Previous
abstract 
Next abstract
Differentially rotating magnetized disks are subject to the Balbus-Hawley instability. With the local approximation of the comoving frame, we study the stability of the disk with seed magnetic fields in either poloidal or toroidal direction. The Balbus-Hawley instability has no stability threshold with respect to {B$. Inclusion of the resistivity (kinematic viscosity is not essential), however, leads to the appearance of instability threshold. In the Keplerian disk with uniform vertical magnetic field with Alfv\'en speed $\upsilon_A$, the criterion for the instability is $\upsilon_A^2(k_z^2 \upsilon_A^2-3\Omega^2)+(\eta/4\pi)^2 k_z^2\Omega^2 < 0$, where $\Omega$ is the angular velocity, $k_z$ the wavenumber in $z$-direction, and $\eta$ the resistivity. When $\eta \ne 0$, there is a critical seed magnetic field $B_{zc} = (\rho/12\pi)^{1/2} \eta k_z$ below which the Balbus-Hawley mode is stabilized. Since the resistivity depends on the instability-induced turbulent magnetic fields as $\eta=\eta( \delta B^2 )$, the $\alpha$ parameter of the angular momentum transport of the disk, $\alpha= \delta B^2 /(4\pi\rho c_s^2)$ is determined by the marginal stability condition. Using the resistivity expression by Ichimaru (1975), the marginal stability condition yields $\alpha = (6/\beta_0^{1/2}) (k_{\perp}/k_z)_{max}$ where $(k_{\perp}/k_z)_{max}$ is the ratio of $k_{\perp}$ and $k_z$ evaluated using the wavenumber at maximum growth, and $\beta_0$ is the plasma $\beta$ due to the unperturbed $B_z$ fields. The $\alpha$ value for the toroidal magnetic field is the order of $1/\beta_0$.
The above theory will be compared with the results of 2D and 3D MHD simulation.
