Coupling between a surface-wave spectrum and an internal wave: modulational interaction

Using a simple three-layer model of the ocean, we study a generation mechanism for the lowest internal-wave mode by nonlinear coupling to modulations of the surface-wave spectrum. We first examine the case of a narrow-band surface-wave spectrum, applying a method developed by Alber (1978) to derive a transport equation for the spectral density. Alber demonstrated that, when the spectral width (in the main wave direction) exceeds some critical value, the spectrum is stable against modulational perturbation (i.e. the Benjamin–Feir-type instability is suppressed). We show, however, that, for a stratified ocean, a modulational instability may persist because of a coupling between a ‘modulational mode’ of the surface-wave spectrum and an internal wave. The growth rate is calculated for a simple model of the angular distribution of the spectrum. It turns out that an important parameter is 〈(∇ξ)²〉/Δθ, the ratio between the averaged square of the wave steepness, and the angular width of the spectrum.For appreciable growth one must have roughly \[ 2\times 10^{-3}\lesssim kd \langle (\nabla\zeta)^2\rangle / \Delta\theta, \] where k is a characteristic wavenumber for the surface-wave spectrum, and d is the depth of the thermocline (50-100 m). This condition is probably too limiting for the above-mentioned modulational instability to be of any practical interest in the oceans.We also consider the broad-band case of modulational interaction, and show the connection with incoherent three-wave interactions.