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 $O(10^{-2})$.
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