Objectives of the project

The development and application of field theoretical formulations in order to highlight the intrinsic trend to self-organization in fluids and plasmas is almost without precedent, therefore the Project will treat a really new subject of research. If successful, this will represent a considerable change in our understanding of the fluid physics, in the concepts and terms to formulate new theories on fluids, in our technical methods.

The objectives of the Project are:

1. To develop a coherent, consistent theory of the physics of fluids and plasmas close to the coherent stationary states, in terms of a classical field theory. This is complimentary but vastly more powerful than the classical description restricted to streamfunction, velocity, vorticity.
2. Analytical derivation of the differential equations governing stationary organized states, a family of which the sinh-Poisson equation is a member.
3. Detailed characterization of the stationary (vortical) states, for fluids, atmosphere, magnetized plasma, non-neutral plasma. This means velocity and vorticity profiles, spatially symmetric structures of vortices, degree of metastability. This is possible because the FT approach provides differential equations for the stationary states. In particular, explicit relationships connecting the main quantitative characteristics of the atmospheric vortex (hurricane, typhoon) when stationarity is a good approximation. These are equations between: the radius of maximal extension, the radius of the eye-wall and the maximum azimuthal velocity of the wind. We have already proposed two equations and they appear to work very well for several known tropical cyclones.
4. Exploration of the connection: axial anomaly / vorticity concentration (in plasma). This may be useful for tornado formation, etc.
5. Exploration of the connection: self-induced fermion mass in nucleon Flavor Dynamics / viscous limit to the ideality of the fluids. In our preprint arXiv.org/1001.0151 we show that our bosonic Lagrangian for the Euler fluid is equivalent at low energy with the Thirring (1+1)D model. Actually, the equivalence is with the full 4D fermionic model of Nambu-Jona-Lasinio.
6. Extension to 2D magnetohydrodynamics. Search of the presently unknown equation of the asymptotic states.
7. Extension to three-dimensional fluids/plasmas. Analytic derivation of the force-free equation. And understanding the consequence of the fact that the Chern-Simons term cannot be written in 3D. It suggests that, starting with whatever initial flow state, the fluid will evolve asymptotically to a state consisting of a dense set of singularities.
8. A theoretical basis for the "contour dynamics" method used in Navier-Stokes fluids. Explicit expressions for the hyperelliptic Riemann surfaces derived from our Lagrangian for the Euler fluid, but extended to a N=2 supersymmetric model.

Applications:

• A new, original and powerful theoretical basis (together with its analytical instruments) to construct models to explain processes important for particular systems:
• High confinement regime in tokamak: implies ring-type vorticity, naturally obtained by the Abelian model for magnetized plasma
• density pinch in tokamak: implies vorticity concentration and might be described by the adiabatic change of the coefficient of the Chern-Simons term in the non-Abelian Lagrangian for magnetized plasma. We will provide the explanation of the role of the effective Larmor radius.
• experiments in rotating fluid tanks. We will provide quantitative results on the vortex dynamics on an irrotational background flow (experiments on rotating water tanks). This consists of derivation of variants of the sinh-Poisson equation and finding their solutions. Such a development is only accessible to the field theory of the Euler fluid.
• experiments on large scale vortical patterns in linear plasma machines. We can explain the hollow density profile and we will provide the quantitative characterization of the vortex, on the basis of the non-Abelian Lagrangian for magnetized plasma.
• vortex crystals in non-neutral plasmas: we can explain the loss of stability of electron density rings and we will provide quantitative characterization of the metastability of the crystals. The model is non-Abelian Lagrangian for magnetized plasma.

The fluid/plasma description by point like vortices, which is the starting point of the field theoretical approach, is also analyzed from a different perspective. We propose a statistical study of the strongly nonlinear phase in turbulence evolution, where both stochastic and quasi-coherent characteristics appear and compete. The aim is to understand the tendency of organization in fluids also from the early stage.

The study is based on our results on the statistics of test particle trajectories in stochastic fields. We have shown that non-standard statistics of trajectories can appear in 2-dimensional divergence less stochastic velocity fields. This is the effect of the particle intrinsic trapping or eddying in the stochastic field. Trapping determines memory effects, anomalous transport regimes, non-Gaussian distribution and a high degree of coherence [Vlad, Spineanu 2004]. Preliminary results suggest that this stochastic trapping has a strong effect on the evolution of turbulence and that it is the fundamental reason for the formation of quasi-coherent structures.

These results will be developed for the study of interacting point like vortices. This is a special test particle problem for which the Eulerian correlation of the velocity field is not a given function but it is (nonlinearly) obtained from the deterministic two-point interaction and from the distribution of elementary vortices. The statistical properties of the trajectories will be determined. The nonlinear effects produced by trajectory eddying will be analyzed. We expect that these results could lead to the development of a new Lagrangian approach for the evolution of turbulence beyond the quasilinear stage.