Modulational instability of wave trains in a rotating ocean

The problem of modulational stability of quasi-monochromatic wave-trains propagating in a rotating fluid which provides both the small-scale Boussinesq dispersion and large-scale Coriolis dispersion is studied. We derive two-dimensional non-linear Schrödinger (NLS) equation from the basic set of Boussinesq equations for shallow water waves taking into account the Coriolis force caused by Earth’ rotation. For unidirectional waves propagating in one direction only the considered set of equations reduces to the Gardner–Ostrovsky equation which is applicable only within a finite range of wavenumbers. It is shown that the narrow-band wave-trains are modulationally stable for relatively small wavenumbers k kc, where kc is some critical wavenumber. The derived NLS equation is applicable even for waves with very small wavenumbers up to zero. The detailed analysis of coefficients of the NLS equation is presented both for one-dimensional and two-dimensional cases. The conditions of self-modulation and self-focussing are determined and presented graphically on the plane of parameters. Application of results obtained to the real oceanic conditions is discussed