Vortex dynamics

The equations governing the evolution of quantum vortex defects subject to twist are derived in standard hydrodynamic form. Vortex defects emerge as solutions of the Gross-Pitaevskii equation, that by Madelung transformation admits a hydrodynamic description. Here, we consider a vortex defect subject to superposed twist due to the rotation of the phase of the wave function. We prove that, when twist is present, the corresponding Hamiltonian is non-Hermitian and determine the effect of twist on the energy expectation value of the system. We show how twist diffusion may trigger linear instability, a property directly related to the non-Hermiticity of the Hamiltonian. We derive the correct continuity equation and, by applying defect theory, we obtain the correct momentum equation. Finally, by coupling twist kinematics and vortex dynamics we determine the full set of hydrodynamic equations governing quantum vortex evolution subject to twist.

It is my great honour to welcome you on behalf of the Bureau of IUTAM to this Symposium on Hamiltonian dynamics, vortex structures and turbulence. The Symposium has been in preparation for two years, and I congratulate our hosts here at the Steklov Institute of the Russian Academy of Sciences for having prepared an excellent and wide-ranging programme, and for having succeeded in attracting such a distinguished gathering to debate problems in fluid dynamics many of which have a long history, yet still today present many challenges of a fundamental nature. The letters IUTAM, as you all know, stand for the International Union of Theoretical and Applied Mechanics. This Union is one of the International Scientific Union members of ICSU, the International Council for Science, which this year celebrates its 75th anniversary. The roots of IUTAM itself go back to the early Congresses in Mechanics, the first of which was held in Delft in the Netherlands, in 1924. IUTAM was formally established as an International Union at the 7th Congress, which was held in London in 1948. The 13th Congress of Theoretical and Applied Mechanics was held here in Moscow in 1972, under the Presidency of the great Mushkhelishvili. The most recent 21st Congress was held in Warsaw in 2004, and the next will be held in Adelaide, South Australia, in 2008.
Professor  Keith Moffatt, Vice-President, IUTAM

Vortices most often represent some kind of contact with an other, most often another world or realm, another dimension or plan of existence, another state of being. As a result of a divine mechanical process, according to necessity or a (meta)physical law, the vortices participate both to the coming-into-being of all things and to the passing-out-of-being of the world(s). Thanks to a “twisted”, i.e. dialectical logic, the vortex in all its forms join both the cosmogonical and the eschatological narratives of origin/ end of the world. Whirling, helicoidal or curling elements, whirlpools, spirals and rotatory or spinning motions, centrifugal forces flinging things outwards and eddies sucking them inwards, all these forms of vortices often are enough to signal a breach between worlds or between states of the universe. Apparently chaotic, all these forms of vortices nonetheless contribute to the transition from an undifferentiated state of the universe (the primordial repose) to an organized state (provided with structure, hierarchy, order ), from nebulous time of origins to structured time (provided with rhythm and duration), from disorder to order, from chaos to kósmos.
In this paper we will examine early Greek myths of origin and Presocratic views (Anaximenes, Anaximander, Anaxagoras, Empedocles, Leucippus, Democritus) on the significant role played by vortices in the context of cosmogonical processes as a necessary and unique source of διακόσμησις. Acting as a “moving cause” whose rapidity produces a separation of opposites, vortices jointly cause the resulting masses to move towards their “appropriate” place in the universe and thus the passage between multiplicity and unity.

Furthermore, vortices act in multiple forms and ways as cosmic agents of the will of the gods, as cosmic regulators and instruments of divine law/justice. Faced with the danger of the annihilation of the cosmic order which was established and preserved by the Olympians, the gods intervene as guardians of laws and justice, as slayers of the transgressions of the reign of Zeus and the equilibrium of the world. They neutralize and punish any act of hybris that threatens to harm them by means of certain divine instruments, their own attributes and, originally, specific divine gifts each of which has been invested as a sign of its functions and the authority it is supposed to exercise. Attributes of the gods, divine signs and agents of their will with regard to the ordering of the world, instruments of divine justice and cosmic regulators of the equilibrium of the world and divine laws advocated by the gods, these divine instruments manifest themselves through or often borrow shapes, movements and whirling patterns, in virtue, perhaps, of the antiquity of these "primitive" motives going back to the times of the origins.
We will see the example of Zeus' lightning bolt: it is with the aid of this true relay between heaven and earth that Zeus signifies to the mortals who are the recipients his anger or his divine will. Zeus exercises his authority and preserves the order of the world with the help of the turning motions of his thunderbolt, evoking the tempestuous force and cosmic energy of the origins. Let us note, in connection with the lightning bolt, the inherent alliance between curve and line: the thread which goes straight (and right to the target) through the spiral or twisting curves of its transetherian movement. For, here as elsewhere, with the vortex and all the figures which derive, the essential is in the form and in the movements which traverse it.