Angular momentum and action in surface gravity waves: Application to wave‐current interaction

Abstract

When small‐amplitude surface gravity waves progress in deep water, all the fluid particles are observed to orbit in circles within the depth of wave influence. Each fluid particle therefore has angular momentum with respect to the center of its orbit, and the angular momentum vectors are directed parallel to the crests and troughs or perpendicular to the wave number. The angular momentum per unit volume is calculated for each fluid particle, and then, by vertical integration, the angular momentum per unit horizontal area is computed. The total energy and the magnitude of the angular momentum, both per unit area or both per unit volume, are found to be proportional, the factor of proportionality being the wave frequency; specifically, angular momentum magnitude equals energy divided by frequency. Wave action, which has become an increasingly popular quantity for interpreting surface gravity wave problems and is defined as wave energy divided by frequency, is the same thing as the magnitude of the angular momentum. A general equation for the conservation of angular momentum along wave rays is given, and it contains on the right‐hand side two torque terms; one changes the magnitude and the other changes the direction of the angular momentum. In applying this equation to the wave‐current refraction problem, there is only one torque, which changes the direction of the angular momentum, and it is explicitly determined as a function of the horizontal shear in the current. The magnitude of the angular momentum, and therefore also the wave action, will be conserved along the rays if there are no torques that could alter it, as in pure wave‐current refraction. Our conservation equation for angular momentum is easily adapted to describing wave generation and dissipation by including the appropriate torques on the right side, and this may prove to be helpful for calculations of wave evolution.