3-DIMENSIONALIZATION OF BAROTROPIC VORTICES ON THE F-PLANE

We examine the stability characteristics of a two-dimensional flow whi ch consists initially of an inflexionally unstable shear layer on an f -plane. Under the action of the primary instability, the vorticity in the shear-layer initially coalesces into two Kelvin-Helmholtz vortices which subsequently merge to form a single coherent vortex. At a seque nce of times during this process, we test the stability of the two-dim ensional flow to fully three-dimensional perturbations. A somewhat nov el approach is developed which removes inconsistencies in the secondar y stability analyses which might otherwise arise owing to the time-dep endence of the two-dimensional flow. In the non-rotating case, and bef ore the onset of pairing, we obtain a spectrum of unstable longitudina l modes which is similar to that obtained previously by Pierrehumbert & Widnall (1982) for the Stuart vortex, and by Klaassen & Peltier (198 5, 1989, 1991) for more realistic flows. In addition, we demonstrate t he existence of a new sequence of three-dimensional subharmonic (and t herefore helical) instabilities. After pairing is complete, the second ary instability spectrum is essentially unaltered except for a doublin g of length- and timescales that is consistent with the notion of spat ial and temporal self-similarity. Once pairing begins, the spectrum qu ickly becomes dominated by the unstable modes of the emerging subharmo nic Kelvin-Helmholtz vortex, and is therefore similar to that which is characteristic of the post-pairing regime. Also in the context of non -rotating flow, we demonstrate that the direct transfer of energy into the dissipative subrange via secondary instability is possible only i f the background flow is stationary, since even slow time-dependence a cts to decorrelate small-scale modes and thereby to impose a short-wav e cutoff on the spectrum. The stability of the merged vortex state is assessed for various values of the planetary vorticity f. Slow rotatio n may either stabilize or destabilize the columnar vortices, depending upon the sign off, while fast rotation of either sign tends to be sta bilizing. When f has opposite sign to the relative vorticity of the tw o-dimensional basic state, the flow becomes unstable to a new mode of instability that has not been previously identified. Modes whose energ y is concentrated in the vortex cores are shown to be associated, even at non-zero f, with Pierrehumbert's (1986) elliptical instability. Th rough detailed consideration of the vortex interaction mechanisms whic h drive instability, we are able to provide physical explanations for many aspects of the three-dimensionalization process.