Testing Lagrangian Theories of Internal Wave Spectra. Part II: Varying the Number of Waves

Abstract

It has been proposed by Allen and Joseph, Hines, and Chunchuzov that the kinematic advection produced by superpositions of sinusoidal Lagrangian gravity waves confined to lower vertical wavenumbers m provides an explanation for the quasi-universal m?3 Eulerian spectral tails commonly found at higher m in the oceans and the atmosphere. In support of these theories, Hines has proposed a prototype wave spectrum claimed to meet criteria for Lagrangian linearity and the production of m?3 Eulerian spectra. Although the shape of the Lagrangian spectrum is claimed not to play a major role in this process, Hines has argued that moderately large numbers of waves are required to ensure quasilinear behavior in the Lagrangian frame. The present results demonstrate that, for amplitudes consistent with measurements of saturated waves in the middle atmosphere and for wavenumbers consistent with Hines? prototype, adiabatic excesses do not diminish with increasing numbers of waves; in contrast, consistency with adiabatic constraints is only achieved in the limit of a single wave, for which the advective nonlinearity u · ? vanishes. Moreover, fields with strong singularities yield Eulerian tail slopes as large as ?1.6, whereas those with lesser violations of adiabatic constraints yield Eulerian spectral tail slopes that are much steeper (more strongly negative) than ?3. The implications for theories based on superpositions of Lagrangian sinusoidal waves, for the Hines quasilinear criteria, and for the Hines Doppler-spread theory and parameterization are addressed. The results are also relevant for experimentalists interested in spectral analysis of internal wave fields.