An instability due to the nonlinear coupling of p-modes to g-modes: Implications for coalescing neutron star binaries

A weakly nonlinear fluid wave propagating within a star can be unstable to three-wave interactions. The resonant parametric instability is a well-known form of three-wave interaction in which a primary wave of frequency $omega_a$ excites a pair of secondary waves of frequency $omega_b+omega_c simeq omega_a$. Here we consider a nonresonant form of three-wave interaction in which a low-frequency primary wave excites a high-frequency p-mode and a low-frequency g-mode such that $omega_b+omega_c >>omega_a$. We show that a p-mode can couple so strongly to a g-mode of similar radial wavelength that this type of interaction is unstable even if the primary wave amplitude is small. As an application, we analyze the stability of the tide in coalescing neutron star binaries to p-g mode coupling. We find that the equilibrium tide and dynamical tide are both p-g unstable at gravitational wave frequencies f_{gw} > 20 Hz and drive p-g pairs to significant energies on very short timescales (much less than the orbital decay time). Resonant parametric coupling to the tide is, by contrast, either stable or drives modes at a much smaller rate. We do not solve for the saturation of the instability and therefore cannot say precisely how it influences neutron star binaries. However, we show that if even a single daughter mode saturates near its wave breaking amplitude, the p-g instability of the equilibrium tide: (i) induces significant orbital phase errors ($Delta phi$ > 1 radian) that accumulate primarily at low frequencies (f_{gw} < 50 Hz) and (ii) heats the neutron star core to T~10^{10} K. Since there are >100 unstable daughters, $Delta phi$ and T are potentially much larger than these values. Tides might therefore significantly influence the gravitational wave signal and electromagnetic emission from neutron star binaries at much larger orbital separations than previously thought.