The effects of flow on resistive instabilities in magnetohydrodynamics

  • Lei Hou

    Student thesis: Doctoral Thesis


    Resistive instabilities in MHD have been understood as one of the most important problems in astrophysical and controlled thermonuclear fusion research during the past four decades. This class of instability results from the introduction in the ideal MHD equations of a small but finite electrical resistivity. The addition of this term results in a relaxation of the "fluxfreezing" constraint thereby enabling the plasma to change its magnetic field topology to a state of lower energy.

    The model studied in this thesis is the plane magnetic current layer in which a gravitational acceleration acting normally to the magnetic surfaces is introduced to simulate the effects of magnetic field line curvature of more realistic geometries. The two main types of resistive instability considered are the resistive tearing mode and the resistive (gravitational) interchange (or G-) mode. The equations describing the linearized set of MHD equations have been derived for a viscous plasma in the presence of a sub-Alfvenic equilibrium flow along the confining magnetic field. In the limit of small electrical resistivity, the analysis of the stability problem follows the standard procedure of division of the current layer into a narrow resistive boundary layer about the "resonant" plane, where dissipative effects are important, together with an ideal, infinitely-conducting outer region where flux-freezing is a good approximation.

    The determination of the normalized growth rate P of the instability presents itself in the form of a non-standard eigenvalue problem where the (complex) eigenvalue P appears both in the governing differential equation and in the boundary conditions. The dependence of P on the various parameters of the problem has been determined by two different methods of numerical solution. The first method employs a Fourier transform approach to reduce the boundary-layer equation to a third-order ordinary differential equation. The second approach applies to the boundary-layer equation directly and appeals to residue calculus. This alternative method of solution provides a means of verification of the numerical results.

    A detailed investigation of the eigenvalue loci in the complex P-plane as a function of the equilibrium flow parameter and other physical parameters has been carried out. This study is supported by an asymptotic analysis of the boundary-layer equation for large values of the flow parameter and comparison is made with the numerical results.

    The main contributions in this study include:

    (i) A detailed understanding of the physical model and the derivation of the differential equations. The linearized stability of the plane current layer configuration has been carefully analysed.
    (ii) For high temperature thermonuclear fusion plasmas, the magnetic Lundquist number is high and solution of the MHD system has been carried out by means of a standard boundary layer analysis to determine the dependence of the growth rate of the instability on the other parameters.
    (iii) Numerical solution of the system of equations valid in the boundary layer has been initially carried out using Fourier transform techniques. An alternative method of solution of the boundary-layer equations has been implemented as a means of verification.
    (iv) To explore the complicated eigen-loci structure, a very delicate numerical investigation has been undertaken. We obtain a good understanding on the loci spectrum bifurcation, visco-G mode exchange behaviour.
    (v) By using a singular perturbation technique we have estimated the growth rate analytically in the large flow limit which confirms the numerical results and predicts the asymptotics of the growth rate.
    Date of AwardMay 1994
    Original languageEnglish
    SponsorsEuropean Commission
    SupervisorRichard Paris (Supervisor)

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