Estimating the size of the regular region of a topographically trapped vortex

Abstract

We formulate a dynamically consistent two-layer quasigeostrophic model of geophysical flow using the concept of background currents which are characterized by a constant potential vorticity minimizing the energy. An incident current over a delta-like isolated topographic feature generates a topographically trapped vortex in the bottom layer with a singular elliptic point, and one with a regular elliptic point in the upper layer. Such vortices are finite regions of recirculation which occur in the vicinity of isolated topographic features. The corresponding Hamiltonian equations of motion for a fluid particle are known to produce chaotic advection under the presence of the periodic incident current. When a periodic incident current is superimposed, fluid is entrained and detrained from the neighbourhood of the vortex and chaotic particle motion occurs. At a small amplitude periodic incident current we have a near-integrable case of weak chaotization in narrow stochastic layer near separatrix. At a finite amplitude periodic incident current we have the case of strong chaos, which will be investigated here. For the bottom layer of the flow, there is always a regular region (regular vortex core) around the singular elliptical point even in the case of a finite amplitude periodic incident current. Particles in this regular region have regular trajectories and will remain there permanently, whereas particles with chaotic trajectories will be emanated from the vortex by the incident current. Using the Chirikov criterion of resonance overlap, we estimate the radius of this regular vortex core and a range of the optimal frequencies, i.e. those frequencies of the incident current which provide a maximal possible stochastization of fluid particle trajectories.

Keywords:

• Vortex
• Chaotic advection
• Resonance
• Chirikov resonance overlap criterion
• Background currents

Acknowledgements

Support for this study has been provided by the Russian Fund for Fundamental Studies Grants No. 10–05–00646–, 10–05–00770–, 11–05–91052, and FEB RAS Grants No. 09–I–17–07, 09–II––07–002, 09–III–A–07–319, 10–III––07–147, 10–III––07–155.