Can bottom friction suppress ‘freak wave’ formation?

Abstract

The paper examines the effect of the bottom stress on the weakly nonlinear evolution of a narrow-band wave field, as a potential mechanism of suppression of ‘freak’ wave formation in water of moderate depth. Relying upon established experimental studies the bottom stress is modelled by the quadratic drag law with an amplitude/bottom roughness-dependent drag coefficient. The asymptotic analysis yields Davey–Stewartson-type equations with an added nonlinear complex friction term in the envelope equation. The friction leads to a power-law decay of the spatially uniform wave amplitude. It also affects the modulational (Benjamin–Feir) instability, e.g. alters the growth rates of sideband perturbations and the boundaries of the linearized stability domains in the modulation wavevector space. Moreover, the instability occurs only if the amplitude of the background wave exceeds a certain threshold. Since the friction is nonlinear and increases with wave amplitude, its effect on the formation of nonlinear patterns is more dramatic. Numerical experiments show that even when the friction is small compared to the nonlinear term, it hampers formation of the Akhmediev/Ma-type breathers (believed to be weakly nonlinear ‘prototypes’ of freak waves) at the nonlinear stage of instability. The specific predictions for a particular location depend on the bottom roughness ks in addition to the water depth and wave field characteristics.