Kelvin-Helmholtz versus Hall magnetoshear instability in astrophysical flows

We study the stability of shear flows in a fully ionized plasma. Kelvin-Helmholtz is a well-known macroscopic and ideal shear-driven instability. In sufficiently low-density plasmas, also the microscopic Hall magnetoshear instability can take place. We performed three-dimensional simulations of the Hall-magnetohydrodynamic equations where these two instabilities are present, and carried out a comparative study. We find that when the shear flow is so intense that its vorticity surpasses the ion-cyclotron frequency of the plasma, the Hall magnetoshear instability is not only non-negligible, but it actually displays growth rates larger than those of the Kelvin-Helmholtz instability.

Figure 1

Numerical box displaying the imposed velocity flow $U_y\left(x\right)$ and the external magnetic field $B_0\widehat{\mathbf{z}}$. The shaded patches correspond to regions with intense shear. Each axis ranges from 0–$2\pi$.

Figure 2

Distribution of ${\omega }_z$ on the $x,y$ plane for the MHD run ($x$ is the horizontal axis and each axis ranges from 0 to $2\pi$) at an early time $t=1$ (left frame) and also at a later time $t=10$ (right frame), where the Kelvin-Helmholtz instability has entered a nonlinear stage. Light (dark) regions correspond to structures of strong positive (negative) vorticity ${\omega }_z$.

Figure 3

Root mean square value of ${\langle U_x^2\rangle }_{y,z}(x_0,t)$ vs time in a lin-log plot for the MHD run (i.e., $\epsilon =0$). The thin black trace corresponds to $x_0=\pi /2$ and the thick gray trace to $x_0=3\pi /2$. Superimposed we show a fit corresponding to ${\gamma }_{kh}\approx 1.8$.

Figure 4

Black lines show ${(\omega /v_A{k}_z)}^2$ versus $k$ for the normal modes in the absence of shear (i.e., ${\omega }_{sh}=0$). The upper branch corresponds to whistlers, while the lower one corresponds to the ion-cyclotron mode. The gray lines show the modified solutions in the presence of shear, more specifically for ${\omega }_{sh}=-10$. The negative branch corresponds to unstable evolutions, since ${\omega }^2<0$.

Figure 5

Distribution of $J_z$ on the $x,y$ plane for the HMHD run ($x$ is the horizontal axis and each axis ranges from 0 to $2\pi$) at an early time $t=1$ (left frame) and also at a later time $t=7$ (right frame). Light (dark) regions correspond to structures of strong positive (negative) electric current density $J_z$.

Figure 6

Distribution of ${\omega }_z$ on the $(x,y)$ plane for the HMHD run ($x$ is the horizontal axis and each axis ranges from 0 to $2\pi$) at an early time $t=1$ (left frame) and also at a later time $t=7$ (right frame). Light (dark) regions correspond to structures of strong positive (negative) vorticity ${\omega }_z$.

Figure 7

Root mean square value of the magnetic field, i.e., the square root of ${\langle \left|\mathbf{B}\right|}^2{\rangle }_{y,z}(x_0=3\pi /2,t)$ vs. time in a lin-log plot for the HMHD run ($\epsilon =0.4$). Superimposed we show a fit corresponding to ${\gamma }_{hmsi}\approx 1.7$. In gray trace we also show the square root of ${\langle \left|\mathbf{B}\right|}^2{\rangle }_{y,z}(x_0=\pi /2,t)$.

Figure 8

Ratio of instability rates versus $R=|{\omega }_{sh}|/{\omega }_{ci}$.