Extending the Range of the Inductionless Magnetorotational Instability

The magnetorotational instability (MRI) can destabilize hydrodynamically stable rotational flows, thereby allowing angular momentum transport in accretion disks. A notorious problem for the MRI is its questionable applicability in regions with low magnetic Reynolds number. Using the WKB method, we extend the range of applicability of the MRI by showing that the inductionless versions of the MRI, such as the helical MRI and the azimuthal MRI, can easily destabilize Keplerian profiles $\propto r^{-3/2}$ if the radial profile of the azimuthal magnetic field is only slightly modified from the current-free profile $\propto r^{-1}$. This way we further show how the formerly known lower Liu limit of the critical Rossby number $\mathrm{Ro}\approx -0.828$ connects naturally with the upper Liu limit $\mathrm{Ro}\approx +4.828$.

Figure 1

Dependence of the optimal critical Rossby numbers ${\mathrm{Ro}}_{\mathrm{opt}}^{\pm }$ on Rb when $\mathrm{Pm}=0$ and $\mathrm{N}\to 0$.

Figure 2

SMRI and HMRI for $n=0$, $\mathrm{Ro}=-3/4$, $\mathrm{Ha}=30$. Black: $\mathrm{Rb}=-0.74$, $\mathrm{Re}\approx 19876$. Red: $\mathrm{Rb}=-0.75$, $\mathrm{Re}\approx 21294$. Green: $\mathrm{Rb}=-0.77$, $\mathrm{Re}\approx 23935$. Blue: $\mathrm{Rb}=-0.91$, $\mathrm{Re}\approx 38553$. Middle of the blue loop: $\mathrm{Pm}=0.0002$, $S=\mathrm{Ha}{\mathrm{Pm}}^{1/2}\approx 0.42$, $\mathrm{Rm}=\mathrm{RePm}\approx 7.71$.

Figure 3

Domains of SMRI, HMRI, and AMRI for $\mathrm{Ha}=30$, $\mathrm{Ro}=-3/4$, $\mathrm{Re}=4000$, $\mathrm{Rb}=-0.755$, and (a) $n=-1$ (black), $n=-2$ (blue), $n=-3$ (green), $n=-4$ (red); and (b) $n=0$ (red), and $n=1$ (black).