Generalized Symmetries in Shallow Water

TL;DR

This work demonstrates that generalized symmetries, notably two subsystem symmetries, naturally arise in linearized shallow water dynamics with spatially varying Coriolis parameter. By constructing an action principle and applying Noether's theorem, it shows that the first subsystem symmetry enforces the local conservation of potential vorticity $oxed{ oldsymbol{Q}_0 = H oldsymbol{ abla} imes oldsymbol{u} - f oldsymbol{ abla} imes oldsymbol{ abla} imes oldsymbol{u} }$ and generates a Kac-Moody type current algebra controlled by $oldsymbol{ abla} f$, while the second subsystem symmetry (valid for constant $f$) yields another family of Noether charges that can generate new solutions, including geostrophic-balanced states. An action-based treatment clarifies the symplectic structure and exposes how these charges label and constrain solution spaces, with geostrophic adjustment interpreted as an energy redistribution mediated by the symmetry-generated flows. The results bridge fluid mechanics with subsystem symmetry concepts from exotic field theories, suggesting practical uses in solution generation, memory effects, and the study of IR/UV phenomena in real geophysical flows, and point to future extensions to broader fluid models and IR triangle structures.

Abstract

Recent developments have extended the concept of global symmetries in several directions, offering new perspectives across a wide range of physical systems. This work shows that generalized global symmetries naturally emerge in shallow water systems. In particular, we demonstrate that two subsystem symmetries-previously studied primarily in exotic field theories-arise intrinsically in the dynamics of shallow water flows. A central result is that the local conservation of potential vorticity follows directly from the first subsystem symmetry, revealing that the classic Kelvin circulation theorem is rooted in these symmetries. Notably, the associated charge algebra forms a Kac-Moody current algebra, with the level determined by the spatial variation of the Coriolis parameter. Beyond the first subsystem symmetry, we also identify a second one, construct the corresponding Noether charges, and explore their potential applications.