D Hermans

On the existence of the continuous spectrum of ideal MHD in a 2D magnetostatic equilibrium

The continuous spectrum of a 2D magnetostatic equilibrium with y-invariance is derived. It is shown that the continuous spectrum is given by an eigenvalue problem on each magnetic surface and is related to the different behaviour of the equilibrium quantities in different magnetic surfaces. The special case of a uniform poloidal magnetic field in a 1D equilibrium that is stratified with height, has been considered in detail and it is found that there is no continuous spectrum. It is shown that this result relies completely on the artificial property that the behaviour of the equilibrium quantities along a magnetic field line is independent of the field line considered. As a consequence the non-existence of a continuous spectrum in a 1D equilibrium with a uniform magnetic field cannot be used to argue that the continuous spectrum has no physical relevance.

Linear Spectrum of Magnetostatic and Thermally Conducting Planar Plasmas

The linear spectrum of 1‐D magnetostatic and thermally conducting equilibrium plasmas is analyzed. It is found that the influence of thermal conduction is fundamentally different on the various parts of the linear spectrum of ideal magnetohydrodynamics and that it is by far the most profound for the slow magnetoacoustic part. The ideal Alfven continuous spectrum is unaffected by thermal conduction. However, the ideal slow continuous spectrum is replaced by the isothermal slow continuous spectrum. This new continuous spectrum owes its existence to thermal conduction but is independent of κ and involves a different range of continuum frequencies. In addition to these two continuous parts, the spectrum consists of discrete slow and fast magnetoacoustic modes and thermal modes. The point eigenvalues of the fast magnetoacoustic modes are slightly distorted in proportion to κ. However, the point eigenvalues of the slow magnetoacoustic modes lie on well‐defined curves in the complex plane that are independent of κ and controlled by the ideal slow and isothermal slow continua. The discrete slow magnetoacoustic spectrum hangs, as it were, on the ideal slow and isothermal slow continua and is determined by the nonuniformity of the equilibrium. The thermal modes are the result of the inclusion of the nonideal effect of thermal conduction in the energy equation.

Linearly overstable magnetic convection in 1D compressible and non-uniform plasmas

Abstract The linear overstable magnetic convection in a non-uniform and compressible plasma slab is computed using a finite element eigenvalue code. Overstable modes are found to be part of the slow-gravity subspectrum and can be stabilized by increasing the magnetic field strength. The periods of overstable oscillations are determined by the isothermal frequency. Results for two different sequences of equilibrium configurations are presented.

Overstable Convection in a Non-Uniform Magnetic Field

The linear spectrum of a non-homogeneous, compressible and thermally conducting planar plasma with a vertical gravitational field and a sheared horizontal magnetic field is studied. It is shown that the spectrum has two continuous parts. The Alfven continuum of linear ideal MHD is unaffected by radiative conduction, but the slow continuum is removed and replaced by a new continuous part which has been called isothermic continuum. Purely exponentially growing and overstable normal modes have been determined numerically and the distribution of their eigenfrequencies in the complex plane has been studied. The eigenfrequencies lie on specific curves in the complex plane which are partially determined by the isothermic continuum.