  We consider a rotating non-hydrostatic flow with arbitrary stratification and
argue that 1) the appropriate form of potential vorticity (PV) for this system
is in terms of isopycnal deviation and 2) the decomposition into energetically
orthogonal solutions is fundamentally a PV-inversion.
  The new closed-form expression for available potential vorticity (APV) is
expressed in terms of isopycnal deviation, following the ideas in Wagner &
Young (2015). This form of APV linearizes to quasigeostrophic PV (QGPV) after
discarding the nonlinear stretching term and a height nonlinearity, the latter
of which is not present in constant stratification. This formulation leads to
positive definite definitions of potential enstrophy and total energy expressed
in terms of isopycnal deviation, from which the quadratic versions emerge at
lowest order. It is exactly these quantities diagonalized by the linear
eigenmodes.
  Internal-gravity waves, geostrophic motions, inertial oscillations, and a
mean density anomaly form the energetically and enstrophically orthogonal
constituents of flow. The complete state of the fluid can be represented in
terms of these physically realizeable modes and determined from the derived
projection operators using the horizontal velocity and density anomaly. The
projection of the fluid state onto the non-hydrostatic wave modes, reveals that
one must first account for the PV portion of the flow before recovering the
wave solutions.
  We apply the physical insights of the decomposition to a mesoscale eddy
showing how strict adherence to adiabatic rearrangement places strong
constraints on the vertical structure of such eddies, including a skew towards
stronger cyclonic eddies in the upper water-column. Finally, the expression for
APV is shown to reproduce the height nonlinearity of shallow-water PV, a well
know feature that breaks the cyclone-anticyclone symmetry in QGPV.
