Semi-Ellipse Theorem for the Heterogeneous Rotating Flow with Respect to Three-Dimensional Disturbances

Abstract

The semi-ellipse theorem for the heterogeneous rotating flow with respect to the two-dimensional disturbance is extended to the three-dimensional disturbance. Two assumptions of positive density gradient and positive Rayleigh-Synge’s discriminant Φ are made. It is shown that the complex angular phase-velocity of any unstable mode must lie within the semi-ellipse whose major axis equals b−a, while its minor axis changes in length from
[1+4(a⁄b)R(1+\sqrt1−4R)⁻²]^−1⁄2(b−a) if (Φ⁄ρ₀)_m\gtrsim2ab
to b−a as (Φ⁄ρ₀)_m decreases below 2ab, R being (Ω₀²⁄Ω₀^′2)_m(ρ₀′⁄ρ₀r)_m, ρ₀(r) the density, Ω₀(r) the angular velocity of rotating flow, a and b its lower and upper bounds, respectively, and r the radial distance. The prime denotes differentiation with respect to r and the suffix m means the minimum value. R must be less than 1/4 by the necessary condition for instability.