Numerical simulations of stellar winds: polytropic models

We discuss steady-state transonic outflows obtained by direct numerical solution of the hydrodynamic and magnetohydrodynamic equations. We make use of the Versatile Advection Code, a software package for solving systems of (hyperbolic) partial differential equations. We proceed stepwise from a spherically symmetric, isothermal, unmagnetized, non-rotating Parker wind to arrive at axisymmetric, polytropic, magnetized, rotating models. These represent 2D generalisations of the analytical 1D Weber-Davis wind solution, which we obtain in the process. Axisymmetric wind solutions containing both a `wind' and a `dead' zone are presented. Since we are solving for steady-state solutions, we efficiently exploit fully implicit time stepping. The method allows us to model thermally and/or magneto-centrifugally driven stellar outflows. We particularly emphasize the boundary conditions imposed at the stellar surface. For these axisymmetric, steady-state solutions, we can use the knowledge of the flux functions to verify the physical correctness of the numerical solutions.Comment: 11 pages, 6 figures, accepted for Astron. Astrophys. 342, to appear 199

Unstable magnetohydrodynamical continuous spectrum of accretion disks. A new route to magnetohydrodynamical turbulence in accretion disks

We present a detailed study of localised magnetohydrodynamical (MHD) instabilities occuring in two--dimensional magnetized accretion disks. We model axisymmetric MHD disk tori, and solve the equations governing a two--dimensional magnetized accretion disk equilibrium and linear wave modes about this equilibrium. We show the existence of novel MHD instabilities in these two--dimensional equilibria which do not occur in an accretion disk in the cylindrical limit. The disk equilibria are numerically computed by the FINESSE code. The stability of accretion disks is investigated analytically as well as numerically. We use the PHOENIX code to compute all the waves and instabilities accessible to the computed disk equilibrium. We concentrate on strongly magnetized disks and sub--Keplerian rotation in a large part of the disk. These disk equilibria show that the thermal pressure of the disk can only decrease outwards if there is a strong gravitational potential. Our theoretical stability analysis shows that convective continuum instabilities can only appear if the density contours coincide with the poloidal magnetic flux contours. Our numerical results confirm and complement this theoretical analysis. Furthermore, these results show that the influence of gravity can either be stabilizing or destabilizing on this new kind of MHD instability. In the likely case of a non--constant density, the height of the disk should exceed a threshold before this type of instability can play a role. This localised MHD instability provides an ideal, linear route to MHD turbulence in strongly magnetized accretion disk tori.Comment: 20 pages, 10 figures, accepted for publication in Astronomy & Astrophysic

Comment on 'Alfven Instability in a Compressible Flow' [Phys. Rev. Lett. 101, 245001 (2008)]

In a recent paper [Y. Taroyan, Phys. Rev. Lett 101, 245001 (2008) it is claimed that a new MHD instability of the Alfven wave has been found: incompressible Alfven modes propagating in a compressible spatially varying flow were apparently exponentially amplified. We show that the results of the work follow from an impossible equilibrium and therefore should be disregarded

Waves and Instabilities in Accretion Disks: MHD Spectroscopic Analysis

A complete analytical and numerical treatment of all magnetohydrodynamic waves and instabilities for radially stratified, magnetized accretion disks is presented. The instabilities are a possible source of anomalous transport. While recovering results on known hydrodynamicand both weak- and strong-field magnetohydrodynamic perturbations, the full magnetohydrodynamic spectra for a realistic accretion disk model demonstrates a much richer variety of instabilities accessible to the plasma than previously realized. We show that both weakly and strongly magnetized accretion disks are prone to strong non-axisymmetric instabilities.The ability to characterize all waves arising in accretion disks holds great promise for magnetohydrodynamic spectroscopic analysis.Comment: FOM-Institute for plasma physics "Rijnhuizen", Nieuwegein, the Netherlands 12 pages, 3 figures, Accepted for publication in ApJ

