  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.
