NONLINEAR MHD KELVIN-HELMHOLTZ INSTABILITY IN A PIPE

Nonlinear MHD Kelvin-Helmholtz（K-H）instability in a pipe is treated with the deriva-tive expansion method in the present paper The linear stability problem was discussed in the past byChandrasekhar（1961）and Xu et al.（1981）.Nagano（1979）discussed the nonlinear MHDK-H instability with infinite depth.He used the singular perturbation method and extrapolated the ob-tained second order modifier of amplitude vs.frequency to seek the nonlinear effect on the instabilitygrowth rate γ.However,in our view,such an extrapolation is inappropriate.Because when the instabili-ty sets in,the growth rates of higher,order terms on the right hand side of equations will exceed the cor-responding secular producing terms,so the expansion will still become meaningless even if the secularproducing terms are eliminated.Mathematically speaking,it’s impossible to derive formula（39）when γ02 is negative in Nagano’s paper.Moreover,even as early as γ02→O+,the expansion be-comes invalid because the 2nd order modifier γ2（in his formula（56））tends to infinity.This weak-ness is removed in this paper,and the result is extended to the case of a pipe with finite depth.