Rossby wave extra invariant in the Galerkin approximation

The non-linear system of Rossby waves or plasma drift waves is known to have a unique adiabatic-like extra invariant in addition to the energy and enstrophy. This invariant is physically significant because its presence implies the generation of zonal flow. The latter is known to slow down the anomalous transport of temperature and particles in nuclear fusion with magnetic confinement. However, the derivation of the extra invariant - unlike the energy and enstrophy - is based on the continuum of resonances, while in numerical simulations there are only finite number of resonances. We show that precisely the same invariant takes place in the Galerkin approximations (even of low order, with a few ODEs). To show this we make variation of boundary conditions, when the solution is periodic in different directions. We also simplify the derivation of the extra conservation.