R. Gruber

Continuous spectra of a cylindrical magnetohydrodynamic equilibrium

It is shown that, for a cylindrical equilibrium of a perfectly conducting plasma, the magnetohydrodynamic spectrum contains at most two continua.

A new finite element approach to the normal mode analysis in magnetohydrodynamics

Abstract The eigenvalue problem arising in the one-dimensional normal mode analysis of fixed boundary magnetohydrodynamic stability is solved by a finite element method. Piecewise constant, discontinuous basis functions are used for two components of the displacement vector because this permits an accurate representation of the nearly divergence-free property of the modes being treated. In spite of the simple basis functions, the accuracy is greatly improved compared to that obtained with piecewise linear, continuous basis functions. Important features of the spectrum, such as infinitely degenerated eigenvalues, accumulation points, and continua are well represented by the method. The method used is equally well applicable to the free boundary stability problem.

Numerical Study of the Ideal-MHD Stability Limits in Oblate Spheromaks

The ideal-MHD stability properties of a class of hollow oblate spheromaks are studied numerically with a spectral code. An internal kink instability is found whenever the safety factor on axis, qo, rises above one. Its growth rate is only a weak function of the pressure. By shaping the current profile such that, simultaneously, q0 = 1 and the Mercier criterion is satisfied, full stability against internal perturbations is obtained up to a maximum value of β, this value increasing with the aspect ratio. For a configuration without a central hole, the maximum β is 1.5% and for an aspect ratio of 2.34, which is the largest that has been studied, it reaches 40%. Without a conducting shell, all equilibria are unstable to free-boundary modes, the fastest-growing mode being that associated with a global tilting motion. These modes, in contrast to the internal modes, are stabilized as the aspect ratio decreases. In the limiting case of no hole, the force-free equilibrium is only unstable with respect to n = 1 displacement which can be stabilized by a conducting shell located at a distance from the plasma boundary of 30% of the plasma minor radius.

Studium der Eigenschwingungen eines zylindrischen Plasmas mit der Methode der finiten Elemente

ZusammenfassungDie Eigenschwingungen eines zylindrischen homogenen Plasmas können analytisch untersucht und in drei Klassen eingeteilt werden. Die Eigenwerte der ersten Klasse sind gut getrennt, jene der zweiten haben einen Häufungspunkt, und der Eigenwert der dritten Klasse ist unendlich entartet. Es wird untersucht, wie und unter welchen Bedingungen die Eigenwerte und-funktionen des mit finiten Elementen diskretisierten Problems gegen die wahre Lösung streben.AbstractThe eigenmodes of a cylindrical homogeneous plasma may be calculated analytically. There are three classes of solutions. The eigenvalues of the first class are well separated, those of the second class have a point of accumulation. The eigenvalue of the third class is infinitely degenerated. The eigenvalues and eigenmodes of the corresponding discrete finite element problem tend towards the true solution. The convergence properties are discussed.

Linear Mhd Stability Studies of Helically Symmetric Equilibria with Hera

Helically symmetric equilibria without net longitudinal current and their linear MHD stability are studied with the HASE quasi-analytical equilibrium code and the HERA 2D spectral code. The modes investigated comprise low-m-number internal, external, high-m-number modes obtained with the large-k-number option of the HERA code and localized (k → ∞) ballooning modes. Stable straight Heliotron equilibria with beta up to = 0.01 and stable straight Heliac equilibria with beta up to = 0.29 are obtained. The correspondence between instability as obtained from the ballooning-mode equation in part of the plasma volume and from finite-k analysis is discussed.

Transformation of ERATO into a δW code

Abstract The spectral code ERATO has been transformed into a stability code based on the energy principle δ W . It only gives the stability index, i.e. the “yes or no” answer to the stability of an axisymmetric toroidal plasma equilibrium. Compared to the full ERATO code, the computing time is reduced by a factor of 4. Storage requirements and input/output operations reduced by a factor of 5.

Axisymmetric MHD stability of elongated tokamaks

The stability of axisymmetric modes is studied by using the general ideal-MHD code ERATO. The parameters investigated are the triangularity and the elongation of the tokamak meridian cross-section and the distance between the plasma and an external conducting wall. Both aspect ratio, R/a = 3, and safety factor on the axis, q0 = 1, are kept fixed. The validity of simpler models is tested. Wall stabilization is found to be effective for axisymmetric modes.

Comparison of MHD Stability Results Obtained with the BETA 3D and HERA 2D Codes

Comparison of MHD stability results obtained for helically symmetric equilibria with two codes, HERA and BETA, showed excellent agreement in growth rates and marginal stability parameters. While HERA is the helical version of the ERATO linearized 2D stability code, BETA obtains nonlinear 3D equilibria and their (nonlinear) stability properties. The cases studied comprise internal, i.e., fixed plasma boundary, resonant and nonresonant modes in low-shear equilibria with intermediate and small ..beta.. values.

Stabilité magnétohydrodynamique des modes ballooning dans un tokamak

We implanted in the ERATO spectral code [1] the stability criterion for ballooning modes (toroidal wave numbern→∞) [2]. The study of ballooning waves located around a given magnetic surface reduces itself to the investigation of the sign of the potential energy on this surface; this method avoids the determination of eigenvalues which is expensive in computing time. The stability criterion obtained by this way has been applied to a high-beta equilibria family proposed for JET. The stability limit found is more constraining than the limit for low-n internal modes.

Resonant absorption of Alfvén waves in Fat Tori with circular cross-section

Keywords: LRP 170 Reference CRPP-REPORT-1980-001 Record created on 2008-04-18, modified on 2017-05-12