We present a complete analysis of all wave modes in a cold pair plasma, significantly extending standard textbook treatments. Instead of identifying the maximal number of two propagating waves at fixed frequency w we introduce a unique labelling of all 5 mode pairs described by the general dispersion relation w(k), starting from their natural ordering at small wavenumber k. There, the 5 pairs start off as Alfv{\'e}n (A), fast magnetosonic (F), modified electrostatic (M) and electromagnetic O and X branches, and each w(k) branch smoothly connects to large wavenumber resonances or limits. For cold pair plasmas, these 5 branches show avoided crossings, which become true crossings at exactly parallel or perpendicular orientation. Only for those orientations, we find a changed connectivity between small and large wavenumber behaviour. Analysing phase and group diagrams for all 5 wave modes, distinctly different from the Clemmow{\textendash}Mullaly{\textendash}Allis representation, reveals the true anisotropy of the A, M and O branches.

}, doi = {10.1017/S0022377819000102}, author = {Keppens, R. and Goedbloed, J. P.} } @article {4239, title = {A fresh look on waves in ion-electron plasmas}, journal = {Frontiers in Astronomy and Space Sciences}, volume = {2019}, year = {2019}, month = {03/2019}, pages = {00011}, abstract = {Exploiting the general dispersion relation describing all waves in an ideal ion-electron fluid, we revisit established treatments on wave families in a cold ion-electron plasma. These contain the magnetohydrodynamic Alfv{\'e}n and fast waves at low frequencies, long wavelengths, but are enriched by short wavelength resonance behaviors, electrostatic and electromagnetic mode types, and cut-off frequencies distinguishing propagating from evanescent waves. Our theoretical treatment exploits purely polynomial expressions, which for the cold ion-electron case only depend on 2 parameters: the ratio of masses over charges u and the ratio E of the electron gyro frequency to the combined ion-electron plasma frequency. We provide a complete description of all waves, which stresses the intricate variation of all five branches of eigenfrequencies w(k, @) depending on wavenumber k and angle @ between wavevector and magnetic field B. Corresponding 5-mode phase and group diagrams provide insight on wave transformations and energy transport. Special cases, like the high frequency modes in magneto-ionic theory following from Appleton-Hartree dispersion relations, are naturally recovered and critically discussed. Faraday rotation for electromagnetic waves is extended to all propagation angles @. The discussion covers all cold ion-electron plasma waves, up into the relativistic regime.}, doi = {10.3389/fspas.2019.00011}, author = {Keppens, R. and Goedbloed, J. P.} } @article {4376, title = {Waves in a warm pair plasma: a relativistically complete two-fluid analysis}, journal = {Journal of Plasma Physics}, volume = {85}, year = {2019}, month = {08/2019}, pages = {905850408}, abstract = {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{\textendash}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.

}, doi = {10.1017/S0022377819000552}, author = {Keppens, R. and Goedbloed, J. P. and Durrive, J. B.} } @article {3884, title = {MHD instabilities in astrophysical plasmas: very different from MHD instabilities in tokamaks!}, journal = {Plasma Physics and Controlled Fusion}, volume = {60}, year = {2018}, pages = {014001}, abstract = {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 {\textquoteright}intuitive{\textquoteright} description based on the energy principle that is very misleading for most astrophysical plasmas. The {\textquoteright}intuitive{\textquoteright} picture almost directly singles out the dominant stabilizing field line bending energy of the Alfv{\'e}n 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 Alfv{\'e}n 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 Alfv{\'e}n wave is pushed into insignificance under these conditions to give rise to a host of instabilities that do not occur in laboratory plasmas.

}, doi = {10.1088/1361-6587/aa89fe}, author = {Goedbloed, J. P.} } @article {3974, title = {The Spectral Web of stationary plasma equilibria. II. Internal modes}, journal = {Physics of Plasmas}, volume = {25}, year = {2018}, pages = {032110}, abstract = {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 Alfv{\'e}n instabilities of the force-free configuration that would have been very hard to predict by other methods. The second example concerns the Rayleigh{\textendash}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,{\textellipsis}$\infty$ 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 Alfv{\'e}n 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,{\textellipsis} of unstable MRIs is located on one continuous solution path, but also on infinitely many separate loops ({\textquotedblleft}pancakes{\textquotedblright}) 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.

