Uniform potential vorticity flows are examined. In the quasi-geostrophic system, conservation of total energy and conservation of available potential energy on plane rigid horizontal boundaries imply a restriction on energy exchanges as a result of scale interactions. It is shown that for the Eady problem instability is always associated with energy transfer both up and down the vertical wavenumber spectrum although energy transfer from small to large three-dimensional wavenumbers may occur over a finite range of the spectrum. An inertial theory of two-dimensional turbulence is also presented. The formal analysis, based on Leith's diffusion approximation, predicts two inertial subranges: −5/3 and −1 power dependences on the horizontal wavenumber for available potential energy on horizontal boundaries. In the former range, available potential energy on horizontal boundaries cascades at a constant rate toward higher wavenumbers; in the latter range, the depth-integrated total energy cascades at a c... Abstract Uniform potential vorticity flows are examined. In the quasi-geostrophic system, conservation of total energy and conservation of available potential energy on plane rigid horizontal boundaries imply a restriction on energy exchanges as a result of scale interactions. It is shown that for the Eady problem instability is always associated with energy transfer both up and down the vertical wavenumber spectrum although energy transfer from small to large three-dimensional wavenumbers may occur over a finite range of the spectrum. An inertial theory of two-dimensional turbulence is also presented. The formal analysis, based on Leith's diffusion approximation, predicts two inertial subranges: −5/3 and −1 power dependences on the horizontal wavenumber for available potential energy on horizontal boundaries. In the former range, available potential energy on horizontal boundaries cascades at a constant rate toward higher wavenumbers; in the latter range, the depth-integrated total energy cascades at a c...