Global Non-Axisymmetric Hall Instabilities in a Rotating Plasma

TL;DR

This paper investigates how Hall-MHD physics modifies global, non-axisymmetric flow-driven instabilities in a differentially rotating cylinder. By combining an electron-MHD (EMHD) limit for whistler waves with full Hall-MHD calculations, it reveals two instability branches: a fast-growing whistler branch and a modified ion-cyclotron-like branch, both energized by shear. The EMHD analysis uncovers a corotation-driven over-reflection mechanism that can destabilize k=0 modes beyond local dispersion predictions, while Hall-MHD calculations show asymmetry with respect to the sign of the axial field and field-geometry–dependent global modes, including azimuthal-field–driven instabilities at larger $d_i$. These global Hall-MHD modes persist at stronger magnetic fields than standard MHD MRI modes, indicating potentially enhanced angular-momentum transport in weakly ionized disks and informing laboratory plasma experiments; nonlinear, global simulations are suggested for quantifying momentum transport.

Abstract

Non-axisymmetric, flow-driven instabilities in the incompressible Hall-MHD model are studied in a differentially rotating cylindrical plasma. It is found that in the Hall-MHD regime, both whistler waves and ion-cyclotron waves can extract energy from the flow shear, resulting in two distinct branches of global instability. The non-axisymmetric whistler modes grow significantly faster than non-axisymmetric, ideal MHD modes. A discussion of the whistler instability mechanism is presented in the large-ion-skin-depth, `electron-MHD' limit. It is observed that the effect of the Hall term on the non-axisymmetric modes can be appreciable when $d_i$ is on the order of a few % of the width of the cylindrical annulus. Distinct global modes emerge in the Hall-MHD regime at significantly stronger magnetic fields than those required for unstable global MHD modes.

Figures (15)

• Figure 1: Plot of the positive solutions of $(\omega^2-\omega_A^2) = \pm kd_i \omega \omega_A$ as a function of $kd_i$. $\Omega_{ci}\equiv |V_A|/d_i$ is the ion cyclotron frequency, and $\omega_A\equiv k_\parallel B/\sqrt{\mu_0\rho}$ is the Alfvén frequency. Here, $k_\parallel=k\cos(\pi/6)$
• Figure 2: Numerically computed growth rates and frequencies of the fastest growing radial mode in a disc with vertical magnetic field in the electron MHD limit with $m=1,k=\pi/4$ as a function of $\mathrm{H}\propto kB_z$ at several aspect ratios.
• Figure 3: Plots of the eigenfunction $\psi$ associated with the fastest growing modes from the vertical field case at $A=0.5$, $\mathrm{H}=0.08$ (left), and $A=8$, $\mathrm{H}=0.08$ (right). The dashed red lines denote the point $\bar{\omega}=0$ for each mode. Note the difference in radial scales.
• Figure 4: Numerically computed growth rates and frequencies of the fastest growing radial mode in a disc with azimuthal magnetic field in the electron MHD limit with $m=1,k=\pi/4$ as a function of $\mathrm{H}\propto B_\phi$ evaluated at $s=1$ at several aspect ratios.
• Figure 5: Left: Plot of the square of the effective refractive index for whistler waves, $Q(\omega,s)$ for the fastest growing $m=1,k=0$ mode with $d_iV_{A,\phi}=0.58$. The shaded region is where $Q(\omega,s)<0$, and the zeros of $Q$ are where $\bar{\omega}^2 = |\boldsymbol{k}|^2d_i^2\omega_A^2$ The real and imaginary parts of $\psi$ are also plotted. Right: plot of the growth rate for $m=1,k=0$ modes in the EMHD model as a function of $\mathrm{H}_\phi$ for aspect ration $A=0.5$ along with analytic estimate based on asymptotics equation \ref{['eq: gamma_one_wall']}.