}, doi = {10.1063/1.5019838}, author = {Goedbloed, J. P.} } @article {3975, title = {The Spectral Web of stationary plasma equilibria. I. General theory}, journal = {Physics of Plasmas}, volume = {25}, year = {2018}, pages = {032109}, abstract = {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{\textendash}Rotenberg force operator, G, and the Doppler{\textendash}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.

}, doi = {10.1063/1.5019831}, author = {Goedbloed, J. P.} } @inbook {4055, title = {The Role of Magnetic Fields in AGN Activity and Feedback}, booktitle = {Cosmic Magnetic Fields}, series = {Canary Islands Winter School of Astrophysics}, year = {2018}, pages = {87{\textendash}122}, publisher = {Cambridge University Press}, organization = {Cambridge University Press}, chapter = {4}, address = {Cambridge, MA, USA}, abstract = {Active galactic nuclei (AGNs), the luminous, compact core regions of galaxies where accretion occurs onto supermassive black holes, can dramatically influence their entire host galaxy evolution by a process referred to as AGN feedback. Energy feedback to the galaxy is the result of combined radiation fields and directed outflows, and especially radio-loud active galaxies show pronounced jets and lobes. Their synchrotron radio emission indicates that dynamically important magnetic fields are at play in AGN jet collimation, stability, energy transfer to the intergalactic medium and their overall morphological appearance. Current knowledge on the launching mechanisms for such highly energetic relativistic jets, as well as the near black-hole accretion processes themselves, all invoke magnetic fields as active agents in angular momentum, mass and energy redistributions. In this review, we cover aspects of AGN feedback and the role played by magnetic fields, almost necessarily studied at vastly different length and timescales. We emphasize how typical large-scale galaxy interaction studies rely on parametric prescriptions for feedback, while detailed dedicated studies for near black-hole dynamics and relativistic jet propagation exist which take full account of magnetic field influences. We discuss representative hydro to magnetohydrodynamic (MHD) numerical simulations that exploit analogies with less energetic X-ray binary sources or even protostellar accretion-ejection systems, emphasize relativistic MHD descriptions, and point out that magnetic fields in accretion disks yield many linear instability routes to turbulence that have scarcely been recognized in the astrophysical community. In combination, they serve to show that magnetic field influences in AGN accretion, jet launch, energy feedback, and overall evolution are still far from completely understood, although many aspects have been disclosed by advanced analytical and numerical relativistic MHD studies. Motivation: Astrophysical Jets Radio galaxies confront us with dramatic views on energy redistributions at all scales, as mediated by central massive black holes lurking in their nucleus. A clear example is provided by the elliptical galaxy NGC5532, a nearby (red shift z = 0.0237, type S0) galaxy where the stellar distribution is in sharp contrast with its double-jetted appearance in radio images.}, doi = {10.1017/9781316160916.005}, author = {Keppens, R. and Porth, O. and Goedbloed, J. P.}, editor = {S{\'a}nchez Almeida, J. and Mart{\'\i}nez Gonz{\'a}lez, M. J.} } @article {2286, title = {Comment on "Continuum modes in rotating plasmas: General equations and continuous spectra for large aspect ratio tokamaks" [Phys. Plasmas 18, 092103 (2011)]}, journal = {Physics of Plasmas}, volume = {19}, number = {6}, year = {2012}, month = {Jun}, pages = {064701}, type = {Editorial Material}, abstract = {It is shown that some of the main results of the recent paper by Lakhin and Ilgisonis [Phys. Plasmas 18, 092103 (2011)], viz. the derivation of the equations for the continuous spectra of poloidally and toroidally rotating plasmas and their special solution for large aspect ratio tokamaks with large parallel flows were obtained before by Goedbloed, Belien, van der Holst, and Keppens [Phys. Plasmas 11, 28 (2004)]. A further rearrangement of the system of equations for the coupled Alfven and slow continuous spectra clearly exhibits: (a) coupling through a single tangential derivative, which is a generalization