The Spectral Web of stationary plasma equilibria. II. Internal modes

The new method of the Spectral Web to calculate the spectrum of waves and instabilities of plasma equilibria with sizeable flows, developed in the preceding Paper I [Goedbloed, Phys. Plasmas 25, 032109 (2018)], is applied to a collection of classical magnetohydrodynamic instabilities operating in cylindrical plasmas with shear flow or rotation. After a review of the basic concepts of the complementary energy giving the solution path and the conjugate path, which together constitute the Spectral Web, the cylindrical model is presented and the spectral equations are derived. The first example concerns the internal kink instabilities of a cylindrical force-free magnetic field of constant α subjected to a parabolic shear flow profile. The old stability diagram and the associated growth rate calculations for static equilibria are replaced by a new intricate stability diagram and associated complex growth rates for the stationary model. The power of the Spectral Web method is demonstrated by showing that the two associated paths in the complex ω-plane nearly automatically guide to the new class of global Alfvén instabilities of the force-free configuration that would have been very hard to predict by other methods. The second example concerns the Rayleigh–Taylor instability of a rotating theta-pinch. The old literature is revisited and shown to suffer from inconsistencies that are remedied. The most global n = 1 instability and a cluster sequence of more local but much more unstable n=2,3,…∞ modes are located on separate solution paths in the hydrodynamic (HD) version of the instability, whereas they merge in the MHD version. The Spectral Web offers visual demonstration of the central position the HD flow continuum and of the MHD Alfvén and slow magneto-sonic continua in the respective spectra by connecting the discrete modes in the complex plane by physically meaningful curves towards the continua. The third example concerns the magneto-rotational instability (MRI) thought to be operating in accretion disks about black holes. The sequence n=1,2,… of unstable MRIs is located on one continuous solution path, but also on infinitely many separate loops (“pancakes”) of the conjugate path with just one MRI on each of them. For narrow accretion disks, those sequences are connected with the slow magneto-sonic continuum, which is far away though from the marginal stability transition. In this case, the Spectral Web method is the first to effectively incorporate the MRIs into the general MHD spectral theory of equilibria with background flows. Together, the three examples provide compelling evidence of the computational power of the Spectral Web Method.</p

Waves in a warm pair plasma: a relativistically complete two-fluid analysis

We present an ideal two-fluid wave mode analysis for a pair plasma, extending an earlier study for cold conditions to the warm pair plasma case. Starting from the completely symmetrized means for writing the governing linearized equations in the pair fluid rest frame, we discuss the governing dispersion relation containing all six pairs of forward and backward propagating modes, which are conveniently labelled as S, A, F, M, O and X. These relate to the slow (S), Alfven (A) and fast (F) magnetohydrodynamic waves, include a modified (M) electrostatic mode, as well as the electromagnetic O and X branches. In the dispersion relation, only two parameters appear, which define the pair plasma magnetization E2 E[0, infinity] and the squared pair plasma sound speed v2, measured in units of the light speed c. The description is valid also in the highly relativistic regime, where either a high magnetization and/or a relativistic temperature (hence sound speed) is reached. We recover the exact relativistic single-fluid magnetohydrodynamic expressions for the S, A and F families in the low wavenumber–frequency regime, which can be obtained for any choice of the equation of state. We argue that, as in a cold pair plasma, purely parallel or purely perpendicular propagation with respect to the magnetic field vector B is special, and near-parallel or near-perpendicular orientations demonstrate avoided crossings of branches at computable wavenumbers and frequencies. The complete six-mode phase and group diagram views are provided as well, visually demonstrating the intricate anisotropies in all wave modes, as well as their transformations. Analytic expressions for all six wave group speeds at both small and large wavenumbers complement the analysis.</p

MHD instabilities in astrophysical plasmas: very different from MHD instabilities in tokamaks!

The extensive studies of MHD instabilities in thermonuclear magnetic confinement experiments, in particular of the tokamak as the most promising candidate for a future energy producing machine, have led to an \u27intuitive\u27 description based on the energy principle that is very misleading for most astrophysical plasmas. The \u27intuitive\u27 picture almost directly singles out the dominant stabilizing field line bending energy of the Alfvén waves and, consequently, concentrates on expansion schemes that minimize that contribution. This happens when the wave vector k 0 of the perturbations, on average, is perpendicular to the magnetic field B. Hence, all macroscopic instabilities of tokamaks (kinks, interchanges, ballooning modes, ELMs, neoclassical tearing modes, etc) are characterized by satisfying the condition k 0 -|- B, or nearly so. In contrast, some of the major macroscopic instabilities of astrophysical plasmas (the Parker instability and the magneto-rotational instability) occur when precisely the opposite condition is satisfied: k 0 | | B. How do those instabilities escape from the dominance of the stabilizing Alfvén wave? The answer to that question involves, foremost, the recognition that MHD spectral theory of waves and instabilities of laboratory plasmas could be developed to such great depth since those plasmas are assumed to be in static equilibrium. This assumption is invalid for astrophysical plasmas where rotational and gravitational accelerations produce equilibria that are at best stationary, and the associated spectral theory is widely, and incorrectly, believed to be non-self adjoint. These complications are addressed, and cured, in the theory of the Spectral Web, recently developed by the author. Using this method, an extensive survey of instabilities of astrophysical plasmas demonstrates how the Alfvén wave is pushed into insignificance under these conditions to give rise to a host of instabilities that do not occur in laboratory plasmas.</p

