  Many definitions of moist potential vorticity (PV) have been proposed to
extend the dry theory of Ertel PV. None of the moist PV definitions seem to
have all of the desirable properties of the dry Ertel PV. For instance, dry PV
is not only a globally conserved quantity, but also a material invariant that
is conserved along fluid parcel trajectories. Therefore, an open question
remains: is there a moist PV that is a material invariant, if clouds and phase
changes of water are present? In prior studies, definitions of moist PV have
been proposed based on physical and mathematical intuition. Here, a systematic
approach is used. In particular, a particle relabeling symmetry is devised for
a moist atmosphere and then Noether's theorem is employed to arrive at the
associated conservation laws for moist PV. A priori, it is not clear whether
this systematic approach will be viable, since it relies on variational
derivatives in Hamilton's principle, and phase changes introduce singularities
that could potentially prevent derivatives at cloud edge. However, it is shown
that the energy and the Lagrangian density are sufficiently smooth to allow
variational derivatives, in a moist Boussinesq system with reversible phase
transitions between water vapor and liquid cloud water.. From the particle
relabeling symmetry, a moist Kelvin circulation theorem is found, along with a
moist PV conservation law that applies not for each individual parcel but for
parcel-integrated PV, integrated over certain local volumes.
