Field theoretical methods applied to fluid and plasma problems

 The 2D Euler fluid

The ideal (non-dissipative) two-dimensional fluid shows at relaxation from a turbulent state the formation of a coherent pattern of flow. It consists of two vortices of opposite signs (a dipole) and this state is stable. This evolution has been well confirmed by numerical simulations and it was found that the streamfunction verifies the sinh-Poisson equation. This is an exactly integrable equation, which indicates a deep and exceptional stucture. The fact that the ideal fluid can reach such a state is strange and suggests some fundamental order. The classical approaches, based on conservation equations, are less efficient in exploring this aspects.

The field theoretical method is the appropriate framework: it identifies the states as extrema of an action functional.

Field theory for the Euler equation

Derivation of the (sinh-Poisson) equation for the asymptotic states of the Euler equation. Examination of the equations of motion in close proximity of the self-dual state.

Liouville equation and the snake of density in tokamak

The "snake of density" is a phenomenon still not understood: around the magnetic surface q=1 in tokamak a perturbation of density (that may be generated by a pellet) evolves by concentration into a localized formation that has 3D helical shape, regular, stable i.e. resisting to saw-teeth longer than expected. The same process takes place with heavy impurities and even with current density.

We do not attempt a solution but show that the equation supposed to describe the 2D current distribution exhibits a solution that has required qualities: it owes its robustness to its (exact) integrability structure and is a functional extremum. The process of concentration is the evolution to this extremum and no perturbation can prevent the system to return to this path.

Ring-type vorticity distribution

There are flow structures with ring aspect, generated in oceans, in the south of Africa or in Carribean sea. They are very stable, for years. Since one of the possible origins of stability is the topological constraint, we examine a field theoretical model whose solutions can be rings of vorticity. It is the Abelian version of the Field Theory for a 2D fluid and indeed possesses topological properties. Even if in real fluids the preservation of a topological invariant cannot be precise, there may be a remnant robustness.

Comment on conditions of application of Field Theoretical models to plasma and fluid problems

The Field theoretical formulation can be characterized as very powerful: they dispose of an analytical apparatus that has been developed for decades and has been confronted with deep and complex problems. On the other hand, they are easily applied to identify stable structures as action functional extrema but the equations of motion are in general as complicated as those of the standard fluid/plasma approaches. With one eception, by far the most interesting one: the self-duality. It seems that the nature chooses self-duality as the asymptotic state of any evolution.

We examine several field theoretical frameworks that seem adequate for plasma/fluid problems.

For an important "caveat emptor" see below.

 Fluids, Field Theory and Constant Mean Curvature surfaces

A presentation on the connection fluids/surfaces

This presentation (Marseille 2013) has been made early in the evolution of my knowledge on this connection. It mixes the subjects that are connected with Field Theory and its applications.

 Turbulence of unitons

Structures in turbulence, chiral model and unitons

The statistical theory of point like vortices is described by a Field Theory with mixed spinors Chern-Simons gauge field and long range, Coulombian, interaction. At self-duality the model can be mapped onto a chiral model and further it is equivalent with the Constant Mean Curvature surfaces. We try to pull back from the common asymptotic (identically identified by all these formulations) to include turbulence.

A text on all that.

Attempts to formulate a Field Theoretical model for plasma in strong magnetic field and for the planetary atmosphere

A model equation proposed for these physical systems with short range of interaction

The plasma in strong magnetic field and the planetay atmosphere have intrinsic lengths: the Larmor radius and respectively the Rossby radius. This makes a fundamental difference relative to the Field Theoretical formulation for the Euler fluid. For the latter, there is no intrinsic length (conformal invariance). The first idea is to introduce a mass to the photon of the Chern-Simons gauge field (without however introducing the Maxwell term since we do not expect short range coherency). This mass (inverse Larmor/Rossby lengths) can result from the equivalent of a Higgs phenomenon, i.e. involving a particular self-interaction of the matter field. But then, the model (Non-Abelian, Chern Simons, self interaction of order 6) leads to an equation that has NO real solutions. If we want to keep a similarity with the sinh-Poisson equation we must modify the Bogomolnyi procedure. We have done this, in a way which does not have any physical justification except our desire to keep close to the Euler model. Then we obtain an equation.

This equation has been largely ignored or rejected for the reason stated above: it is not derived as a consistent Field Theory but it is created when we decided to modify the analytical path toward Self-Duality of the Euler FT model. The system, in any case, would have lost the self-duality and a residual energy should have to remain.

We have applied this equation to the tropical cyclone and to magnetically confined plasma.

Strangely enough, it works.

A text on this derivation. We have formulated clearly our CAVEAT, above.