The Spectral Web of stationary plasma equilibria. I. General theory

A new approach to computing the complex spectrum of magnetohydrodynamic waves and instabilities of moving plasmas is presented. It is based on the concept of the Spectral Web, exploiting the self-adjointness of the generalized Frieman–Rotenberg force operator, G, and the Doppler–Coriolis gradient operator parallel to the velocity, U. The problem is solved with an open boundary, where the complementary energy Wcom represents the amount of energy to be delivered to or extracted from the system to maintain a harmonic time-dependence. The eigenvalues are connected by a system of curves in the complex ω-plane, the solution path and the conjugate path (where Wcom is real or imaginary) which together constitute the Spectral Web, having a characteristic geometry that has to be clarified yet, but that has a deep physical significance. It is obtained by straightforward contour plotting of the two paths. The complex eigenvalues, within a specified rectangle of the complex ω-plane, are found by fast, reliable, and accurate iterations. Real and complex oscillation theorems, replacing the familiar tool of counting nodes of eigenfunctions, provide an associated mechanism of mode tracking along the two paths. The Spectral Web method is generalized to toroidal systems and extended to include a resistive wall by accounting for the dissipation in such a wall. It is applied in an accompanying Paper II [J. P. Goedbloed, Phys. Plasmas 25, 032110 (2018).] to a multitude of the basic fundamental instabilities operating in cylindrical plasmas.</p

MS 671 Anthropology

COURSE DESCRIPTION This course is “An introduction to cultural anthropology, with application to Christian evangelization and mission.” From before the beginning (Genesis 1), our Triune God has been in mission, reaching out in self-giving, other-embracing love (John 1 & I John 1). We now call this the missio Dei, “the mission of God” to remind us that it is not our mission, nor does mission belong to the church. How God does mission is best understood in the sending of Jesus the Christ (Ephesians 2:11-15). As the Father has sent Jesus, so Jesus sends us (John 20:21). It is in the life of Jesus here on earth that we most clearly see our example for mission: Jesus emptied himself, came down to the people, developed relationships and learned the local language and culture for 30 years; and then he began to preach about the Kingdom of God (Matthew 4:17; Luke 4:43) and perform acts of mercy and justice (Luke 4:18-21) that the people saw as signs and wonders of the presence of God among them (Luke 4:22,32,36). The acts of Jesus Christ through the Holy Spirit (the proper title of Acts) continued in the apostles, deacons and those who were called later, like Paul. The apostles began in mission with the Jews (Acts 2-7). The deacons reached out a little further to the half- Jews (Acts 8) and then to the proselytes (Acts 8). Finally, Peter reluctantly reached out to those who were not Jewish, half-Jewish or even wanna-be Jews but were full fledged Gentiles (Acts 10). Still, it was the multi-cultural church at Antioch (not the monocultural church at Jerusalem) that commissioned and sent out the first missionaries: Paul and Barnabas (Acts 13). What they did changed not just the composition of the church but the life of the church. Jesus the Messiah (Christos) became Jesus the Lord (Kyrios) (Acts 28), because that is what the Greeks and Romans were looking for. New understandings of Jesus brought fuller meaning to the cosmic event of the death and resurrection of Jesus. Jesus was interpreted anew as the Logos (John 1), the Pleroma (Colossians 1:19, 2:9-10), the Mystery and Wisdom of God (Ephesians). Every generation deserves a fresh reading of the gospel. Who will speak now to the migrant generation of this globalized world? A critical view of anthropology as it relates to theology will lead us to our main concern, and that is missiology. There will be a conscious effort in our course to maintain a trialogue between anthropology, theology and missiology. Our goal is to gain insights about missiological issues and concerns.https://place.asburyseminary.edu/syllabi/3533/thumbnail.jp