Bell-Plesset(BP)effect caused perturbation growth plays an important role in better understanding of characteristics of the convergence effect.Governing equations for multi-mode perturbation growth on a cylindrically convergent interface are derived.The second-order weakly nonlinear(WN)solutions for two-mode perturbations at the interface which is subject to uniformly radical motion are obtained.Our WN theory is consistent with the numerical result in terms of mode-coupling effect in converging Richtmyer-Meshkov instability.Nonlinear mode-coupling effects will cause irregular deformation of the convergent interface.The mode-coupling behavior in convergent geometry depends on the mode number,Atwood number A and convergence ratio Cr.The A =-1.0 at the interface results in larger perturbation growth than A = 1.0.The growth of generated perturbation modes from two similar modes at the initial stage are smaller than that from two dissimilar modes.