Equal-diffusivity magnetorotational channel (${\omega }_A^2=A(2\Omega -A),\gamma =A-\nu k_z^2$)

For $0<A<2\Omega$ and equal kinematic viscosity and magnetic diffusivity, a vertical wavevector admits $\mathbf{v}=(V,V,0)$ and $\mathbf{b}=(i{\omega }_{A}V/A,-i{\omega }_{A}V/A,0)$ in Alfvén-speed units. Substitution into the magnetorotational instability equations gives the displayed frequency and growth rate. In a Keplerian disk, growth of this polarization requires $B_z^2/({\mu }_0\rho )>5\nu \Omega /4$. This is a bound for this selected channel and its wavelength, not a universal threshold for every MRI mode in an unbounded domain.

For complex velocity amplitudes $v_x=v_y=V$ and magnetic amplitudes $b_x=-b_y=B$ in Alfvén-speed units, phase averaging gives the displayed stress. For fixed horizontal velocity energy, equal radial and azimuthal components maximize their positive Reynolds stress. Opposite equal magnetic components maximize the outward tension contribution of the Maxwell stress tensor, since $-B_x{B}_y$ is then positive. These conclusions follow from $2|ab|\le |a{|}^2+|b{|}^2$. The vertical channel transports angular momentum outward and extracts energy from decreasing angular velocity.