  The r-modes of neutron stars can be driven unstable by gravitational
radiation. While linear perturbation theory predicts the existence of this
instability, linear theory can't provide any information about the nonlinear
development of the instability. The subject of this paper is the weakly
nonlinear regime of fluid dynamics. In the weakly nonlinear regime, the
nonlinear fluid equations are approximated by an infinite set of oscillators
which are coupled together so that terms quadratic in the mode amplitudes are
kept in the equations of motion. In this paper, the coupling coefficients
between the r-modes are computed. The stellar model assumed is a polytropic
model where a source of buoyancy is included so that the Schwarzschild
discriminant is nonzero. The properties of these coupling coefficients and the
types of resonances possible are discussed in this paper. It is shown that no
exact resonance involving the unstable $l=m=2$ r-mode occur and that only a
small number of modes have a dimensionless coupling constant larger than unity.
However, an infinite number of resonant mode triplets exist which couple
indirectly to the unstable r-mode. All couplings in this paper involve the
l>|m| r-modes which only exist if the star is slowly rotating. This work is
complementary to that of Schenk et al (2002) who consider rapidly rotating
stars which are neutral to convection.
