Dissipative Instability of Magnetohydrodynamic Sausage Waves in a Compressional Cylindrical Plasma: Effect of Flow Shear and Viscosity Shear

Abstract

The shear flow influences the stability of magnetohydrodynamic (MHD) waves. In the presence of a dissipation mechanism, flow shear may induce an MHD wave instability below the threshold of the Kelvin–Helmholtz instability, which is called dissipative instability. This phenomenon is also called negative energy wave instability because it is closely related to the backward wave, which has negative wave energy. Considering viscosity as a dissipation mechanism, we derive an analytical dispersion relation for the slow sausage modes in a straight cylinder with a discontinuous boundary. It is assumed that the steady flow is inside and dynamic and bulk viscosities are outside the circular flux tube under photospheric condition. When the two viscosities are weak, it is found that for the slow surface mode, the growth rate is proportional to the axial wavenumber and flow shear, consistent within the incompressible limit. For a slow body mode, the growth rate has a peak at a certain axial wavenumber, and its order of magnitude is similar to surface mode. The linear relationship between the growth rate and the dynamic viscosity established in the incompressible limit develops nonlinearly when the flow shear and/or the two viscosities are sufficiently strong.