  This paper presents (Lagrangian) variational formulations for single and
multicomponent semi-compressible fluids with both reversible
(entropy-conserving) and irreversible (entropy-generating) processes.
Semi-compressible fluids are useful in describing \textit{low-Mach} dynamics,
since they are \textit{soundproof}. These models find wide use in many areas of
fluid dynamics, including both geophysical and astrophysical fluid dynamics.
Specifically, the Boussinesq, anelastic and pseudoincompressible equations are
developed through a unified treatment valid for arbitrary Riemannian manifolds,
thermodynamic potentials and geopotentials. By design, these formulations obey
the 1st and 2nd laws of thermodynamics, ensuring their thermodynamic
consistency. This general approach extends and unifies existing work, and helps
clarify the thermodynamics of semi-compressible fluids. To further this goal,
evolution equations are presented for a wide range of thermodynamic variables:
entropy density $s$, specific entropy $\eta$, buoyancy $b$, temperature $T$,
potential temperature $\theta$ and a generic entropic variable $\chi$; along
with a general definition of buoyancy valid for all three semicompressible
models and arbitrary geopotentials. Finally, the elliptic equation is developed
for all three equation sets in the case of reversible dynamics, and for the
Boussinesq/anelastic equations in the case of irreversible dynamics; and some
discussion is given of the difficulty in formulating the elliptic equation for
the pseudoincompressible equations with irreversible dynamics.
