Kelvin--Helmholtz instability in a cool solar jet in the framework of Hall magnetohydrodynamics: A case study

• Published in: arXiv.org
• Main Authors: Zhelyazkov, I, Dimitrov, Z
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 04.01.2018
• Subjects: Angular velocity; Critical flow; Flow-density-speed relationships; Kelvin-Helmholtz instability; Light speed; Mach number; Magnetic flux; Magnetohydrodynamics; Parameters; Physics - Solar and Stellar Astrophysics; Physics - Space Physics; Plasma frequencies; Wave dispersion
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1706.03683

We investigate the conditions under which the magnetohydrodynamic (MHD) modes in a cylindrical magnetic flux tube moving along its axis become unstable against the Kelvin--Helmholtz (KH) instability. We \textbf{use} the dispersion relations of MHD modes \textbf{obtained} from the linearized Hall MHD equations for cool (zero beta) plasma \textbf{by assuming} real wave numbers and complex angular wave frequencies\textbf{/complex wave phase velocities}. The dispersion equations are solved numerically at fixed input parameters and varying values of the ratio \(l_\mathrm{Hall}/a\), where \(l_\mathrm{Hall} = c/\omega_\mathrm{pi}\) (\(c\) being the speed of light, and \(\omega_\mathrm{pi}\) the ion plasma frequency) and \(a\) is the flux tube radius. It is shown that the stability of the MHD modes depends upon four parameters: the density contrast between the flux tube and its environment, the ratio of external and internal magnetic fields, the ratio \(l_\mathrm{Hall}/a\), and the value of the Alfvén Mach number \textbf{defined as the ratio of the tube axial velocity to Alfvén speed inside the flux tube}. It is found that at high density contrasts, for small values of \(l_\mathrm{Hall}/a\), the kink (\(m = 1\)) mode can become unstable against KH instability at some critical Alfvén Mach number (or equivalently at critical flow speed), but a threshold \(l_\mathrm{Hall}/a\) can suppress the onset of the KH instability. At small density contrasts, however, the magnitude of \(l_\mathrm{Hall}/a\) does not affect noticeably the condition for instability occurrence---even though it can reduce the critical Alfvén Mach number. It is established that the sausage mode (\(m = 0\)) is not subject to the KH instability.