Symmetries and conservation laws for internal gravity waves

Publisher Website
Google Scholar
Abstract
The nonlinear evolution of stably stratified incompressible fluid in the ’’f‐plane’’ (or non‐rotating) is constrained by the existence of several conservation laws; e.g., those of density, potential vorticity, (horizontal) linear momenta and (vertical) angular mometum. Only five integrals of motion are, to lowest order, quadratic in the vorticity from the equilibrium state: energy, enstrophy (square of potential vorticity deviation), (horizontal) linear pseudo‐momenta and (vertical) angular pseudo‐momentum. These conservation laws are found to be related to the symmetries of the system, namely, invariance under time translations, change of the particles label in each isopycnal, horizontal translations and rotations around a vertical axis, respectively. The nonlinear equations are studied for three different descriptions of the system: Eulerian, Lagragian and mixed (Eulerian in the horizontal and Lagragian in the vertical). In particular, it is discussed whether the eigenvectors of the linear evolution operator form an orthogonal and complete basis or not. If they do, then a phase‐space expansion of the dynamical variables and the corresponding evolution equations for the amplitudes are easily and exactly obtained. Neither a hypothesis on the magnitude of these amplitudes nor a reduction of the model equations are needed in this latter case.

This publication has 0 references indexed in Scilit: