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 by Chandrasekhar(1961)and Xu et al...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 by Chandrasekhar(1961)and Xu et al.(1981).Nagano(1979)discussed the nonlinear MHD K-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 instability growth 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 secular producing terms are eliminated.Mathematically speaking,it's impossible to derive formula(39) when γ_0~2 is negative in Nagano's paper.Moreover,even as early as γ_0~2→O^+,the expansion be- comes invalid because the 2nd order modifier γ_2(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.展开更多
基金The project is supported by the National Natural Science Foundation of China.
文摘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 by Chandrasekhar(1961)and Xu et al.(1981).Nagano(1979)discussed the nonlinear MHD K-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 instability growth 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 secular producing terms are eliminated.Mathematically speaking,it's impossible to derive formula(39) when γ_0~2 is negative in Nagano's paper.Moreover,even as early as γ_0~2→O^+,the expansion be- comes invalid because the 2nd order modifier γ_2(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.
