On the convergence of the normal form transformation in discrete Rossby and drift wave turbulence

We study numerically the region of convergence of the normal form transformation for the case of the Charney-Hasagawa-Mima (CHM) equation to investigate whether certain finite amplitude effects can be described in normal coordinates. We do this by taking a Galerkin truncation of four Fourier modes making part of two triads: one resonant and one non-resonant, joined together by two common modes. We calculate the normal form transformation directly from the equations of motion of our reduced model, successively applying the algorithm to calculate the transformation up to $7^\textrm{th}$ order to eliminate all non-resonant terms, and keeping up to $8$-wave resonances. We find that the amplitudes at which the normal form transformation diverge very closely match with the amplitudes at which a finite-amplitude phenomenon called $precession$ $resonance$ (Bustamante $et$ $al.$ 2014) occurs, characterised by strong energy transfers. This implies that the precession resonance mechanism cannot be explained using the usual methods of normal forms in wave turbulence theory, so a more general theory for intermediate nonlinearity is required.