Transonic magnetohydrodynamic flows.

Stationary flows of an ideal plasma with translational symmetry along the (vertical) a axis are considered, and it is demonstrated how they can be described in the intrinsic (natural) coordinates (xi, eta, theta) where xi is a label of flux and stream surfaces, eta is the total pressure and theta is the angle between the horizontal magnetic (and velocity) field and the x axis. Three scalar nonlinear equilibrium equations of mixed elliptic-hyperbolic type for theta(xi, eta), xi(eta, theta) and eta(theta, xi) respectively are derived. The equilibrium equation for theta(xi, eta) is especially useful, and has considerable advantages compared with the coupled system of algebraic-differential equations that are conventionally used for studying plasma flows. In particular, for this equation the location of the regions of ellipticity and hyperbolicity can be deter mined a priori. Relations between the equilibrium equation for theta(xi, eta) and the nonlinear hodograph equation for xi(eta, theta) are elucidated. Symmetry properties of the intrinsic equilibrium equations are discussed in detail and their self-similar solutions are described. In particular, magnetohydrodynamic counterparts of several classical flows of an ideal fluid (the Prandtl-Meyer flows around a corner, the spiral flows and the Ringleb flows around a plate, etc.) are found. Stationary flows described in this paler can be used for studying both astrophysical and thermonuclear plasmas

Mode coupling in two-dimensional magnetohydrodynamic flows

The spectrum of incompressible waves and instabilities of two-dimensional plasma geometries with background flow is calculated. The equilibrium is solved numerically by the recently developed program FLow Equilibrium Solver (FLES). The spectra of the equilibria are computed by means of another ne vcr program, the INcompressible 2-dimensional FLow Eigenvalue Solver (IN2FLES). Magnetic instabilities and instabilities driven by the the two-dimensionality and the flow are found. For linear equilibria, the eigenvalues for elliptical geometries remain close to the curves on which the eigenvalues for circular geometries lie. These curves may be found for unbounded domains by a calculation in Fourier space [see Lifschitz, A. In: Proceedings of 1995 International Workshop on Operator Theory and Applications (ed. R. Mennicken and C. Tretter), pp. 97-117, Birkhauser, Boston, 1998]. Here the relation between a new continuous spectrum of unbounded domains and the discrete spectrum of bounded domains is investigated. Finally, it is found that the two-dimensionality and the background flow may lead to an overstable cluster point

The spectrum of MHD flows about X-points.

A recently proposed method to calculate the spectrum of linear, incompressible, unbounded plasma flows is applied to magnetohydrodynamic flows about X points. The method transforms the two-dimensional spectral problem in physical space into a one-dimensional problem in Fourier space. The latter problem is far easier to solve. application of this method to X-point plasma flows results in two kinds of essential spectra. One kind corresponds to stable perturbations and the other one to perturbations that become overstable whenever the square of the poloidal Alfven Mach number becomes larger than 1. Apart from these two spectra, no other spectral values were found

Growth and saturation of the Kelvin-Helmholtz instability with parallel and anti-parallel magnetic fields

We investigate the Kelvin-Helmholtz instability occuring at the interface of a shear flow configuration in 2D compressible magnetohydrodynamics (MHD). The linear growth and the subsequent non-linear saturation of the instability are studied numerically. We consider an initial magnetic field aligned with the shear flow, and analyze the differences between cases where the initial field is unidirectional everywhere (uniform case), and where the field changes sign at the interface (reversed case). We recover and extend known results for pure hydrodynamic and MHD cases with a discussion of the dependence of the non-linear saturation on the wavenumber, the sound Mach number, and the Alfvenic Mach number for the MHD case. A reversed field acts to destabilize the linear phase of the Kelvin-Helmholtz instability compared to the pure hydrodynamic case, while a uniform field suppresses its growth. In resistive MHD, reconnection events almost instantly accelerate the buildup of a global plasma circulation. They play an important role throughout the further non-linear evolution as well, since the initial current sheet gets amplified by the vortex flow and can become unstable to tearing instabilities forming magnetic islands. As a result, the saturation behaviour and the overall evolution of the density and the magnetic field is markedly different for the uniform versus the reversed field case

Magnetohydrodynamics spectrum of gravitating plane plasmas with flow.

