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

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 Alfv\'en Mach number \textbf{defined as the ratio of the tube axial velocity to Alfv\'en 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 Alfv\'en 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 Alfv\'en Mach number. It is established that the sausage mode ($m = 0$) is not subject to the KH instability.