Nonlinear MHD Kelvin-Helmholtz instability in a pipe

Nonlinear MHD Kelvin-Helmholtz (K-H) instability in a pipe is treated with the derivative expansion method in the present paper. The linear stability problem was discussed in the past by Chandrasekhar (1961)[1] and Xu et al. (1981).[6]Nagano (1979)[3] discussed the nonlinear MHD K-H instability with infinite depth. He used the singular perturbation method and extrapolated the obtained second order modifier of amplitude vs. frequency to seek the nonlinear effect on the instability growth rate γ. However, in our view, such an extrapolation is inappropriate. Because when the instability sets in, the growth rates of higher order terms on the right hand side of equations will exceed the corresponding secular producing terms, so the expansion will still become meaningless even if the secular producing terms are eliminated. Mathematically speaking, it's impossible to derive formula (39) when γ 0 2 is negative in Nagano's paper.[3]Moreover, even as early as γ 0 2 → O+, the expansion becomes invalid because the 2nd order modifier γ2 (in his formula (56)) tends to infinity. This weakness is removed in this paper, and the result is extended to the case of a pipe with finite depth.