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

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 \varPhi 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+\sqrt{1-4R})^-2]^-1/2(b-a) if (\varPhi/ρ₀)_m≳2ab to b-a as (\varPhi/ρ₀)_m decreases below 2ab, R being (\varOmega₀²/\varOmega₀^'2)_m(ρ₀'/ρ₀r)_m, ρ₀(r) the density, \varOmega₀(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.