  It has been shown in \cite{Tong:2022gpg} that shallow water in the Euler
description admits a dual gauge theory formulation. We show in the Lagrange
description this gauge symmetry is a manifestation of the 2 dimensional
area-preserving diffeomorphisms. We find surface charges associated with the
gauge symmetry and their algebra, and study their physics in the shallow water
system. In particular, we provide a reinterpretation of the Kelvin circulation
theorem in terms of conserved charges. In the linear shallow water case, the
charges form a u(1) current algebra with level proportional to the Coriolis
parameter over the height of the fluid. We also study memory effect for the
gauge theory description of the linearized shallow water and show Euler, Stokes
and Darwin drifts can be understood as a memory effect and/or change of the
surface charges in the gauge theory description.
