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We study numerically the region of convergence of the normal formtransformation for the case of the Charney-Hasagawa-Mima (CHM) equation toinvestigate whether certain finite amplitude effects can be described in normalcoordinates. We do this by taking a Galerkin truncation of four Fourier modesmaking part of two triads: one resonant and one non-resonant, joined togetherby two common modes. We calculate the normal form transformation directly fromthe equations of motion of our reduced model, successively applying thealgorithm to calculate the transformation up to $7^\textrm{th}$ order toeliminate all non-resonant terms, and keeping up to $8$-wave resonances. Wefind that the amplitudes at which the normal form transformation diverge veryclosely match with the amplitudes at which a finite-amplitude phenomenon called$precession$ $resonance$ (Bustamante $et$ $al.$ 2014) occurs, characterised bystrong energy transfers. This implies that the precession resonance mechanismcannot be explained using the usual methods of normal forms in wave turbulencetheory, so a more general theory for intermediate nonlinearity is required.

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