  Motivated by problems arising in geophysical fluid dynamics, we investigate
resonant and near resonant wave interactions in nonlinear wave equations with
quadratic nonlinearity, We place a special focus on interactions between slow
wave modes, with zero frequency in the linear limit, and fast modes. These
regularly occur in geophysical fluid systems with conserved potential vorticity
or similar conserved quantities. A general multi-scale asymptotic expansion is
used to show how the higher order nonlinear interaction coefficients are
derived as a combination of the first order terms arising at the triad
interaction level. From the general nth-order interaction coefficient we
present a proof by induction how the limiting effect for particular
combinations of slow and fast modes pushes their interactions to a slower
timescale, and we show how this is linked to the form of the conserved
quantities. We compare near resonant expansions with exact resonant expansions,
and show how near-resonances allow higher order expansions to be reduced to
just the most dominant contributions. These contributions then occur at one
order higher in the expansion when compared to the analogous exact resonance
expansion.
