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 23 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 23 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 Á = vanishes. Moreover, fields with strong singularities yield Eulerian tail slopes as large as 21.6, whereas those with lesser violations of adiabatic constraints yield Eulerian spectral tail slopes that are much steeper (more strongly negative) than 23. 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.

An analysis of the Hines and Warner–McIntyre–Scinocca non‐orographic gravity wave drag parametrizations

“…In a broader context, the present results suggest that an̂− 2 frequency spectrum for wave energy density is the only choice compatible with the much better constrained −3 vertical wavenumber spectrum. Medvedev and Klaassen (1995), Klaassen and Sonmor (2006) and Klaassen (2009aKlaassen ( , 2009b have pointed out serious issues with the physical and mathematical foundations of Hines Doppler-spread theory, so it is natural to ask why Hines DSP has been successfully applied to represent the effects of non-orographic GWD in middle atmosphere climate models. This is clearly because the DSP parametrization scheme employs a simplification of Doppler-spread theory that mainly produces wave drag as a consequence of enhanced (or premature) critical-layer absorption.…”

“…Hines, ) which assume that waves are conservative, linear (non‐interacting) and sinusoidal in the Lagrangian frame, yet become nonlinear when transformed to the Eulerian frame. Klaassen and Sonmor (2006) and Klaassen (, ) tested the foundation of those Lagrangian wave theories, and demonstrated that the Hines picture of wave–wave interactions (Doppler‐spreading) was based on a physically flawed theoretical model that involves singular Lagrangian to Eulerian transformations that violate both conservation of mass and thermodynamic energy. This immediately raises the question of why the Hines Doppler‐spread parametrization has been able to produce reasonable middle atmosphere circulations (personal communication, J. Alexander, 2009).…”

Impact of a Stochastic Nonorographic Gravity Wave Parameterization on the Stratospheric Dynamics of a General Circulation Model

“…The DST is formulated in a Lagrangian frame, in which the Doppler shift from the background flow is removed from the dispersion relation (Allen & Joseph, ), but the added dependence from the background shear due to the Eulerian to Lagrangian transform is not considered (Broutman et al, ). The dynamics of the DSP have been studied in detail by Klaassen (, ), who demonstrated that the observed Eulerian tails of the GW spectrum cannot be reproduced following the arguments of the DST. The transform from the Lagrangian to the Eulerian reference frame of the multiwave spectrum is found to be singular, as continuity, adiabaticity, and hydrostasy are violated.…”

Gravity Waves in Planetary Atmospheres: Their Effects and Parameterization in Global Circulation Models

“…The DST is based on the assumption that waves are conservative and non-interacting in the Lagrangian frame of reference and become nonlinear when transformed to the Eulerian frame [49]. The physical and mathematical foundations of the DST were tested in the works of Klaassen and Sonmor [50], Klaassen [51], and Klaassen [52]. It was shown that the entire picture of Doppler spreading was based on a physically flawed model violating conservation of mass and energy.…”