# Variational nonlinear WKB in the Eulerian frame

## TL;DR

This paper establishes a variational principle for nonlinear WKB in the Eulerian frame, revealing circulation invariants and applying the framework to model high-frequency acoustic waves interacting with large-scale flows.

## Contribution

It demonstrates that nonlinear WKB in the Eulerian frame is variational and identifies its symmetry group, which was previously unknown.

## Key findings

- Variational principle for extended Eulerian WKB equations established.
- Loops of relabeling transformations form a symmetry group.
- Derived a variational model for high-frequency acoustic waves in flows.

## Abstract

Nonlinear WKB is a multiscale technique for studying locally-plane-wave solutions of nonlinear partial differential equations (PDE). 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 high-frequency acoustic waves interacting with a larger-scale compressible isothermal flow.