Modulational instability of co-propagating internal wavetrains under rotation

Weakly-nonlinear unidirectional long internal waves in a non-rotating frame are well described by the Korteweg-de Vries equation (KdV). Within the KdV framework, all isolated monochromatic wavetrains are stable to modulational instability. However, analysis of a coupled nonlinear Schrödinger equation system (CNLS) has shown that all systems of two co-propagating monochromatic wavetrains in the KdV are modulationally unstable. To take into account the effect of the background rotation of the Earth on long internal waves, this analysis is extended here to derive the CNLS for the rotation-modified KdV, or Ostrovsky, equation. Rotation stabilises wavetrain pairs when the wavelengths of both waves comprising the wavetrains are longer than the linear wave with maximum group velocity. The particular case when the wavetrains have different wavenumbers but the same linear group speed is emphasised.