Resumen del producto

Laboratory experiments suggest that the instability of a circular flow with alternating vorticity around a cylinder may culminate in emission of a few vortex dipoles. These observations lead to the following questions: What are the conditions for a flow around a cylindrical island to be stable? If the flow is unstable, what are the growth rates of its unstable modes? What vortical patterns may emerge due to the instability? This topic is of oceanographic significance. Indeed, island-trapped inertial or subinertial waves induced by tidal or planetary waves and tidal rectification may cause the formation of mass-transporting currents around islands. Coastal waters near islands are usually rich in chemical and biological material, while vortex dipoles are normally quite robust and can serve as carriers of the trapped material. To answer the above-posed questions we study analytically and numerically the instability and longterm evolution of two-dimensional circular flows with alternating vorticity around a circular cylinder (island). The base flow is given by two concentric neighbouring rings of uniform but different vorticity, with the inner ring touching the cylinder. First, the inviscid linear instability of such flows to the perturbations of the free edges of the two rings is studied. At a fixed ratio of the vorticities in the rings, the governing parameters of the problem are the outer radii of the inner and outer rings scaled with the cylinder radius. For each azimuthal mode m = 1, 2, …, we determine analytically the regions in the twodimensional parameter space where this mode is linearly stable/unstable. Mode m = 1 is always stable. In the physically most meaningful case of zero net circulation (i.e. when the total energy of the flow is finite), for each mode m > 1, we identify two regions of instability, a so-called regular instability region where mode m is unstable along with the lower and/or higher modes, and a region of unique instability where mode m and only this mode is unstable. Typical of the unique instability regions (at m > 2) is that the inner ring is much thinner than the outer rind, while the growth rate of a specific mode is smaller than that in the regular instability region by orders of magnitude. In the limit wherein the outer radii of the rings go to infinity, the boundaries of the stability/instability regions approach asymptotically the stability bounds for ‘shielded’ monopoles established by Flierl [1]. After the conditions of linear instability are established, the non-linear, long-term evolution of unstable flows and the formation of new vortical patterns are studied numerically by conducting inviscid contourdynamics simulations and high Reynolds number finite-element simulations with the initial conditions being set based on the analytical results for modes m = 2, 3 and 4. The inviscid and viscid simulations initialized with the parameters characteristic of the regular instability regions yield the emission of vertical dipoles, which either propagate away from the cylinder or tend to form coherent vortex multipoles around the cylinder. The number of emerged dipoles is equal to the number of the most unstable mode. For the initial conditions representative of the unique instability regions, the contour-dynamics simulations do not reveal dipole emission. In the finite-element simulations, two phases of evolution can be distinguished. In the first phase, the viscous widening of the thin inner ring is faster than the growth of instability. This, in effect, moves the point representing the flow in the parameter space from the unique instability region into the regular instability region. Thus, in the second phase, the instability development leads to the emission of a number of dipoles.

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