of the geodesic curvature; (b) the "transonic" transitions of the equilibrium, which need to be carefully examined in order to avoid entering hyperbolic flow regimes where the stability formalism breaks down. A critical discussion is devoted to the implications of this failure, which is generally missed in the tokamak literature, possibly as a result of the wide-spread use of the sonic Mach number of gas dynamics, which is an irrelevant and misleading parameter in "transonic" magnetohydrodynamics. Once this obstacle in understanding is removed, further application of the theory of trans-slow Alfven continuum instabilities to both tokamaks, with possible implications for the L-H transition, and astrophysical objects like "fat" accretion disks, with a possible new route to magnetohydrodynamic turbulence, becomes feasible. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3694872]}, keywords = {EQUILIBRIA, MAGNETOHYDRODYNAMIC FLOWS, STABILITY, SYSTEMS, WAVES}, isbn = {1070-664X}, doi = {10.1063/1.3694872}, url = {http://scitation.aip.org/content/aip/journal/pop/19/6/10.1063/1.3694872}, author = {Goedbloed, J. P.} } @article {2159, title = {New approach to magnetohydrodynamics spectral theory of stationary plasma flows}, journal = {Plasma Physics and Controlled Fusion}, volume = {53}, year = {2011}, month = {Jul}, pages = {074001}, type = {Review}, abstract = {While the basic equations of MHD spectral theory date back to 1958 for static plasmas (Bernstein et al 1958 Proc. R. Soc. A 244 17) and to 1960 for stationary plasma flows (Frieman and Rotenberg 1960 Rev. Mod. Phys. 32 898), progress on the latter subject has been slow since it suffers from lack of analytical insight concerning the structure of the spectrum. One of the reasons is the usual misnomer of {\textquoteright}non-self adjointness{\textquoteright} of the stationary flow problem. Actually, self-adjointness of the occurring operators, namely the generalized force operator and the Doppler-Coriolis gradient operator-i rho v.del, was proved right away by Frieman and Rotenberg. Based on the reality of the two quadratic forms corresponding to these operators, we here construct (a) an effective method to compute the solution paths in the complex omega plane on which the eigenvalues are situated, (b) the counterpart of the oscillation theorem for eigenvalues of static equilibria (Goedbloed and Sakanaka 1974 Phys. Fluids 17 908) for the eigenvalues of stationary flows, based on the monotonicity of the alternating ratio, or alternator, of the boundary values of the displacement. and the total pressure perturbation Pi. This enables one to map out the complete spectrum of eigenvalues in the complex.-plane. The intricate topology of the solution paths is discussed for the fundamental examples of Rayleigh-Taylor, Kelvin-Helmholtz and combined instabilities.}, keywords = {HYDROMAGNETIC STABILITY, INSTABILITIES, MHD SPECTROSCOPY, MODES, PINCH, STABILIZATION, WAVES}, isbn = {0741-3335}, doi = {10.1088/0741-3335/53/7/074001}, author = {Goedbloed, J. P.} } @article {1988, title = {MAGNETOHYDRODYNAMIC MODELING FOR FUSION PLASMAS}, journal = {Fusion Science and Technology}, volume = {57}, number = {2T}, year = {2010}, note = {ISI Document Delivery No.: 592PSTimes Cited: 0Cited Reference Count: 14}, month = {Feb}, pages = {137-147}, type = {Proceedings Paper}, abstract = {The magnetohydrodynamic model for fusion plasma dynamics governs the large-scale equilibrium properties, and sets the most stringent constraints on the parameter space accessible without violent disruptions. In conjunction with linear stability analysis in the complex tokamak geometry, the MHD paradigm is also routinely being used to diagnose recurring wave modes and identify potential MHD mode triggers of consequent non-MHD phenomena. On the other hand, it is currently computationally feasible to perform fully nonlinear simulations in tokamak geometry, and determine nonlinear, long-term (i.e. on resistive time scales) evolutions for individual MHD dominated plasma scenarios. It can be expected that this success continues its evolution towards a fully integrated computational analysis of the experimental campaigns, certainly in view of the desired steady-state self-burning plasmas.}, keywords = {ASTROPHYSICAL PLASMAS, EQUILIBRIA, MHD, SIMULATION, STABILITY}, isbn = {1536-1055}, url = {