Local study of helical magnetorotational instability in viscous Keplerian disks

Abstract

In this paper, regarding the recent detection of significant azimuthal magnetic field in some accretion disks such as protostellar (Donati et al. in Nature 438:466, 2005), the multi-fluid model has been employed to analysis the stability of Keplerian rotational viscous dusty plasma system in a current-free helical magnetic field structure. Using the fluid-Maxwell equations, the general dispersion relation of the excited modes in the system has been obtained by applying the local approximation method in the linear perturbation theory. The typical numerical analysis of the obtained dispersion relation in the high-frequency regime shows that the presence of azimuthal magnetic field component in Keplerian flow has a considerable role in the stability conditions of the system. It also shows that the magnetic field helicity has a stabilization role against the magnetorotational instability (MRI) in the system due to contraction of the unstable wavelength region and decreasing the maximum growth rate of the instability. In this sense, the stabilization role of the viscosity term is more considerable for HMRI (instability in the presence of azimuthal magnetic field component) than the corresponding MRI (instability in the absence of azimuthal magnetic field component). Moreover, considering the discovered azimuthal magnetic field in these systems, the MRI can be arisen in the over-all range of dust grains construction values in contract with traditional MRI. This investigation can greatly contribute to better understanding the physics of some astrophysical phenomena, such as the main source of turbulence and angular momentum transport in protostellar and the other sufficiently ionized astrophysical disks, where the azimuthal magnetic field component in these systems can play a significant role.

Appendix

Dispersion relation of the magnetized rotational viscous dusty plasma system has been reported as

