# Plan

The work plan contains analytical and numerical work to be carried out in close relationship with the extensive accumulation of experimental data.

I. A thorough evaluation and integration of the achievements until present. This will allow us to elaborate a draft of a structure in which the analytical choices (Lagrangians), the calculations, the practical results should be integrated. Here are the steps of such a construction.

I-1. A more clear identification of the physical fluid/plasma variables in the field theoretical formulation.

1. Meaning of the generators of the algebra : generator of the su(2) Cartan subalgebra; ladder generators, and their connection with the concentration or rarefaction of point-like vortices equivalent with the increase or decrease of the physical vorticity.
2. Conservation laws. In physical systems they are related to energy, vorticity, enstrophy, possibly higher order quantities. In field theory formulations the conservation laws are a consequence of the invariance properties of the Lagrangian, most notably the conformal invariance, where it exists (Liouville and sinh-Poisson case). We have to study carefully the mapping between these two series of conservation laws, expressed in different languages.
3. Physical parameters: the few parameters of the Lagrangian (e.g. the constant multiplying the Chern-Simons term in the Lagrangian and the Higgs parameter) are reflected further in physical parameters like the Rossby or Larmor radius, asymptotic vorticity, etc.

I-2. A more detailed investigation of the practical results that have been obtained until now or that are accessible by immediate application of what we have obtained until now (including the numerical methods already available): atmospheric vortex, plasma vortical motions, etc. This will allow us to have a critical evaluation of the comparison experiment/theory at this moment.

II. Investigation of the field theoretical models in close connection with the physical models

II-1. Systematic construction of the Lagrangians. For example, the scalar field self-interaction in the Lagrangians for systems with intrinsic length (describing the atmosphere, magnetized plasma, non-neutral plasma, etc.) may be inferred but, alternatively, it results from the observation that the theory enjoys an unexpected N=2 super-symmetry. This fixes in an elegant way the form of the Lagrangian. It appears that the super-symmetry is not so strange in this context. The equations of motion connects the "magnetic field" derived from the potential of vortex interactions and the density of the vortices, and there is the possibility of a transformation between these two representations of the same physical object : local physical vorticity. Or, in the field theoretical formulation the magnetic field has a bosonic nature and the point-like vortices have a fermionic nature, which makes the supersymmetry a natural and very interesting line of investigation. The result would be a more systematic way of finding adequate form of the Lagrangian density for particular physical systems. But, more importantly and far-reaching, the confirmation of the possibility to derive the expression of Lagrangian densities for our models from a very general super-symmetric Yang-Mills Lagrangian by taking adequate restrictions would be a significant result that may improve our understanding of the deep mathematical properties of the theory of fluids. We note that this line of fundamental considerations would only be accessible when the description of the fluids is formulated as a field theory.

II-2. Investigation of the equations of motion derived in the field theory, close to the state of self-duality, i.e. close to the relaxation of fluid/plasma into coherent flows. This will permit to understand better the "shape" of the manifold of the action functional near the extremum, to explain the observed slowing down in the organization of fluids close to stationary states and the multitude of quasi-solutions that are obtained in experimental and numerical studies. Some of the states found (experimentally or numerically) close to stationarity are actually metastable (like the crystals of vortices in atmospheric tornadoes or in non-neutral plasmas) and evolve to neighbor lowest states, in general monopolar vortices. This should be possible to be seen by examining the neighborhood of the extrema of the action, looking for quasi-degenerate direction (along which the system evolves very slowly) or to local minima of the action. An extensive numerical investigation is required, accompanied by evaluation of thresholds and barriers separating metastable states from the lowest action states, estimated from observations or experiments that are available.

II-3. Investigation of the new information that can be obtained from

1. the currents calculated from the Lagrangian, in particular the possibility of vorticity pinch. Also, the conservation laws derived from the currents.
2. the meaning of the topological properties of vortex solution, where they exists (ring-type vortices, Liouville vortices), and understanding the absence of topological constraints in the other cases (magnetized plasma).
3. Connection between the topological bounds on the energy and the physical threshold observed in the formation of stable coherent structures in particular cases. In the same idea, understanding if the dropouts observed at vortex coalescence in fluids can be explained by topological limits on the final states.
4. Understanding the reduction from non-Abelian to Abelian field theoretical models and how this can explain the distinct flow pattern obtained in the two cases. This may have a considerable importance for understanding observed physical phenomena: in 2D strongly magnetized plasma in circular geometry the vorticity of one sign is expelled toward the periphery, keeping in the center a monopolar vortex of opposite sign. From this it may be possible that a ring-type vorticity distribution is obtained, but with a strong concentration which makes the sheared velocity layer unstable to Kelvin-Helmholtz instability and later to filamentation (this has been noticed in numerical simulations of both planetary atmosphere and non-neutral plasma). It would result a realistic scenario for the Edge Localized Modes in tokamak H states (while the present explanation based on peeling-ballooning modes faces the difficulty that has a low ability to select a precise filamentary structure).