J. Fluid Mech. (1995), vol. 289, pp. 319405 Copyright @ 1995 Cambridge University Press 379 Wavenumber transport: scattering of small-scale internal waves by large-scale wavepackets By DAVID L. BRUHWILER'AND TASSO J. KAPER2 Grumman Research and Development Center, 4 Independence Way, Princeton, NJ 08540, USA 2Department of Mathematics, Boston University, 11 1 Cummington Street, Boston, MA 02215, USA (Received 2 August 1994 and in revised form 29 November 1994) In this work, we treat the problem of small-scale, small-amplitude, internal waves interacting nonlinearly with a vigorous, large-scale, undulating shear. The ampli- tude of the background shear can be arbitrarily large, with a general profile, but our analysis requires that the amplitude vary on a length scale longer than the wavelength of the undulations. In order to illustrate the method, we consider the ray- theoretic model due to Broutman & Young (1986) of high-frequency oceanic internal waves that trap and detrap in a near-inertial wavepacket as a prototype problem. The near-inertial wavepacket tends to transport the high-frequency test waves from larger to smaller wavenumber, and hence to higher frequency. We identify the essential physical mechanisms of this wavenumber transport, and we quantify it. We also show that, for an initial ensemble of test waves with frequencies between the inertial and buoyancy frequencies and in which the number of test waves per frequency interval is proportional to the inverse square of the frequency, a single nonlinear wave-wave interaction fundamentally alters this initial distribution. Af- ter the interaction, the slope on a log-log plot is nearly flat, whereas initially it was -2. Our analysis captures this change in slope. The main techniques em- ployed are classical adiabatic invariance theory and adiabatic separatrix crossing theory. 1. Introduction Resonant wave-wave interactions are ubiquitous in nature. They occur in plasmas, in shallow water, between surface waves, in the atmosphere, and in the ocean, to list a few of the many places. In this work, we focus on a class of oceanic wave- wave interactions involving small-scale waves and vigorous, large-scale, oscillating background flows. We develop a method to quantitatively analyse the wavenumber transport for the small-scale waves as they interact with the velocity field of the background flow. The method relies on ray theory as well as adiabatic separatrix crossing theory. In order to illustrate the method's utility, we apply it directly to the idealized, ray-theoretic model formulated and studied in Broutman & Young (1986, hereinafter referred to as B&Y), for the strongly nonlinear interaction between high-frequency, short-wavelength internal waves and localized packets of progressive near-inertial waves.