  Using a manifestly invariant Lagrangian density based on Clebsch fields and
suitable for geophysical fluid dynamics, the conservation of mass, entropy,
momentum and energy, and the associated symmetries are investigated. In
contrast, it is shown that the conservation of Ertel's potential vorticity is
not associated with any symmetry of the equations of motion, but is instead a
trivial conservation law of the second kind. This is at odds with previous
studies which claimed that potential vorticity conservation relates to a
symmetry under particle-relabeling transformations. From the invariant
Lagrangian density, a canonical Hamiltonian formulation is obtained in which
Dirac constraints explicitly include the (possibly time-dependent) metric
tensor. In this case, it is shown that all Dirac constraints are primary and of
the second class, which implies that no local gauge symmetry transformations of
Clebsch fields exist. Finally, the corresponding non-canonical Hamiltonian
structure with time-dependent strong constraints is derived using tensor
components. The existence of Casimir invariants is then investigated in
arbitrary coordinates for two choices of dynamical variables in phase space.
