Limit states of internal wave beams

Abstract

Internal gravity waves propagate away from a localized monochromatic disturbance inside beams, which develop around a St Andrew's Cross in two dimensions and around a double cone in three dimensions. The structure of the beams depends on three mechanisms, which couple together the different directions of propagation of the waves within the fluid. These mechanisms are associated with the size of the disturbance, the start-up of the motion and the viscosity of the fluid, respectively. The present paper considers each mechanism in isolation, for three-dimensional generation. The analysis is asymptotic and relies on far-field and large-time approximations. For each mechanism, three expressions of the waves are found: one, exact for an extended disturbance, that involves all the wavenumber vectors satisfying the dispersion relation; and two others, respectively uniform and non-uniform asymptotic expansions, that involve only the wavenumber vectors associated with group velocity vectors pointing toward the observer. For each mechanism, profiles of pressure and velocity are presented. A new time-independent characterization of the waves is introduced, in terms of the intensity or average energy flux; it is applied to the definition of the beam width. For an extended disturbance this width is a constant, the diameter of the disturbance. For an impulsive start-up the width increases linearly with the distance from the disturbance, and decreases in inverse proportion to the time elapsed since the start-up. For a viscous fluid the width increases as the one-third power of the distance. In all three cases, for a disturbance of multipolar order $2^n$, at constant strength, the power output varies in inverse proportion to the $(2n+1)$th power of the beam width.