  The equations of reversible (inviscid, adiabatic) fluid dynamics have a
well-known variational formulation based on Hamilton's principle and the
Lagrangian, to which is associated a Hamiltonian formulation that involves a
Poisson bracket structure. These variational and bracket structures underlie
many of the most basic principles that we know about geophysical fluid flows,
such as conservation laws. However, real geophysical flows also include
irreversible processes, such as viscous dissipation, heat conduction, diffusion
and phase changes. Recent work has demonstrated that the variational
formulation can be extended to include irreversible processes and
non-equilibrium thermodynamics, through the new concept of thermodynamic
displacement. By design, and in accordance with fundamental physical
principles, the resulting equations automatically satisfy the first and second
law of thermodynamics. Irreversible processes can also be incorporated into the
bracket structure through the addition of a dissipation bracket. This gives
what are known as the single and double generator bracket formulations, which
are the natural generalizations of the Hamiltonian formulation to include
irreversible dynamics. Here the variational formulation for irreversible
processes is shown to underlie these bracket formulations for fully
compressible, multicomponent, multiphase geophysical fluids with a single
temperature and velocity. Many previous results in the literature are
demonstrated to be special cases of this approach. Finally, some limitations of
the current approach (especially with regards to precipitation and nonlocal
processes such as convection) are discussed, and future directions of research
to overcome them are outlined.
