The non-axisymmetric instability of a cylindrical shear flow containing an azimuthal magnetic field

Abstract

The stability of a differentially rotating, cylindrical fluid body containing an azimuthal magnetic field is investigated by solving the linear eigenvalue problem for non-axisymmetric perturbations. The model system consists of a perfectly conducting, ideal, incompressible fluid contained within cylindrical boundaries. It is found that exponentially growing modes are always present when the angular velocity decreases outwards, unless the magnetic field exceeds a certain strength. In the weak-field limit, growth rates approaching the Oort constant A can be attained. In the absence of diffusion, the instability grows preferentially at arbitrarily small scales. A purely magnetic instability can also be present, and persists when the magnetic field is arbitrarily strong. The growing modes are found to depend on the presence of at least one radial boundary. The structure of the spectrum of discrete eigenvalues is discussed in relation to the Alfvén continua, and the limit points of eigenvalues at large axial wavenumber are obtained. Analogous behaviour is found in a system with Cartesian geometry, which more accurately describes models currently being used to study non-linear behaviour in accretion discs.