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.
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