J. Plasma Physics (2005), vol. 40, part 3, pp. 1–5. c  2005 Cambridge University Press DOI: 10.1017/S0000000000000000 Printed in the United Kingdom 1 First and Second Order Accurate Implicit Difference Schemes for the Numerical Resolution of the Generalized Charney-Obukhov and Hasegawa-Mima Equations T. Kaladze, J. Rogava, L. Tsamalashvili, M. Tsiklauri, I. Vekua Institute of Applied Mathematics, Tbilisi State University, 2 University Str., 0143 Tbilisi, Georgia (Received 10 July 2005) Abstract. In the present work, first and second order accurate implicit difference schemes for the numerical solution of the nonlinear generalized Charney-Obukhov and Hasegawa-Mima equations with scalar nonlinearity are constructed. On the basis of numerical calculations accomplished by means of these schemes, the dy- namics of two-dimensional nonlinear solitary vortical structures is studied. The problem of stability for the first order accurate semi-discrete scheme is investigated. The dynamic relation between solutions of the generalized Charney-Obukhov and Hasegawa-Mima equations is established. It is shown that, in spite of the existing opinion, the scalar nonlinearity in case of the generalized Hasegawa-Mima equation develops monopolar anticyclone, while in case of the generalized Charney-Obukhov equation develops monopolar cyclone. 1. Statement of the problem and implicit difference schemes Generalized Charney-Obukhov (GChO) and Hasegawa-Mima (GHM) equations de- scribe propagation dynamics of nonlinear solitary vortical structures in geophysical flows and magnetized plasmas, respectively. In the frame of reference moving with velocity v along the axis OX, these dimensionless equations can be written in the following form: ∂ (Δψ − γψ) ∂t + β ∂ψ ∂x − v ∂ (Δψ − γψ) ∂x + J (ψ, Δψ) ± αψ ∂ψ ∂x =0, (1) where α, β and γ are positive constants defined through physical characteristics of the medium; “Plus” sign at α defines the GChO equation when γ =1,α = β = v R is the dimensionless Rossby velocity, and ψ is the variable part of fluid depth. “Minus” sign at α corresponds to the GHM equation when γ =1,β = v d is the dimensionless drift velocity, and ψ is the perturbed potential. The Jacobian (Poisson bracket) J (ψ, Δψ)= ∂ψ ∂x ∂ Δψ ∂y − ∂ψ ∂y ∂ Δψ ∂x describes the contribution of so-called vectorial nonlinearity, while the Korteweg-de Vries (KdV) type last term in Eqs. (1) describes the contribution of so-called scalar nonlinearity.