  A linear decomposition of states underpins many classical systems. This is
the case of the Helmholtz decomposition, used to split vector fields into
divergence-free and potential components, and of the dry Boussinesq system in
atmospheric dynamics, where identifying the slow and fast components of the
flow can be viewed as a decomposition. The dry Boussinesq system incorporates
two leading ingredients of mid-latitude atmospheric motion: rotation and
stratification. In both cases the leading order dynamics are linear so we can
rely on an eigendecomposition to decompose states.
  Here we study the extension of dry Boussinesq to incorporate another
important ingredient in the atmosphere: moisture and clouds. The key challenge
with this system is that nonlinearities are present at leading order due to
phase boundaries at cloud edge. Therefore standard tools of linear algebra,
relying on eigenvalues and eigenvectors, are not applicable. The question we
address in this paper is this: in spite of the nonlinearities, can we find a
decomposition for this moist Boussinesq system?
  We identify such a decomposition adapted to the nonlinear balances arising
from water phase boundaries. This decomposition combines perspectives from
partial differential equations (PDEs), the geometry, and the conserved energy.
Moreover it sheds light on two aspects of previous work. First, this
decomposition shows that the nonlinear elliptic PDE used for potential
vorticity and moisture inversion can be used outside the limiting system where
it was first derived. Second, we are able to rigorously justify, and interpret
geometrically, an existing numerical method for this elliptic PDE. This
decomposition may be important in applications because, like its linear
counterparts, it may be used to analyze observational data. Moreover, by
contrast with previous decompositions, it may be used even in the presence of
clouds.
