On the decrease of kinetic energy with depth in wave–current interactions

Abstract

We study wave-current interactions in two-dimensional water flows with constant vorticity over a flat bed. We establish decay rates for the velocity beneath spatially periodic surface waves without any restrictions on the wave amplitude. The approach relies on complex function theory and overcomes the intricacies inherent to nonlinear flow patterns by taking advantage of specific structural properties of the governing equations for water waves.

On the energy of nonlinear water waves

“…[27] for numerical investigations that provide some insight into this question). Rigorous results concerning monotonicity properties of the kinetic energy for nonlinear waves do exist [7,12,13]; however, these establish an exponential decrease of the kinetic energy with respect to vertical depth (for fluid motion beneath the wave trough) and do not pertain to the behaviour along the free-surface. Relation (4.11) is therefore a new expression that is worthy of further analytical, and numerical, investigations for nonlinear wave solutions.…”

“…The theory underlying ocean wave energy is a nascent field of scientific research, having been developed in recent decades, and suffers from the fundamental limitation that most of the state-of-the-art is strongly contingent on invoking linear approximations [6]. While there do exist some results characterizing energy properties for nonlinear water waves (see, for example, the 'classical' papers [7][8][9][10][11], and more recent developments [12,13]), due to the technical complexities inherent in nonlinear analysis, the literature is sparse in terms of both quantity and scope. Recent advances in mathematical analysis have enabled progress in tackling fundamental questions concerning nonlinear waves, and this article presents new results concerning the excess kinetic and potential energies for exact nonlinear periodic and travelling irrotational water waves.…”

On three-dimensional free surface water flows with constant vorticity

“…Notwithstanding that a complete understanding of two-dimensional rotational water flows is still elusive many significant analytical advances had been made in the last two decades. They concern several quantitative and qualitative aspects like: existence results by means of bifurcation tools [7,19,33,34,52,58], qualitative properties of the free surface [6,9,12,32,59] as well as of the flow beneath [8,10,1], stratification [18,21,25,26,35,36,37], observed phenomena like stagnation points and critical layers [15,19,24,25,66].…”