Analysis of azimuthal magnetorotational instability of rotating magnetohydrodynamic flows and Tayler instability via an extended Hain-Lüst equation

We consider a differentially rotating flow of an incompressible electrically conducting and viscous fluid subject to an external axial magnetic field and to an azimuthal magnetic field that is allowed to be generated by a combination of an axial electric current external to the fluid and electrical currents in the fluid itself. In this setting we derive an extended version of the celebrated Hain-Lüst differential equation for the radial Lagrangian displacement that incorporates the effects of the axial and azimuthal magnetic fields, differential rotation, viscosity, and electrical resistivity. We apply the Wentzel-Kramers-Brillouin method to the extended Hain-Lüst equation and derive a comprehensive dispersion relation for the local stability analysis of the flow to three-dimensional disturbances. We confirm that in the limit of low magnetic Prandtl numbers, in which the ratio of the viscosity to the magnetic diffusivity is vanishing, the rotating flows with radial distributions of the angular velocity beyond the Liu limit, become unstable subject to a wide variety of the azimuthal magnetic fields, and so is the Keplerian flow. In the analysis of the dispersion relation we find evidence of a new long-wavelength instability which is caught also by the numerical solution of the boundary value problem for a magnetized Taylor-Couette flow.

Figure 1

The growth rate $\mathrm{Re}\left(\lambda \right)/\mathrm{\Omega }$ given by (33) versus the dimensionless axial wave number $\widehat{k}=kr$ for $\mathrm{Re}=100,{\mathrm{Ha}}_{\theta }=10,m=1,\widehat{q}=0,\mathrm{Ro}=-3/4$. From upper left to lower right, $\mathrm{Rb}$ is varied from $-1$ to 1. As $\mathrm{Rb}$ increases, the value of $\widehat{k}$ corresponding to the maximum growth rate increases from $\widehat{k}=0$ to finite but nonzero value and ultimately this $\widehat{k}\to \infty$.

Figure 2

The growth rate $\mathrm{Re}\left({\lambda }_2\right)$ of (36) versus ${\mathrm{Ha}}_{\theta }$ when $\mathrm{Re}={10}^4, m=1, \widehat{k}=\widehat{q}=0, \mathrm{Ro}=-3/4$, and $\mathrm{Rb}=-1$. The right panel is the close-up view of the left one near ${\mathrm{Ha}}_{\theta }=0$, demonstrating a certain strength of magnetic field needed for instability.

Figure 3

The growth rate $\mathrm{Re}\left(\lambda \right)$ to ${\mathrm{Ha}}_{\theta }$ when $\mathrm{Re}={10}^4, m=1, \widehat{k}\to \infty , \alpha =1, \mathrm{Ro}=-3/4$, and $\mathrm{Rb}=0$ according to (38). The left panel shows that large ${\mathrm{Ha}}_{\theta }$ increase the growth rate, and the right panel is the amplification of the left one when ${\mathrm{Ha}}_{\theta }$ is small, which demonstrates that a certain strength of magnetic field is needed for instability.

Figure 4

The growth rate to magnetic Rossby number $\mathrm{Rb}$ for $\mathrm{Re}={10}^4, {\mathrm{Ha}}_{\theta }=100, m=1, \mathrm{Ro}=-3/4$, and $\widehat{q}=0$ (upper panel) or $\widehat{q}=1$ (lower panel) according to (33). The solid line is $\widehat{k}=0$ mode, the dotted one is the $\widehat{k}\to \infty$ mode, and the dashed line stands for the growth rate maximized over $\widehat{k}$, whose left part tends to the $\widehat{k}=0$ mode and the right part tends to the $\widehat{k}=\infty ,\alpha =1$ mode.

Figure 5

Growth rate calculated with the use of the Hain-Lüst dispersion relation (32) in projection to the Rossby plane ($\mathrm{Rb},\mathrm{Ro}$) for $\mathrm{Re}={10}^4, {\mathrm{Ha}}_{\theta }={10}^2, \widehat{q}=1, m=1$, and (from upper-left to lower-right panel): $\widehat{k}=0.01$, 0.4, 0.7, 0.8, 0.9, 1, 1.1, 1.3, 1.8, 2.5, 5, and 10. The white domains represent stability.

Figure 6

The regions of the Tayler instability (blue) with the boundary (56) for $\widehat{q}=0$ and (top row from left to right) $\widehat{k}=100, \widehat{k}=\sqrt{3}+0.1$, and $\widehat{k}=\sqrt{3}$ and (bottom row from left to right) $\widehat{k}=\sqrt{3}-0.1, \widehat{k}=0.3$, and $\widehat{k}\to 0$.

Figure 7

The regions of the Tayler instability (blue) with the boundary (56) for $\widehat{q}=3$ and (from left to right) $\widehat{k}=100, \widehat{k}=0.2$, and $\widehat{k}\to 0$.

Figure 8

AMRI regions (above the neutral stability curves) in the $({\mathrm{Ha}}_{{\theta }_1},{\mathrm{Re}}_1)$ plane for $\mathrm{Rb}=\mathrm{Ro}=-1, m=1, q=3r_0^{-1}, r=1.5r_0$ and (left to right) $\mathrm{Pm}=10, \mathrm{Pm}=1$, and $\mathrm{Pm}={10}^{-6}$ found with the use of the growth rates maximized over $k$ of the roots of the dispersion relation (32) with the parameters specified by (62).

