Instability and advection properties
of two-dimensional vortices

The aim of the project is to study both analytically and numerically the stability behaviour of two-dimensional vortices and the associated (chaotic) advection ('mixing') of scalars. The results will be compared with results of other numerical methods and with experiments performed at the Vortex Dynamics Group. The questions that are put are: How is the vortex's (in)stability influenced by bottom or free-surface topographic effects? Can we characterize the related mixing properties of the vortex structure?
It is appropriate to mention that the relative flow in a rotating fluid with a parabolic free-surface (as is the case in the laboratory) is dynamically equivalent to geophysical flows near the earth's poles. Studying the dynamics and mixing properties of a vortex on such a topography may thus give some insight in the dynamical behaviour of the earth's stratospheric polar vortices. In particular, this may lead to a better understanding of the (limited!) horizontal mixing between the ozone depleted region above Antarctica - also known as the ozone hole - and the atmosphere at lower latitudes.

The dynamics of two-dimensional (2D) vortex structures has received increasing attention during the last decade, because of their relevance to geophysical flows. Satellite imagery has revealed the abundant occurrence of large vortices in the atmosphere and the world's oceans. Well-known oceanic examples are the Gulf Stream rings, the Agulhas rings and the vortices shed from coastal currents. These coherent vortices have relatively long lifetimes, and are believed to play an important role in the transport of heat and other properties. In the atmosphere, coherent vortices can be observed in the form of tropical cyclones, and probably the most fascinating example is Jupiter's Great Red Spot on a page under Vortex Dynamics Group - a huge vortex known to exist for at least 350 years. The motion associated with these vortex structures is to first approximation two-dimensional, which is mainly due to the planetary rotation and to the density stratification in the atmosphere and in the oceans (meaning that the density varies with the depth or height).
In addition to these monopolar vortices, another type of coherent structure has been distinguished: the dipolar vortex, consisting of two counter-rotating vortices. In the absence of any non-uniform background flow, the symmetric dipole translates along a straight trajectory. Examples of dipolar vortices have been observed both in the atmosphere (blocking systems) and in the oceans (vortices near unstable density fronts). Recently, numerical simulations and laboratory experiments have revealed the existence of another 2D vortex type: the tripole. This vortex structure is a symmetric linear configuration of three patches of alternate circulation, characterized by a steady rotation of its axis around the centre of the core vortex.

The occurrence and persistence of coherent vortex structures is intimately linked with a crucial property of 2D turbulence, the so-called 'inverse energy cascade'. In contrast to 3D turbulence, vorticity production due to vortex stretching is absent in 2D flows, and for inviscid fluid motions this implies that kinetic energy is a conserved quantity. It can be shown that this conservation means that the energy goes to larger scales of motion (inverse cascade). In physical space, these effects result in an ever-increasing size of the energy-containing eddies. (In 3D-turbulence a decrease of the scales takes place, as a result of which the flows become more chaotic.)
In practice, the inverse energy cascade results in a self-organization: energy initially distributed over both larger and smaller eddies eventually becomes concentrated in larger coherent vortex structures, which give the flow an 'ordered' appearance. This phenomenon is nicely demonstrated in numerical simulations of 2D turbulence: initially the vorticity and energy was randomly distributed over a wide range of length scales, but in the subsequent stages the flow is seen to gradually 'organize' itself in a number of larger vortices. One thus observes the emergence of both monopolar and dipolar vortices, and in one case even a tripolar vortex structure was seen to arise.
The presence of a (weak) viscosity does not drastically change the phenomenological character of the 2D flow: most of the energy still becomes concentrated in the larger scales, in which dissipation is not active. Two-dimensional flows are therefore only weakly dissipative.

The above text is based on the project proposal of my post-doc. position, written by prof. G.J.F. van Heijst.
He also wrote a nice introduction to
Self-organization of Two-dimensional Flows under Vortex Dynamics Group.

<=== Post-doc. research in Eindhoven pagina.

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last modified: 27 May 2001