  Previously we found that large amplitude $r$-modes could decay
catastrophically due to nonlinear hydrodynamic effects. In this paper we found
the particular coupling mechanism responsible for this catastrophic decay, and
identified the fluid modes involved. We find that for a neutron star described
by a polytropic equation of state with polytropic index $\Gamma=2$, the
coupling strength of the particular three-mode interaction causing the decay is
strong enough that the usual picture of the $r$-mode instability with a flow
pattern dominated by that of an $r$-mode can only be valid for the
dimensionless $r$-mode amplitude less than $O(10^{-2})$.
