfven Mach number, M_a = 2.5) and a weak field (M_a =5). Each flow rapidly evolves until it reaches a nearly steady condition, which is fundamentally different from the analogous gasdynamic state. Both MHD flows relax to a stable, laminar flow on timescales less than or of the order of 15 linear growth times, measured from saturation of the instability. That timescale is several orders of magnitude less than the nominal dissipation time for these simulated flows, so this condition represents an quasi-steady relaxed state. The strong magnetic field case reaches saturation as magnetic tension in the displaced flow boundary becomes sufficient to stabilize it. That flow then relaxes in a straightforward way to the steady, laminar flow condition. The weak magnetic field case, on the other hand, begins development of the vortex expected for gasdynamics, but that vortex is destroyed by magnetic stresses that locally become strong. Magnetic topologies lead to reconnection and dynamical alignment between magnetic and velocity fields. Together these processes produce a sequence of intermittent vortices and subsequent relaxation to a nearly laminar flow condition in which the magnetic cross helicity is nearly maximized. Remaining irregularities consist of a pair of flux tubes straddling the shear layer. Fluctuations within those features are closely aligned, representing Alfven waves propagating locally downstream.

The Timescale Correlation Method: Distances to Planetary Nebulae with Halos

We have carried out high-resolution MHD simulations of the nonlinear evolution of Kelvin-Helmholtz unstable flows in 2½ dimensions. The modeled flows and fields were initially uniform except for a thin shear layer with a hyperbolic tangent velocity profile and a small, normal mode perturbation. These simulations extend work by Frank et al. and Malagoli, Bodo, & Rosner. They consider periodic sections of flows containing magnetic fields parallel to the shear layer, but projecting over a full range of angles with respect to the flow vectors. They are intended as preparation for fully three-dimensional calculations and to address two specific questions raised in earlier work: (1) What role, if any, does the orientation of the field play in nonlinear evolution of the MHD Kelvin-Helmholtz instability in 2½ dimensions? (2) Given that the field is too weak to stabilize against a linear perturbation of the flow, how does the nonlinear evolution of the instability depend on strength of the field? The magnetic field component in the third direction contributes only through minor pressure contributions, so the flows are essentially two-dimensional. In Frank et al. we found that fields too weak to stabilize a linear perturbation may still be able to alter fundamentally the flow so that it evolves from the classical Cats Eye vortex expected in gasdynamics into a marginally stable, broad laminar shear layer. In that process the magnetic field plays the role of a catalyst, briefly storing energy and then returning it to the plasma during reconnection events that lead to dynamical alignment between magnetic field and flow vectors. In our new work we identify another transformation in the flow evolution for fields below a critical strength. That we found to be ~10% of the critical field needed for linear stabilization in the cases we studied. In this very weak field regime, the role of the magnetic field is to enhance the rate of energy dissipation within and around the Cats Eye vortex, not to disrupt it. The presence of even a very weak field can add substantially to the rate at which flow kinetic energy is dissipated. In all of the cases we studied magnetic field amplification by stretching in the vortex is limited by tearing mode, fast reconnection events that isolate and then destroy magnetic flux islands within the vortex and relax the fields outside the vortex. If the magnetic tension developed prior to reconnection is comparable to Reynolds stresses in the flow, that flow is reorganized during reconnection. Otherwise, the primary influence on the plasma is generation of entropy. The effective expulsion of flux from the vortex is very similar to that shown by Weiss for passive fields in idealized vortices with large magnetic Reynolds numbers. We demonstrated that this expulsion cannot be interpreted as a direct consequence of steady, resistive diffusion, but must be seen as a consequence of unsteady fast reconnection.

PRECESSING JETS AND MOLECULAR OUTFLOWS : A 3D NUMERICAL STUDY

The general properties of elliptical planetary nebulae have been very successfully explained by a class of hydrodynamic models known as interacting stellar winds (ISW). PNs also contain a variety of microstructures (Anase, ”FLIERs”) whose origins have remained puzzling. We (Frank, Balick, & Livio 1996) have shown that such flows will naturally occur in the ISW model in momentum-conserving PPN bubbles, and that they can naturally account for the formation of jets and ansae in a way consistent with extant observations.

Time-dependent simulation of oblique MHD cosmic-ray shocks using the two-fluid model

We present a model for the formation of point-symmetric nebulae that relies on the existence of a precessing jet interacting with the interstellar medium (ISM). Using three-dimensional numerical simulations, we investigate the basic gasdynamics inherent to the model. Through synthetic observations of our simulations we show that episodic precessing jets can reproduce the gross morphological structure of point-symmetric nebulae, i.e., a string of discrete clumps in an S-shaped intensity distribution. We also find that the bow shocks of the individual jet segments can merge into a single shock structure that envelops the entire complex of segments. The development of this enveloping shock allows the model to embrace nebulae consisting of discrete point-symmetric clumps as well as those bipolar objects that show nonuniform brightness distributions on their opposing lobes that are point symmetric through the nucleus. By demonstrating that these bipolar planetary nebulae can form from the same mechanism which produces the discrete point-symmetric nebulae, we can include them in the category of point-symmetric objects, thereby increasing their fractional occurrence in planetary nebulae by 75%.

The Evolution and Efficiency of Oblique MHD Cosmic-Ray Shocks: Two-Fluid Simulations

In this paper we explore the physics of time-dependent hydrodynamic collimation of jets from young stellar objects (YSOs). Using parameters appropriate to YSOs, we have carried out high-resolution hydrodynamic simulations modeling the interaction of a central wind with an environment characterized by a toroidal density distribution which has a moderate opening angle of θρ 90°. The results show that for all but low values of the equator-to-pole density contrast the wind/environment interaction produces strongly collimated supersonic jets. The jet is composed of shocked wind gas. Using analytical models of wind-blown bubble evolution, we show that the scenario studied here should be applicable to YSOs and can, in principle, initiate collimation on the correct scales (R 100 AU). Comparison of our simulations with analytical models demonstrates that the evolution seen in the simulations is a mix of wind-blown bubble and jet dynamics. The simulations reveal a number of time-dependent nonlinear features not anticipated in previous analytical studies. These include: a prolate wind shock; a chimney of cold swept-up ambient material dragged into the bubble cavity; a plug of dense material between the jet and bow shocks. We find that the collimation of the jet occurs through both de Laval nozzles and focusing of the wind via the prolate wind shock. Using an analytical model for shock focusing we demonstrate that a prolate wind shock can, by itself, produce highly collimated supersonic jets.Animations from these simulations are available over the internet at http://www.msi.umn.edu/Projects/twj/jetcol.html.