Variational nonlinear WKB in the Eulerian frame
Abstract
Nonlinear WKB is a multiscale technique for studying locally planewave solutions of nonlinear partial differential equations (PDEs). Its application comprises two steps: (1) replacement of the original PDE with an extended system separating the large scales from the small and (2) reduction of the extended system to its slow manifold. In the context of variational fluid theories with particle relabeling symmetry, nonlinear WKB in the mean Eulerian frame is known to possess a variational structure. This much has been demonstrated using, for instance, the theoretical apparatus known as the generalized Lagrangian mean. On the other hand, the variational structure of nonlinear WKB in the conventional Eulerian frame remains mysterious. By exhibiting a variational principle for the extended equations from step (1) above, we demonstrate that nonlinear WKB in the Eulerian frame is in fact variational. Remarkably, the variational principle for the extended system admits loops of relabeling transformations as a symmetry group. Noether's theorem therefore implies that the extended Eulerian equations possess a family of circulation invariants parameterized by S^{1}. As an illustrative example, we use our results to systematically deduce a variational model of highfrequency acoustic waves interacting with a largerscale compressible isothermal flow.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 May 2020
 DOI:
 10.1063/1.5099383
 arXiv:
 arXiv:1902.04221
 Bibcode:
 2020JMP....61e3101B
 Keywords:

 Mathematical Physics;
 Physics  Fluid Dynamics
 EPrint:
 64 pages, preprint