Almost all studies of fluid and plasma problems are based on the use of conservation equations: density, momentum, energy, angular momentum, etc. These are not dynamical equations, which by definition are the Euler-Lagrange equations derived from the extremum of an action functional. For the problem of the existence of preferred states the conservation equations (treated as dynamical equations) are not appropriate. The most adequate approach is variational, the preferred states being determined as configurations of flow that extremize an action functional.
Finding a Lagrangian density for fluids and plasmas is a very difficult problem.
We found however a solution, starting with the two-dimensional case.
For the two-dimensional case there exist models that are equivalent to the physical systems: the 2D ideal (Euler) fluid and the 2D magnetized plasma (and planetary atmosphere) are equivalent with discrete systems consisting of sets of point-like vortices interacting in plane by a potential (Coulombian and respectively short range). As a rich literature shows, these models have been exploited until now within only two approaches : as statistical ensembles and by direct numerical simulations.
Then, apparently, replacing the formulation based on (streamfunction, velocity, vorticity) = (?, v, ?) with the formulation based on positions (xi(t), yi(t)) of discrete vortices does not extend radically our possibilities to describe the fluids. However, we have noted that there is a deep change by this replacement: the discrete model is formulated in terms of matter, field, interaction. Therefore the continuous limit which preserves this structure must be a classical field theory (FT). Therefore it became possible to write Lagrangians for the continuous limits of discrete models, i.e., for fluids and plasmas in 2D.
In the construction of the Lagrangians we have taken into account two aspects. First, the elementary vortices of the discretization behave as classical spins, which means that the scalar field that represents their density in FT must be a classical mixed spinor, or a 2x2 matrix. The fields will then belong to the su(2) algebra. Second, the motion of point-like vortices is Lorentz-type, which means that the field carying the interaction (gauge field) must have the Chern-Simons term in the Lagrangian. We have constructed Lagrangian densities from which the dynamic of fluids, magnetized plasma and planetary atmosphere is derived. We briefly summarize below the results we have obtained until now, with reference to applications. The two-dimensional (2D) ideal incompressible (Euler) fluid. The action functional proved to possess the remarkable property that it can be re-written (according to the Bogomolnyi procedure) as a sum of square terms. The stationary states that represent the extremum of the action are identified as the solutions of the self-duality equations. In this way we have obtained the purely analytical derivation of the sinh-Poisson equation, which governs the coherent vortical flows that the Euler fluid exhibits at relaxation from turbulent states. The current density profiles in 2D meridional section of the tokamak plasma. The physical system does not exhibit an intrinsic space scale nor a condensate of vorticity. In contrast with the model we have developed for the Euler fluid, the model for the current density is Abelian. This has allowed the analytical derivation of the Liouville equation, invoked in tokamak physics as a good approximation of the current profile. Again, the previous derivation of the Liouville equation in plasma theory was based on statistical considerations that parallel those for Euler fluid. Exactly as for the Euler fluid, the statistical approach cannot explain why the entropy is maximized with only one constraint (energy) from the infinite number of invariants of the problem. By contrast, the FT model exhibits the conformal symmetry.A major task was to extend the field theoretical approach to the more complicated problem of asymptotic stationary states of the planetary atmosphere and of the 2D plasma in strong magnetic field (closely related as physical problems). To take into account the finite intrinsic length (Rossby radius and respectively the ion sonic Larmor radius) we had to invoke a classical Higgs mechanism yielding a short range interaction (i.e. a finite mass of the photon). We have written a Lagrangian density for the continuum limit of a model consisting of a discrete set of point-like vortices interacting in plane via a short range potential (model of Stewart and Morikawa in meteorology), taking into account the presence of a condensate of vorticity (Coriolis frequency in atmosphere and ion cyclotron frequency in plasma). The field theoretical model is formulated in su(2) algebra, like for the Euler fluid. However the residual energy after the application of the Bogomolnyi procedure does not seem to have a topological nature and this leaves a wider space for identifying extrema of the self-dual type, i.e. a class of differential equations at stationarity. We have found that a particular form of the differential equation provides very good practical results.
There is still another class of systems that exhibit an intrinsic length (like Larmor radius) or a condensate of vorticity (like cyclotron frequency), but with a particularity: they can be described by an Abelian model. We have obtained the equation for the asymptotic stationary states and its solutions exhibit a ring-type vorticity distribution. Numerous applications are possible.
This is what we have done, but much work is still necessary for each Lagrangian and its applications.
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