The ideal magnetohydrodynamic spectrum of gravitating plane plasmas with equilibrium flow is investigated. Flow makes the spectral problem non-self-adjoint, so that the spectrum can become overstable. The criteria for cluster spectra to appear are derived analytically and both stable and unstable sides of the spectrum are examined numerically. Above certain critical values of the shear flow at the resonant surface, the gravitating interchange modes disappear However, the local extrema of the continua can then take over the cluster spectrum

Application of the Jacobi-Davidson method to spectral calculations in magnetohydrodynamics

For the solution of the generalized complex non-Hermitian eigenvalue problems $Ax=\lambda Bx$ occurring in the spectral study of linearized resistive magnetohydrodynamics (MHD) a new parallel solver based on the recently developed Jacobi-Davidson~\cite{Sleijpen96a} method has been developed. A brief presentation of the implementation of the solver is given here. The new solver is very well suited for the computation of some selected interior eigenvalues related to the resistive Alfv\'{e}n wave spectrum and is well parallelizable. All features of the spectrum are easily and accurately computed with only a few target shifts

Double-resonant fast particle-wave interaction

In future fusion devices fast particles must be well confined in order to transfer their energy to the background plasma. Magnetohydrodynamic instabilities like Toroidal Alfv\'en Eigenmodes or core-localized modes such as Beta Induced Alfv\'en Eigenmodes and Reversed Shear Alfv\'en Eigenmodes, both driven by fast particles, can lead to significant losses. This is observed in many ASDEX Upgrade discharges. The present study applies the drift-kinetic HAGIS code with the aim of understanding the underlying resonance mechanisms, especially in the presence of multiple modes with different frequencies. Of particular interest is the resonant interaction of particles simultaneously with two different modes, referred to as 'double-resonance'. Various mode overlapping scenarios with different q profiles are considered. It is found that, depending on the radial mode distance, double-resonance is able to enhance growth rates as well as mode amplitudes significantly. Surprisingly, no radial mode overlap is necessary for this effect. Quite the contrary is found: small radial mode distances can lead to strong nonlinear mode stabilization of a linearly dominant mode.Comment: 12 pages, 11 figures; Nuclear Fusion 52 (2012

Pressure-driven instabilities in astrophysical jets

Astrophysical jets are widely believed to be self-collimated by the hoop-stress due to the azimuthal component of their magnetic field. However this implies that the magnetic field is largely dominated by its azimuthal component in the outer jet region. In the fusion context, it is well-known that such configurations are highly unstable in static columns, leading to plasma disruption. It has long been pointed out that a similar outcome may follow for MHD jets, and the reasons preventing disruption are still not elucidated, although some progress has been accomplished in the recent years. In these notes, I review the present status of this open problem for pressure-driven instabilities, one of the two major sources of ideal MHD instability in static columns (the other one being current-driven instabilities). I first discuss in a heuristic way the origin of these instabilities. Magnetic resonances and magnetic shear are introduced, and their role in pressure-driven instabilities discussed in relation to Suydam's criterion. A dispersion relation is derived for pressure-driven modes in the limit of large azimuthal magnetic fields, which gives back the two criteria derived by Kadomtsev for this instability. The growth rates of these instabilities are expected to be short in comparison with the jet propagation time. What is known about the potential stabilizing role of the axial velocity of jets is then reviewed. In particular, a nonlinear stabilization mechanism recently identified in the fusion literature is discussed. Key words: Ideal MHD: stability, pressure-driven modes; Jets: stabilityComment: 20 pages, 3 figures. Lecture given at the JETSET European school "Numerical MHD and Instabilities". To be published by Springer in the "Lectures notes in physics" serie

The effect of flows on transverse oscillations of coronal loops

In this paper we study kink oscillations of coronal loops in the presence of flows. Using the thin-tube approximation we derive the general governing equation for kink oscillations of a loop with the density varying along the loop in the presence of flows. This equation remains valid even when the density and flow are time dependent. The derived equation is then used to study the effect of flows on eigenfrequencies of kink oscillations of coronal loops. The implication of the obtained results on coronal seismology is discussed

Transverse oscillations of coronal loops

On 14 July 1998 TRACE observed transverse oscillations of a coronal loop generated by an external disturbance most probably caused by a solar flare. These oscillations were interpreted as standing fast kink waves in a magnetic flux tube. Firstly, in this review we embark on the discussion of the theory of waves and oscillations in a homogeneous straight magnetic cylinder with the particular emphasis on fast kink waves. Next, we consider the effects of stratification, loop expansion, loop curvature, non-circular cross-section, loop shape and magnetic twist. An important property of observed transverse coronal loop oscillations is their fast damping. We briefly review the different mechanisms suggested for explaining the rapid damping phenomenon. After that we concentrate on damping due to resonant absorption. We describe the latest analytical results obtained with the use of thin transition layer approximation, and then compare these results with numerical findings obtained for arbitrary density variation inside the flux tube. Very often collective oscillations of an array of coronal magnetic loops are observed. It is natural to start studying this phenomenon from the system of two coronal loops. We describe very recent analytical and numerical results of studying collective oscillations of two parallel homogeneous coronal loops. The implication of the theoretical results for coronal seismology is briefly discussed. We describe the estimates of magnetic field magnitude obtained from the observed fundamental frequency of oscillations, and the estimates of the coronal scale height obtained using the simultaneous observations of the fundamental frequency and the frequency of the first overtone of kink oscillations. In the last part of the review we summarise the most outstanding and acute problems in the theory of the coronal loop transverse oscillations