Corks, Buoyancy, and Wave-Particle Orbits

Abstract

The motion of open ocean waves and associated water-particle orbits constitute a focused topic that occupies a lasting niche in our arsenal of simple, pedagogically compelling things to offer introductory students in the geosciences. The topic provides a vehicle for introducing the general subject of wave behavior, a universally important phenomenon in the geosciences; it challenges students to understand counterintuitive ideas regarding the difference between wave and water-particle motions, serves as a base for advanced topics (for example, oscillatory ripple formation), and is readily “accessible” to students in that wave motions and particle orbits can be easily demonstrated or “tested” in either lab or field conditions. But unless students are exposed to this topic in a manner that goes well beyond introductory-text explanations, possibly in advanced courses, they are not apt to gain an understanding of ocean-wave behavior beyond that provided by purely kinematic explanations. Because of the pedagogical importance of gravity waves – of which ocean waves are an example – and because ambiguities exist in current introductory-text descriptions of waves, we assemble and summarize a dynamical explanation of their behavior. A notable bonus of this explanation is that it provides a simple, concise introduction to the ideas of buoyancy and gravitational forces – also universally important subjects in the geosciences – and how these forces interact during wave motion.Specifically, the orbit of a water particle beneath a train of ocean waves involves a clockwise motion when viewed from a perspective where the waves are moving from left to right. The vertical component of this motion is associated dynamically with pressure fluctuations about the average pressure state such that the buoyancy force exceeds the gravitational force beneath wave troughs, and the gravitational force exceeds the buoyancy force beneath wave crests. The horizontal component of the motion is associated with these pressure fluctuations wherein the pressure decreases horizontally in the direction of wave motion between wave crests and their leading troughs, and the pressure increases horizontally in the direction of wave motion between wave troughs and their leading crests. These forces alternately accelerate and decelerate the water with simultaneous vertical and horizontal components leading to approximately circular orbits. Peak water (orbit) speeds decrease with increasing wave speed. The attenuation of orbits with increasing depth is related to downward attenuation of the fluctuating pressure field, rather than being attributable to frictional damping.This dynamical explanation can be concisely presented at several levels. The first involves a qualitative description that appeals to graphical representations of the pressure and velocity fields beneath waves. The second appeals to a simple trigonometric representation of waves and the simplest possible form of Euler's equations combined with the approximation of hydrostatic conditions. The third involves solving linearized forms of Euler's equations.