FIG. 6: Growth rate, Re(γ) (panels (a), (b), (c)), and frequency, ω = Im(γ) (panels (d), (e), (f)), vs. kz in the global case with the narrow gap η̂ = 0.85 and power-law angular velocity profile with constant Ro = 3.5 < RoULL in the presence of conducting (blue) and insulating (red) boundary conditions imposed on the cylinders. In all panels, β = 100, Pm = 10−3, while (Ha,Re) = (2000, 2·106) in panels (a) and (d), (Ha,Re) = (2800, 3·106) in panels (b) and (e), and (Ha,Re) = (3200, 3.5·106) in panels (c) and (f). Black dashed curve in each panel is accordingly the growth rate or frequency resulting from the local WKB dispersion relation (Eq. 4) for the same values of the parameters corresponding to that panel and a fixed kr0 = π/δ, where δ = 1− η̂ is the gap width in units of ro. It is seen that in the local and global cases, the shape of the dispersion curves is qualitatively same, with comparable growth rates, frequencies and corresponding wavenumbers, however, the boundary conditions tend to shift the growth rates towards lower axial wavenumbers, while the frequencies towards larger wavenumbers relative to those in the local case. Also, insulating boundaries lower the growth rate about three times. As for the frequencies, they are close to each other for both these boundaries and smaller than those in the local case at a given kz.
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