  The baroclinic modes of quasigeostrophic theory are incomplete and the
incompleteness manifests as a loss of information in the projection process.
The incompleteness of the baroclinic modes is related to the presence of two
previously unnoticed stationary step-wave solutions of the Rossby wave problem
with flat boundaries. These step-waves are the limit of surface
quasigeostrophic waves as boundary buoyancy gradients vanish. A complete normal
mode basis for quasigeostrophic theory is obtained by considering the
traditional Rossby wave problem with prescribed buoyancy gradients at the lower
and upper boundaries. The presence of these boundary buoyancy gradients
activates the previously inert boundary degrees of freedom. These Rossby waves
have several novel properties such as the presence of multiple equivalent
barotropic modes, a finite number of modes with negative norms, and their
vertical structures form a basis capable of representing any quasigeostrophic
state. Using this complete basis, we are able to obtain a series expansion to
the potential vorticity of Bretherton (with Dirac delta contributions). We
compare the convergence and differentiability properties of these complete
modes with various other modes in the literature. We also examine the
quasigeostrophic vertical velocity modes and derive a complete basis for such
modes as well. In the process, we introduce the concept of the quasigeostrophic
phase space which we define to be the space of all possible quasigeostrophic
states. A natural application of these modes is the development of a weakly
non-linear wave-interaction theory of geostrophic turbulence that takes
prescribed boundary buoyancy gradients into account.
