  It has recently been suggested that the tidal deformation of a neutron star
excites daughter p- and g-modes to large amplitudes via a quasi-static
instability. This would remove energy from the tidal bulge, resulting in
dissipation and possibly affecting the phase evolution of inspiralling binary
neutron stars and hence the extraction of binary parameters from gravitational
wave observations. This instability appears to arise because of a large
three-mode interaction among the tidal mode and high-order p- and g-modes of
similar radial wavenumber. We show that additional four-mode interactions enter
into the analysis at the same order as the three-mode terms previously
considered. We compute these four-mode couplings by finding a volume-preserving
coordinate transformation that relates the energy of a tidally deformed star to
that of a radially perturbed spherical star. Using this method, we relate the
four-mode coupling to three-mode couplings and show that there is a near-exact
cancellation between the destabilizing effect of the three-mode interactions
and the stabilizing effect of the four-mode interaction. We then show that the
equilibrium tide is stable against the quasi-static decay into daughter p- and
g-modes to leading order. The leading deviation from the quasi-static
approximation due to orbital motion of the binary is considered; while it may
slightly spoil the near-cancellation, any resulting instability timescale is at
least of order the gravitational-wave inspiral time. We conclude that the
p-/g-mode coupling does not lead to a quasi-static instability, and does not
impact the phase evolution of gravitational waves from binary neutron stars.
