Mass transport at the bottom of propagating surface waves over a rippled bottom

Abstract

The sea surface can be described by means of the superposition of many sinusoidal functions. However, quite often the amplitude of each component turns out to be much smaller than its wavelength, and any component evolves independently of the others. Hence, it is common to investigate the dynamics of a simple monochromatic surface wave. Hereinafter, the flow generated by a monochromatic surface wave within the bottom boundary layer over a rippled sea bed is determined by means of the numerical integration of vorticity and continuity equations. The forcing term that drives the fluid motion within the boundary layer is evaluated assuming that the steepness of the monochromatic surface wave is much smaller than one and considering the first term of the Stokes expansion. Even though the irrotational flow that forces the viscous rotational flow near the sea bottom is symmetric with respect to the ripple crests, Blondeaux and Vittori [“A route to chaos in an oscillatory flow: Feigenbaum scenario,” Phys. Fluids A 3(11), 2492–2495 (1991a)] showed that the symmetry of the flow field is broken when the Reynolds number becomes larger than a threshold value Rδ,t1 that depends on the geometrical characteristics of the ripples. The results of Blondeaux and Vittori [“A route to chaos in an oscillatory flow: Feigenbaum scenario,” Phys. Fluids A 3(11), 2492–2495 (1991a)] suggest that, when the Reynolds number is larger than Rδ,t1 but not too far from it, a steady current is also generated. Hereinafter, the steady velocity component is determined as a function of the ripple characteristics.