Interactions and instabilities of barotropic and baroclinic rossby waves in a rotating, two-layer fluid

Barotropic and baroclinic Rossby waves in a two-layer fluid on a β-plane can interact with each other, and are unstable for all values of M = UK 2 /β, where U is the velocity amplitude of the particular wave, and K is its wavenumber modulus. The direction of energy transfer depends upon the pseudo-wavenumber, here defined as the square root of the sum of the square of the wavenumber and the inverse square of the radius of deformation of the mode; if a wave loses energy, some goes to smaller pseudo-wavenumbers as well as larger pseudo-wavenumbers. This contrasts with Fjortoft's conclusion that the direction of energy transfer in a homogeneous two-dimensional fluid is to larger and smaller wavenumbers. Triads of Rossby waves can interact resonantly at the second order in M. In contrast to the single layer case, three possible triads can occur in cases of geophysical interest, namely (a) pure mode barotropic, (b) pure mode baroclinic, and (c) mixed mode baroclinic-barotropic-baroclinic types. In case (c) there exists a range of barotropic waves which cannot take part in such mixed triads; in the mid-latitude deep ocean, these waves are characterized by periods of less than about 100 days. Some stability properties of finite amplitude barotropic and baroclinic Rossby waves are derived, and some oceanographic applications are discussed. In particular, it is shown that baroclinic eddies possessing horizontal length-scales larger than the baroclinic radius of deformation should decay into both baroclinic and barotropic motions, and it is suggested that these large-scale eddies will transfer their energy to smaller scales only.