On symmetries and conservation laws of the Majda–Biello system

In 2003, A.J. Majda and J.A. Biello derived and studied the so-called reduced equations for equatorial baroclinic–barotropic waves, to which we refer as to the Majda–Biello system. The equations in question describe the nonlinear interaction of long-wavelength equatorial Rossby waves and barotropic Rossby waves with a significant midlatitude projection in the presence of suitable horizontally and vertically sheared zonal mean flows.Below we present a Hamiltonian structure for Majda–Biello system and describe all generalized symmetries and conservation laws for the latter. It turns out that there are only three symmetries corresponding to xx-translations, tt-translations and to a scaling of tt, xx, uu and vv, and four conservation laws, one of which is associated with the conservation of energy, the second conserved quantity is just the Hamiltonian functional and the other two are Casimir functionals of the Hamiltonian operator admitted by our system. Our result provides inter alia a rigorous proof of the fact that the Majda–Biello system has just the conservation laws mentioned in the paper by Majda and Biello.