Figure 9

The maximized over $k$ growth rate $\mathrm{Re}\left({\lambda }_{max}\right)$ in the units of $\mathrm{\Omega }$ versus ${\mathrm{Ha}}_{{\theta }_1}$ according to Eq. (32) with the parameters (62) for the flow with $\mathrm{Ro}=-1, \mathrm{Rb}=-1, \mathrm{Pm}={10}^{-6}, m=1, q=3r_0^{-1}$, and $r=1.5r_0$ when (left) ${\mathrm{Re}}_1=3000$ and (right) ${\mathrm{Re}}_1=500$.

Figure 10

The instability region in the $({\mathrm{Re}}_1,{\mathrm{Ha}}_{{\theta }_1})$ plane for $\mathrm{Ro}=-1/2, \mathrm{Rb}=-1/2, m=1, q=3r_0^{-1}, r=1.5r_0$, and $\mathrm{Pm}=100, \mathrm{Pm}=10, \mathrm{Pm}=1$, and $\mathrm{Pm}=0.1$. The instability domains represented in blue are found with the use of the growth rates maximized over $k$ of the roots of Eq. (32) with the parameters (62).

Figure 11

The optimized over $k$ growth rate versus the Hartmann number ${\mathrm{Ha}}_{{\theta }_1}$ according to Eq. (32) with the parameters (62). The parameters chosen are ${\mathrm{Re}}_1=800, \mathrm{Rb}=-1/2, \mathrm{Ro}=-1/2$, and $\mathrm{Pm}=0.1$ with $m=1, q=3r_0^{-1}$, and $r=1.5r_0$.

Figure 12

The region of AMRI (above the critical lines) in the $({\mathrm{Ha}}_{{\theta }_1},\mathrm{Pm})$ plane when $\mathrm{Rb}=-1, \mathrm{Ro}=-3/4$ and (left) ${\mathrm{Re}}_1={10}^4$ and (right) ${\mathrm{Re}}_1={10}^3$ according to Eq. (32) with the parameters (62). In the former case the instability occurs when $\mathrm{Pm}>0.0046$ with the smallest $\mathrm{Pm}$ corresponding to ${\mathrm{Ha}}_{{\theta }_1}\approx 400$, whereas in the latter when $\mathrm{Pm}>0.048$ with the lowest $\mathrm{Pm}$ corresponding to ${\mathrm{Ha}}_{{\theta }_1}\approx 127$.

Figure 13

The region of AMRI in the $({\mathrm{Ha}}_{{\theta }_1},{\mathrm{Re}}_1)$ plane when $\mathrm{Ro}=-3/4$ and $\mathrm{Rb}=-1, m=1, q=3r_0^{-1}$, and $r=1.5r_0$. The neutral stability curve is obtained by maximizing the growth rate over $k$ for $\mathrm{Pm}=10$ and $\mathrm{Pm}=1$ with the use of Eq. (32) with the parameters (62).

Figure 14

The growth rate $\mathrm{Re}\left(\lambda \right)$ in units of ${\mathrm{\Omega }}_{\mathrm{in}}$ over a range of magnetic Rossby number $\mathrm{Rb}$ for the Keplerian ($\mathrm{Ro}=-3/4$) flow with ${\mathrm{Re}}_{\mathrm{in}}={10}^4, {\mathrm{Ha}}_{\theta }={10}^2, \mathrm{Pm}=0$, and $m=1$, in the case of the long axial wavelength ($\zeta =0.02/\zeta =0.336$ and $k={10}^{-4}{d}^{-1}$, left/middle panels) and short axial wavelength ($\zeta =0.98$ and $k=3.5d^{-1}$, right panel). The dotted blue line comes from the dimensionless Taylor-Couette boundary value problem with the boundary conditions corresponding to the perfectly conducting walls. The red and dashed green lines correspond to the Hain-Lüst dispersion relation (33) and the WKB approximation (79). The radial wave number is set to be $\widehat{q}=\zeta /(1-\zeta )$.

Figure 15

The growth rate $\mathrm{Re}\left(\lambda \right)$ in the units of ${\mathrm{\Omega }}_{\mathrm{in}}$ from BVP (dash-dotted blue), WKB approximation (dashed green), and Hain-Lüst (red) versus the axial wave number $k$ (in units of $d^{-1}$) for perfectly conducting boundaries. We set $\mathrm{Rb}=-1$ with $\zeta =0.02$ (left) and $\zeta =0.366$ (middle) and $\mathrm{Rb}=0$ with $\zeta =0.98$ (right). The parameter space is the same as in Fig. 14.

Figure 16

Growth rate magnitude in the Rossby plane ($\mathrm{Rb},\mathrm{Ro}$) from the boundary value problem with perfectly conducting boundaries for the upper panels and from Hain-Lüst dispersion relation (32) for the lower panels. The geometry correspond to $\zeta =0.02, k={10}^{-4}{d}^{-1}$ (left column), $\zeta =0.366, k={10}^{-4}{d}^{-1}$ (middle column), and $\zeta =0.98, k=3.5d^{-1}$ (right column). The parameter space is the same as in Fig. 14, and stability is represented in white.