Rossby wave interactions and instabilities in a rotating, two-layer fluid on a beta-plane. Part I: Resonant interactions

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Abstract
Triads of Rossby waves in a two-layer fluid on a β-plane can interact resonantly at the second order. In contrast to the single layer case, three possible triads may occur in situations of geophysical interest, namely (a) pure mode barotropic, (b) pure mode baroclinic, and (c) mixed mode baroclinic-barotropic-baroclinic types. The geometrical conditions for two baroclinic waves to form a triad with a given barotropic wave [case (c)] are determined, and it is shown that 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. The direction of energy transfer within the mixed triads is discussed and found to depend upon the ordering of a new quantity, the pseudowavenumber, which is defined for each mode 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; in particular, the “unstable” member of the triad is the wave which possesses the intermediate pseudowavenumber. This last result is in contrast to Fjortoft's conclusion that the direction of energy transfer in a homogeneous two-dimensional fluid is to larger and smaller wavenumbers. It is also shown that Hasselmann's criterion appears to hold for mixed mode Rossby wave resonant triads. With regards to geophysical applications, it is suggested that such intermodal interactions as discussed here can be extended to multi-layer systems, and indeed, to continuously stratified fluids, and provides a mechanism for altering the depth structure of such phenomena as the mid-ocean eddies.

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