  We study the role of nonlinear effects on tidally-excited internal gravity
waves in stellar radiation zones in exoplanetary or binary systems. We are
partly motivated to study tides due to massive short-period hot Jupiters, which
preferentially orbit stars with convective cores, for which wave breaking near
the stellar centre cannot operate. We develop a theory (and test it with
numerical calculations) for the nonlinear excitation of super-harmonic
"secondary" waves (with frequencies $2\omega_p$) by a "primary" tidal wave
(with frequency $\omega_p$) near the interface between the radiation zone and
convective envelope. These waves have the same horizontal phase speeds to
leading order, and this nonlinear effect could contribute importantly to tidal
dissipation if the secondary waves can efficiently damp the primary. We derive
criteria involving the orbital and stellar parameters required to excite these
secondary waves to large amplitudes using a local model of the
radiative/convective interface, which we convert to apply to tides in a
spherical star. We numerically evaluate the critical amplitudes required for
this new nonlinear effect to become important using stellar models, comparing
them to the "conventional" criteria for wave breaking in radiative cores and
the application of WKBJ theory near convective cores. The criteria for this new
effect are easier to satisfy than the conventional measures of nonlinearity in
$1.4$ and $2M_\odot$ stars on the main-sequence. We predict nonlinear effects
to be important even for planetary-mass companions around the latter, but this
effect is probably less important in stars with radiative cores.

Solar and Stellar Astrophysics;Earth and Planetary Astrophysics