  The linear wave and geostrophic (vortex) solutions are shown to be a complete
basis for physical variables $(u,v,w,\rho)$ in a rotating non-hydrostatic
Boussinesq model with arbitrary stratification. As a consequence, the fluid can
be unambiguously separated into linear wave and geostrophic components at each
instant in time, without the need for temporal filtering. The fluid can then be
diagnosed for temporal changes in wave and geostrophic coefficients at each
unique wavenumber and mode, including those that inevitably occur due to
nonlinear interactions.
  We demonstrate that this methodology can be used to determine which physical
interactions cause the transfer of energy between modes by projecting the
nonlinear equations of motion onto the wave-vortex basis. In the particular
example given, we show that an eddy in geostrophic balance superimposed with
inertial oscillations at the surface transfers energy from the inertial
oscillations to internal gravity wave modes. This approach can be applied more
generally to determine which mechanisms are involved in energy transfers
between wave and vortices, including their respective scales.
  Finally, we show that the nonlinear equations of motion expressed in a
wave-vortex basis are computationally efficient for certain problems. In cases
where stratification profiles vary strongly with depth, this approach may be an
attractive alternative to traditional spectral models for rotating Boussinesq
flow.