$$\begin{aligned}& \bigl(\eta k^{2} - i \omega\bigr)^{2} \frac{k^{2}}{k_{z}^{2}} \biggl\{ i k^{2} v_{{Ai}}^{2} \biggl[\eta k^{2} - i \omega \\ & \qquad {} + \frac{k_{z}^{2}}{ k^{2}} \biggl( \frac{\kappa^{2}}{2\varOmega} + \varOmega_{i} \biggr) \frac{ i }{ \omega} \frac{{d}\varOmega}{{dlnr}} \biggr] \\ & \qquad {} +\biggl( \omega \varOmega_{i}^{2} +\omega \varOmega_{e}^{2} \frac{v_{{Ai}}^{2}}{v_{{Ae}}^{2}} + \frac{ i k_{z}^{2} v_{{Ai}}^{2}}{(\eta k^{2} - i \omega)}\biggr)\\ & \quad \quad {}\times (2\varOmega- \varOmega_{e} ) \biggl( \frac{\kappa^{2}}{2\varOmega} + \varOmega_{i} \biggr)+A\biggl[\bigl(\eta k^{2} - i \omega\bigr) \\ & \quad \quad {}\times \biggl(\eta k^{2}- i \omega- \frac{k_{z}^{2}}{k^{2}} i \varOmega_{d} d\biggr) \biggl( \frac{k^{2}}{k_{z}^{2}} \biggr)+( \varOmega_{e} - \varOmega_{d} )\\ & \quad \quad {}\times \biggl( \frac{\kappa^{2}}{ 2\varOmega} + \varOmega_{i} \biggr) +(2\varOmega- \varOmega_{e} ) \biggl( \frac{\kappa^{2}}{ 2\varOmega} - \varOmega_{d} \biggr)\biggr]\biggr\} \\ & \quad \quad {}\times \biggl\{ i k_{z}^{2} v_{{Ai}}^{2} \biggl[ \frac{- i }{\omega} \frac{{d}\varOmega}{ {dlnr}} \frac{( \frac{\kappa^{2}}{2\varOmega} - \varOmega_{e} )}{(\eta k^{2} - i \omega)} \biggl(\eta k^{2} - i \omega\\ & \quad \quad {}- \frac{k_{z}^{2}}{ k^{2}} i \varOmega_{i} d\biggr)+ \frac{k^{2}}{k_{z}^{2} (\eta k^{2} - i \omega)} \biggl(\eta k^{2} - i \omega- \frac{k_{z}^{2}}{ k^{2}} i \varOmega_{e} d\biggr)\\ & \quad \quad {}\times \biggl(\eta k^{2} - i \omega- \frac{k_{z}^{2}}{k^{2}} i \varOmega_{i} d\biggr)\biggr]- \frac{(2\varOmega+ \varOmega_{i} ) i k^{2} v_{{Ai}}^{2}}{(\eta k^{2} - i \omega)}\\ & \quad \quad {}\times \biggl( \frac{\kappa^{2}}{2\varOmega} - \varOmega_{e} \biggr)+\biggl[ \frac{v_{{Ai}}^{2} (\omega \varOmega_{e}^{2} )}{v_{{Ae}}^{2} (\eta k^{2} - i \omega)} \biggl(\eta k^{2} - i \omega\\ & \quad \quad {} - \frac{k_{z}^{2}}{k^{2}} i \varOmega_{i} d\biggr)+ \frac{(\omega \varOmega_{i}^{2} )}{(\eta k^{2} - i \omega)} \biggl(\eta k^{2} - i \omega- \frac{k_{z}^{2}}{k^{2}} i \varOmega_{e} d\biggr)\biggr]\\ & \quad \quad {}+A\biggl[ \frac{(2\varOmega- \varOmega_{d} )}{(\eta k^{2} - i \omega)} \biggl( \frac{\kappa^{2}}{2\varOmega} - \varOmega_{e} \biggr) \biggl(\eta k^{2} - i \omega- \frac{k_{z}^{2}}{k^{2}} i \varOmega_{i} d\biggr)\\ & \quad \quad {}+\biggl( \frac{k^{2}}{ k_{z}^{2}} \biggr) \biggl(\eta k^{2} - i \omega- \frac{k_{z}^{2}}{k^{2}} i \varOmega_{e} d\biggr) \biggl(\eta k^{2} - i \omega\\ & \quad \quad {}- \frac{k_{z}^{2}}{ k^{2}} i \varOmega_{i} d\biggr)-(2 \varOmega+ \varOmega_{i} )\biggl[ \frac{(\eta k^{2} - i \omega- \frac{k_{z}^{2}}{k^{2}} i \varOmega_{d} d)}{ (\eta k^{2} - i \omega)} \\ & \quad \quad {}\times \biggl( \frac{\kappa^{2}}{2\varOmega} - \varOmega_{e} \biggr)- \frac{(\eta k^{2} - i \omega- \frac{k_{z}^{2}}{k^{2}} i \varOmega_{e} d)}{(\eta k^{2} - i \omega)} \\ & \quad \quad {}\times\biggl( \frac{\kappa^{2}}{2\varOmega} - \varOmega_{d} \biggr)\biggr]\biggr]\biggr\} \\ & \quad = \biggl\{ \biggl( \frac{\kappa^{2}}{2\varOmega} - \varOmega_{e} \biggr) \bigl( \bigl( \eta k^{2} - i \omega \bigr) i k^{2} v_{{Ai}}^{2} +\omega \varOmega_{i}^{2} \bigr)\\ & \quad \quad {} + \biggl( \frac{\kappa^{2}}{2\varOmega} + \varOmega_{i} \biggr) \biggl[ i k_{z}^{2} v_{{Ai}}^{2} \biggl( \eta k^{2} - i \omega- \frac{k_{z}^{2}}{k^{2}} i \varOmega_{e} d \biggr) \frac{k^{2}}{ k_{z}^{2}} \\ & \quad \quad {}+ \frac{k_{z}^{2} v_{{Ai}}^{2}}{\omega} \frac{{d}\varOmega}{{dlnr}} \biggl( \frac{\kappa^{2}}{2\varOmega} - \varOmega_{e} \biggr) + \omega \varOmega_{e}^{2} \frac{v_{{Ai}}^{2}}{ v_{{Ae}}^{2}} \biggr]\\ & \quad \quad {} +A \biggl[ ( 2\varOmega- \varOmega_{d} ) \biggl( \frac{\kappa^{2}}{2\varOmega} - \varOmega_{e} \biggr) \biggl( \frac{\kappa^{2}}{2\varOmega} + \varOmega_{i} \biggr) + \biggl( \frac{k^{2}}{ k_{z}^{2}} \biggr) \\ & \quad \quad {}\times\bigl( \eta k^{2} - i \omega \bigr) \biggl[ \varOmega_{d} \biggl( \eta k^{2} - i \omega- \frac{k_{z}^{2}}{k^{2}} i \varOmega_{e} d \biggr)\\ & \quad \quad {} - \varOmega_{e} \biggl( \eta k^{2} - i \omega- \frac{k_{z}^{2}}{k^{2}} i \varOmega_{d} d \biggr) + \biggl( \frac{\kappa^{2}}{2\varOmega} + \varOmega_{i} \biggr)\\ & \quad \quad {}\times \biggl( \eta k^{2} - i \omega- \frac{k_{z}^{2}}{k^{2}} i \varOmega_{e} d \biggr) \biggr] \biggr] \biggr\} \biggl\{ i k_{z}^{2} v_{{Ai}}^{2} \bigl( \eta k^{2} - i \omega \bigr)\\ & \quad \quad {}\times \biggl( \frac{k^{2}}{k_{z}^{2}} \biggr) \biggl( \frac{- i }{\omega} \frac{{d}\varOmega}{{dlnr}} \biggl( \eta k^{2} - i \omega- \frac{k_{z}^{2}}{k^{2}} i \varOmega_{i} d \biggr) \\ & \quad \quad {}- ( 2\varOmega- \varOmega_{e} ) \biggr) -\omega \varOmega_{i}^{2} ( 2 \varOmega- \varOmega_{e} ) - ( 2\varOmega+ \varOmega_{i} ) \\ & \quad \quad {}\times\biggl( i k^{2} v_{{Ai}}^{2} \bigl( \eta k^{2} - i \omega \bigr) +\omega \varOmega_{e}^{2} \frac{v_{{Ai}}^{2}}{v_{{Ae}}^{2}} \biggr) \\ & \quad \quad {}-A \biggl[ \frac{\varOmega_{e}}{\varOmega_{d}} ( 2\varOmega+ \varOmega_{i} ) ( 2 \varOmega- \varOmega_{e} ) \biggl( \frac{\kappa^{2}}{2\varOmega} - \varOmega_{d} \biggr)\\ & \quad \quad {} + \biggl( \frac{k^{2}}{k_{z}^{2}} \biggr) \bigl( \eta k^{2} - i \omega \bigr) \biggl( ( \varOmega_{d} - \varOmega_{e} ) \biggl( \eta k^{2} - i \omega\\ & \quad \quad {}- \frac{k_{z}^{2}}{k^{2}} i \varOmega_{i} d \biggr) + \frac{\varOmega_{e}}{\varOmega_{d}} ( 2\varOmega+ \varOmega_{i} ) \biggl( \eta k^{2} - i \omega\\ & \quad \quad {}- \frac{k_{z}^{2}}{ k^{2}} i \varOmega_{d} d \biggr) \biggr) \biggr] \biggr\} \end{aligned}$$

where $A$ is defined by

$$\begin{aligned}& \frac{v_{{Ai}}^{2}}{v_{{Ad}}^{2}} \biggl( \frac{\omega \varOmega_{d}^{2}}{ ( 2\varOmega {-} \varOmega_{d} ) ( \frac{\kappa^{2}}{2\varOmega} {-} \varOmega_{d} ) + \frac{k^{2}}{k_{z}^{2}} ( \eta k^{2} {-} i \omega ) ( \eta k^{2} {-} i \omega {-} \frac{k_{z}^{2}}{k^{2}} i \varOmega_{d} {d} )} \biggr). \end{aligned}$$