Hamiltonian structure for the Charney-Hasegawa-Mima equation in the asymptotic model regime

We show that the Charney-Hasegawa-Mima (CHM) equation in the asymptotic model (CHM-AM) regime possesses the non-canonical Hamiltonian structure. The CHM-AM corresponds to the CHM equation in the asymptotic limit of length scales large compared to the Rossby deformation radius. It is shown that the Hamiltonian structure of the CHM-AM cannot be derived directly from that of the CHM equation by taking a simple limit of a length scale. Both the two-dimensional (2-D) Euler equation and the CHM-AM are regarded as special cases of a generalized 2-D fluid system, the so-called α-turbulence system. The existence of the Hamiltonian structure of the CHM-AM obtained in this study and that of the 2-D Euler equation implies the existence of the Hamiltonian structure of the α-turbulence system.