Gravity waves' modulational instability under the effect of drag coefficient in the ocean

The research on oceanic gravity waves interacting with a drag coefficient has drawn a lot of attention. The interaction of these waves with a drag coefficient was recently found to be significant when modeling the propagation of these gravity waves. In this framework, the configuration involving the drag coefficient is of special interest. The gravity wave considered here is chosen to be unstable to two kinds (amplitude and phase) of perturbations. Given the complexity of the process to be investigated, it is necessary for us to make use of Miles' theory in order to better model the evolution of these gravity waves propagating in deep water under the effect of drag coefficient, using the deterministic approach (well-known as the Benjamin-Feir method), and the statistical approach (also known as Klimontovich's statistical average method) which is used starting from the Wigner Moyal transform. This study is performed to contribute to the understanding of the drag coefficient to the amplitudes (or phases) modulations of the driven waves: modulations that can sometimes accidentally trigger unpredictable extreme gravity waves.