Variational generalization of the Green–Naghdi and Whitham equations for fluid sloshing in three-dimensional rotating and translating coordinates

This paper derives an averaged Lagrangian functional for dynamic coupling between rigid-body motion and its interior shallow-water sloshing in three-dimensional rotating translating coordinates; with a time-dependent rotation vector. A new set of variational equations (SWEs) generalized Green–Naghdi the fluid 3-D vector translations, also linear momentum angular containing shallow water, are derived from functional, which describes columnar , by using Hamilton’s principle Euler–Poincaré framework. The have form potential vorticity (PV) conservation, can be obtained particle-relabelling symmetry, is combination PV Miles Salmon (1985) Dellar (2005) geophysical dynamics problems, where varies spatially. By applying assumption zero-potential-vorticity flow to Boussinesq-like evolution derived, generalization Whitham coordinates. Moreover, principles appended Luke’s present unified framework hydrodynamic problem interactions gravity-driven potential-flow water waves freely floating rigid-body, dynamically coupled weakly dispersive nonlinear three dimensions.