Conservation style of the extra invariant for Rossby waves

We consider the dynamics of a system of Rossby waves with nonlinear interaction. It has been shown that such a system–besides the energy and momentum (enstrophy)–has an extra invariant, which is conserved approximately in the limit of small wave amplitudes. This invariant implies the anisotropy of the inverse cascade, when the energy is transferred from small scale eddies to the large scale zonal jets. The invariant has a quadratic (in wave amplitudes) term, which has a universal form, and a cubic term, whose form depends on the form of nonlinear interaction between Rossby waves.In this paper, we show that it is impossible to extend the extra invariant by higher order nonlinear terms (fourth order and higher) to obtain an exact conservation law. This fact holds irrespective of the form of nonlinearity (including the three-wave interaction). The extra invariant is similar to the adiabatic invariants in the theory of dynamical systems.We also show that on a “long time scale”, the cubic part of the invariant can be dropped (without sacrificing the accuracy of conservation), so that the extra invariant is given by the universal quadratic part.Finally, we show that the extra invariant mostly represents large scale modes, even to a greater extent than the energy. The presence of a small dissipation destroys the conservation of the enstrophy, which is based on small scale modes. At the same time, the extra invariant, along with the energy, remains almost conserved.