An exact theory of nonlinear waves on a Lagrangian-mean flow

Abstract

An exact and very general Lagrangian-mean description of the back effect of oscillatory disturbances upon the mean state is given. The basic formalism applies to any problem whose governing equations are given in the usual Eulerian form, and irrespective of whether spatial, temporal, ensemble, or ‘two-timing’ averages are appropriate. The generalized Lagrangian-mean velocity cannot be defined exactly as the ‘mean following a single fluid particle’, but in cases where spatial averages are taken can easily be visualized, for instance, as the motion of the centre of mass of a tube of fluid particles which lay along the direction of averaging in a hypothetical initial state of no disturbance. The equations for the Lagrangian-mean flow are more useful than their Eulerian-mean counterparts in significant respects, for instance in explicitly representing the effect upon mean-flow evolution of wave dissipation or forcing. Applications to irrotational acoustic or water waves, and to astrogeophysical problems of waves on axisymmetric mean flows are discussed. In the latter context the equations embody generalizations of the Eliassen-Palm and Charney-Drazin theorems showing the effects on the mean flow of departures from steady, conservative waves, for arbitrary, finite-amplitude disturbances to a stratified, rotating fluid, with allowance for self-gravitation as well as for an external gravitational field. The equations show generally how the pseudomomentum (or wave ‘momentum’) enters problems of mean-flow evolution. They also indicate the extent to which the net effect of the waves on the mean flow can be described by a ‘radiation stress’, and provide a general framework for explaining the asymmetry of radiation-stress tensors along the lines proposed by Jones (1973).