Semicircle Theorem for the Rotating Flow in an Azimuthal Magnetic Field with Respect to Azimuthal Perturbations

Abstract

The instability of the heterogeneous rotating flow in an azimuthal magnetic field is investigated for the flute-mode disturbance. Both the density ρ₀ and the magnetic field intensity H₀⁄r divided by the radial distance are assumed to have non-negative radial gradients. It is shown that the complex angular phase-velocity for any unstable flute-mode must lie within the semicircle in the upper half-plane which has
\sqrt(b−a)²−4(V_a²⁄r²)_m
for diameter. If the square of this expression is negative the rotating flow is stable. Here a and b are, respectively, the lower and the upper bound of angular velocity of the rotating flow, V_a(r) is the Alfvén velocity and the suffix m means the minimum value.