POSITIVE, NEGATIVE AND ZERO WAVE ENERGY AND THE FLOW STABILITY PROBLEM IN THE EULERIAN AND LAGRANGIAN-EULERIAN DESCRIPTIONS

The stability of a horizontal flow on a stratified fluid is studied on a vertical, nonrotating plane; in this manner, integrals of motion related to potential vorticity are out of the scene. The study is thus aided by the conservation of energy, horizontal momentum, and a Casimir functional, defined as an arbitrary function of density plus another arbitrary function times the horizontal vorticity. Two descriptions of the system are used: the usual Eulerian framework (EF) and a mixed Eulerian-Lagrangian one (MF) in which the vertical coordinate is a function of the density. The total flow is, of course, the same in both formalisms, but its partition into a steady basic flow and the perturbation is framework dependent. The perturbation is commonly split into the time-dependent part of the mean flow and a wave; this partition depends also on the framework, i. e., on whether the averages are made as constant depth or constant density. Perturbation energy is conserved, indicating a balance between the variations of mean flow and wave energies. The latter is positive definite in the EF; a growing wave may then be seen as extracting energy from the mean flow. However, in the MF, wave energy may have either sign or vanish; a growing normal mode, for instance, has zero wave energy, which shows that the exchange between wave and mean flow is not, in this formalism, a good indicator of stability. A conserved pseudoenergy (pseudomomentum), quadratic to lowest order in the perturbation, is constructed from the total energy (total momentum) and Casimir integrals. Wave energy (wave momentum) coincides with the pseudoenergy (pseudomomentum) only in the MF. Contrary to the cases of two-dimensional unstratified flow, the quasi-geostrophic model, or a multi-layer system, it is not possible to derive Arnol'd-like stability conditions from the pseudoenergy and pseudomomentum integrals: These can be negative, for any sheared basic flow, if the disturbance has a small enough vertical scale and yet the basic flow remains stable. © 1990 Birkhäuser Verlag.