Metamorphosis of helical magnetorotational instability in the presence of axial electric current

This paper presents numerical linear stability analysis of a cylindrical Taylor-Couette flow of liquid metal carrying axial electric current in a generally helical external magnetic field. Axially symmetric disturbances are considered in the inductionless approximation corresponding to zero magnetic Prandtl number. Axial symmetry allows us to reveal an entirely new electromagnetic instability. First, we show that the electric current passing through the liquid can extend the range of helical magnetorotational instability (HMRI) indefinitely by transforming it into a purely electromagnetic instability. Two different electromagnetic instability mechanisms are identified. The first is an internal pinch-type instability, which is due to the interaction of the electric current with its own magnetic field. Axisymmetric mode of this instability requires a free-space component of the azimuthal magnetic field. When the azimuthal component of the magnetic field is purely rotational and the axial component is nonzero, a new kind of electromagnetic instability emerges. The latter, driven by the interaction of electric current with a weak collinear magnetic field in a quiescent fluid, gives rise to a steady meridional circulation coupled with azimuthal rotation.

Figure 1

Sketch of the problem.

Figure 2

Marginal Reynolds number (a) and the frequency (b) versus wave number for a hydrodynamically unstable flow with $\mu =0.2$ at various helicities $\gamma$ of rotational helical magnetic field with $\alpha =1,\beta =0$, and $\mathrm{Ha}=10.$

Figure 3

Marginal Reynolds number (a) and the frequency (b) versus the wave number for a hydrodynamically stable flow with $\mu =0.3$ at various helicities $\gamma$ of rotational helical magnetic field with $\alpha =1,\beta =0$, and $\mathrm{Ha}=10.$

Figure 4

Critical Reynolds number (a) and the frequency (b) versus the ratio of rotation rates of inner and outer cylinders $\mu$ at various helicities $\gamma$ of rotational helical magnetic field with $\alpha =1,\beta =0$ and $\mathrm{Ha}=10.$

Figure 5

Marginal Reynolds number versus wave number for hydrodynamically unstable $(\mu =0.2)$ and stable $(\mu =0.3)$ flows in the azimuthal magnetic field $(\alpha =0)$ generated only by the axial current in the liquid annulus $(\beta =0)$ with various magnitude $\gamma$ at $\mathrm{Ha}=10.$

Figure 6

Critical Reynolds number versus the ratio of rotation rates of inner and outer cylinders $\mu$ at various helicities $\gamma$ of purely rotational helical magnetic field with $\alpha =1,\beta =\gamma$, and $\mathrm{Ha}=10.$

Figure 7

Marginal Hartmann number versus wave number for purely electromagnetic $(\mathrm{Re}=0)$ stationary $(\omega =0)$ instabilities in the rotational magnetic field with $\alpha =1,\beta =0$ (a), $\alpha =1,\beta =\gamma$ (b), and $\alpha =0,\beta =0$, and $\beta \to \gamma$ (c) for both insulating and perfectly conducting cylinders at various axial currents defined by $\gamma .$

Figure 8

Streamlines (a) and the electric current lines (b) of the critical perturbation for the electromagnetic $(\mathrm{Re}=0)$ pinch-type instability $(\alpha =\beta =0)$.

Figure 9

Critical Hartmann number (a) and wave number (b) for purely electromagnetic $(\mathrm{Re}=0)$ stationary $(\omega =0)$ instabilities versus the axial current parameter $\gamma$ for different magnetic field configurations. The upper and lower branches for each configuration correspond to insulating and perfectly conducting cylinders.

Figure 10

Streamlines (a), isolines of the azimuthal velocity (b), the electric current lines (c), and the meridional magnetic flux lines (d) for the critical perturbation of electromagnetic $(\mathrm{Re}=0)$ rotational instability in the asymptotic case $\alpha \ll \beta =\gamma$.

Figure 11

Critical Hartmann number rescaled with the gap width $\lambda -1$ versus the rescaled inner radius $R_i={(\lambda -1)}^{-1}$ for the pinch instability $(\alpha =\beta =0)$ and for the instability driven by the electric current in a weak axial magnetic field ($\alpha \ll \beta =\gamma$) with both cylinders insulating (i), perfectly conducting (c), inner cylinder insulating and outer cylinder perfectly conducting (i/c). The Hartmann number for the latter instability (on the left axis) is additionally rescaled with the effective current density $\gamma /R_i.$

Figure 12

Critical Hartmann number and the wave number for the electromagnetic pinch instability versus $1-\beta /\gamma$ which defines the strength of free-space azimuthal component the magnetic field relative to the rotational component generated by axial current.