Can Hall effect trigger Kelvin–Helmholtz instability in sub-Alfvénic flows?

Abstract

In the Hall magnetohydrodynamics, the onset condition of the Kelvin-Helmholtz instability is solely determined by the Hall effect and is independent of the nature of shear flows. In addition, the physical mechanism behind the super and sub-Alfvénic flows becoming unstable is quite different: the high frequency right circularly polarized whistler becomes unstable in the super-Alfvénic flows whereas low frequency, left circularly polarized ion-cyclotron wave becomes unstable in the presence of sub-Alfvénic shear flows. The growth rate of the Kelvin-Helmholtz instability in the super-Alfvénic case is higher than the corresponding ideal magnetohydrodynamic rate. In the sub-Alfvénic case, the Hall effect opens up a new, hitherto inaccessible (to the magnetohydrodynamics) channel through which the partially or fully ionized fluid can become Kelvin-Helmholtz unstable. The instability growth rate in this case is smaller than the super-Alfvénic case owing to the smaller free shear energy content of the flow.When the Hall term is somewhat smaller than the advection term in the induction equation, the Hall effect is also responsible for the appearance of a new overstable mode whose growth rate is smaller than the purely growing Kelvin-Helmholtz mode. On the other hand, when the Hall diffusion dominates the advection term, the growth rate of the instability depends only on the Alfvén -Mach number and is independent of the Hall diffusion coefficient. Further, the growth rate in this case linearly increase with the Alfvén frequency with smaller slope for sub-Alfvénic flows.

Charged Grains and Kelvin–Helmholtz Instability in Molecular Clouds

“…( 4) with the growth rate about one-third of the purely growing (top panel) KH mode. The presence of this mode is generic to the Hall MHD flows (Pandey 2018). The overstable mode manifests a dependence on |Z| similar to that of its purely growing counterpart.…”

“…Thus depending on the polarization of waves, magnetic fluctuations may help or hinder the rolling up of the interface. As a result the onset condition of the KH instability will be affected by the Hall diffusion of the magnetic field (Pandey 2018). We shall see that even in sub-Alfvénic flows, low-frequency waves can destabilize the vortex sheet with the growth rate dependent on the charge number |Z| on the dust.…”

“…) which except for the corrected sign before M 2 A + 3 , is the same as Eq. ( 40) of Pandey (2018) if we replace their ω H with ω cd . For M A = 0.5 (Heyer & Brunt 2012), the maximum growth rate of the instability becomes 0.14 ω cd which gives t KHI = 1 Myr when the dust size distribution is dominated by 0.1 µm grains.…”

“…In the sub-Alfvénic case [Fig. ( 4)], the instability, much like in the dustless case (Pandey 2018) is entirely due to the Hall effect, although a purely growing mode (top panel) appears beyond a certain k L D cutoff. Note that with decreasing |Z|, increasingly shorter wavelength fluctuations are KH-unstable.…”

“…In the highly diffusive limit, long-wavelength k L D ≪ 1 waves are excluded (Pandey 2018). The growth rate of the KH instability for various Alfvén Mach numbers in the highly diffusive limit is…”