Comments on “On Internal Waves in a Lagrangian Reference Frame”

Abstract

Broutman et al. (2004, henceforth BGE1) have entered a current discussion that compares and contrasts gravity wave formulations in Lagrangian and Eulerian analyses. They derived a Lagrangian dispersion relation that takes some account of the nonlinear multiwave ambiance that is at issue, and then they showed that there is “a complete equivalence between the Lagrangian ray equations (46) and the Eulerian ray equations (18).” In doing so, they have negated my own speculation (Hines 2002, henceforth H02) that Lagrangian and Eulerian ray paths might differ, but at the same time they have given guarded confirmation of certain conditions I established for minimal Lagrangian nonlinearity, and they appear to anticipate further study of my suggestion that there are advantages to be gained from a Lagrangian approach to ray tracing in corresponding circumstances. I welcome these developments as a positive contribution to the field, but I must respond to one portion of their related discussion. Their analysis was set in the context of studies by Allen and Joseph (1989, henceforth AJ), Hines (2001), and Chunchuzov (2002)—henceforth collectively the AJHCh papers—and they question those studies in their section 5. Specifically, they question whether a certain “nonwavelike” behavior found by the AJHCh papers in Eulerian spectra at large wavenumbers might have been produced by the use of a Lagrangian dispersion relation that differs from their own, with the implication that their own would have been more appropriate and might have produced only “wavelike” Eulerian behavior. The answer is an unreserved no. What BGE1 call “the” Lagrangian dispersion relation is more properly and more fully described as “a” Lagrangian dispersion relation, one that is applicable locally to waves of small space–time scales propagating in the presence of fluctuations of fluid velocity, fluid displacement, density, and pressure that occur at, and only at, much larger scales. Since the AJHCh papers deal with a continuum of scales, the BGE1 dispersion relation really has nothing to say about the AJHCh results: it could not even have been validly employed. Moreover, there is no mechanism by which it could have been employed, validly or not, given the quite different nature of the AJHCh analyses. Nor, if there were, would “wavelike” behavior have been found in the Eulerian results at large wavenumbers. With its ray-tracing point of view, the BGE1 study focuses on local conditions found in the immediate vicinity of a wave packet, and it follows changes of those conditions as the packet progresses along its ray. The AJHCh analyses, on the contrary, take a more global view: they deal with a large volume of space–time in its entirety, and they contemplate a four-dimensional Fourier decomposition of the fluctuations that are found within it. For the result to be wavelike within the operative definition of that word, each wavevector k would have to have been associated with a single frequency k (k) to yield a dispersion relation applicable to the volume as a whole. Such behavior is not found even locally in the BGE1 Eulerian analysis, which produces at best a single intrinsic frequency for a given k wherever and whenever that k is found locally in various packets as they weave their way through the ever-changing background. This intrinsic frequency would have to be Doppler shifted, by a variable amount dependent on the local flow, before being recorded for use as a frequency in the fixed Eulerian coordinates of the AJHCh analyses. How could those analyses—or any others employing a single Eulerian coordinate system—possibly be expected to find a single frequency in association with a chosen k? It is inherent in the nature of multiwave nonlinearity that, if a “wave” defined by its globally applicable k is subject to significant modification through Eulerian nonlinear effects, it cannot have a single corresponding frequency in a single Eulerian frame of reference. Such a “wave”—in truth, a Corresponding author address: Dr. C. O. Hines, 15 Henry Street, Toronto, ON M5T 1W9, Canada. E-mail: hines@stpl.cress.yorku.ca JANUARY 2005 N O T E S A N D C O R R E S P O N D E N C E 251