Variational nonlinear WKB in the Eulerian frame

Burby, J. W.; Ruiz, D. E.

doi:10.1063/1.5099383

Variational nonlinear WKB in the Eulerian frame

Journal Article · Wed May 13 00:00:00 EDT 2020 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.5099383· OSTI ID:1634797

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Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Mathematical Sciences Research Inst., Berkeley, CA (United States)
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Nonlinear WKB is a multiscale technique for studying locally plane-wave 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, here 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 S¹. As an illustrative example, we use our results to systematically deduce a variational model of high-frequency acoustic waves interacting with a larger-scale compressible isothermal flow.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States); Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program; National Science Foundation (NSF)

Grant/Contract Number:: AC04-94AL85000; NA0003525; 89233218CNA000001

OSTI ID:: 1634797

Alternate ID(s):: OSTI ID: 1645081

Report Number(s):: SAND--2020-5532J; 686363

Journal Information:: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 5 Vol. 61; ISSN 0022-2488

Publisher:: American Institute of Physics (AIP)Copyright Statement

Country of Publication:: United States

Language:: English

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Variational nonlinear WKB in the Eulerian frame

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